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#### Regular Expressions and Languages
A regular expression is a pattern describing a set of strings, while a regular language is the set of all strings that can be matched by a regular expression. Regular expressions are used in programming languages and text editors to search for and manipulate strings.
#### Deterministic and Non-Deterministic Turing Machines
A deterministic Turing machine performs one computation on a given input, whereas a non-deterministic Turing machine can perform multiple computations simultaneously. Non-deterministic Turing machines are more powerful, solving certain problems more quickly.
#### Context-Free and Regular Grammars
A context-free grammar generates context-free languages, while a regular grammar generates regular languages. Context-free grammars are more powerful, generating more complex string patterns.
#### Languages and Grammars
A language is a set of strings generated by a grammar, while a grammar is a set of production rules describing string generation from a starting symbol. Languages are the objects of study in computation theory, and grammars are the tools used to generate and analyze them.
#### Context-Sensitive and Recursively Enumerable Languages
A context-sensitive language is generated by a context-sensitive grammar, while a recursively enumerable language is recognized by a Turing machine. Recursively enumerable languages are more powerful, recognizing more complex string patterns.
#### The Chomsky Hierarchy
The Chomsky hierarchy classifies formal languages based on the grammar needed to generate them. It consists of four levels: regular languages, context-free languages, context-sensitive languages, and recursively enumerable languages. Each level is a proper subset of the previous one, except for the first two levels, which are equivalent.
#### One-Way and Two-Way Finite Automata
A one-way finite automaton reads input from left to right, moving its tape head only to the right. A two-way finite automaton can move its tape head left or right. Two-way finite automata are more powerful, recognizing languages that one-way finite automata cannot.
#### Decision and Function Problems
A decision problem has a "yes" or "no" answer, while a function problem requires computing a specific output based on input. Decision problems are the focus of complexity theory, and function problems are the focus of computability theory.
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codecrucks.com
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en
| 0.757721
| 2023-03-24T13:24:06
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https://codecrucks.com/theory-of-computation-question-set-29/
| 0.962373
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# Operators in Python
## Introduction to Operators
Operators in Python are used to perform operations on values, variables, and data structures, called operands, and control the flow of a program. They are essential for manipulating data types and creating flexible expressions.
## Arithmetic Operators
Arithmetic operators are used for basic mathematical operations such as addition, subtraction, multiplication, and division. The following table lists the arithmetic operators available in Python:
| Symbol | Operation |
| --- | --- |
| + | Addition |
| - | Subtraction |
| * | Multiplication |
| / | Division |
| % | Modulus |
| ** | Exponentiation |
| // | Floor division |
Examples of arithmetic operations:
```python
first_num = 5
second_num = 10
result = first_num + second_num # Output: 15
result = first_num - second_num # Output: -5
result = first_num * second_num # Output: 50
result = first_num / second_num # Output: 0.5
```
## Assignment Operators
Assignment operators are used to assign values to variables. Python also supports shorthand assignment operators, or augmented assignments, which allow performing an operation and assigning the result to a variable in a single line of code. The following table lists the assignment operators available in Python:
| Symbol | Operation |
| --- | --- |
| = | Assignment |
| += | Augmented Addition Assignment |
| -= | Augmented Subtraction Assignment |
| *= | Augmented Multiplication Assignment |
| /= | Augmented Division Assignment |
| %= | Augmented Remainder Assignment with the Modulus Operator |
| **= | Augmented Exponent Assignment |
| //= | Augmented Floor Division Assignment |
Examples of assignment operations:
```python
num = 10
num += 5 # Equivalent to num = num + 5
num -= 3 # Equivalent to num = num - 3
num *= 2 # Equivalent to num = num * 2
num /= 2 # Equivalent to num = num / 2
```
## Comparison Operators
Comparison operators are used to compare values and variables in Python. They return a Boolean value of either True or False based on the comparison made. The following table lists the comparison operators available in Python:
| Symbol | Operation |
| --- | --- |
| == | Equal to |
| != | Not equal to |
| > | Greater than |
| < | Less than |
| >= | Greater than or equal to |
| <= | Less than or equal to |
Examples of comparison operations:
```python
first_num = 15
second_num = 20
print(first_num == second_num) # Output: False
print(first_num != second_num) # Output: True
print(first_num > second_num) # Output: False
print(first_num < second_num) # Output: True
```
## Bitwise Operators
Bitwise operators perform operations on binary representations of numbers in Python. They are often used for low-level operations, such as setting or checking individual bits within a number. The following table lists the bitwise operators available in Python:
| Operation | Explanation |
| --- | --- |
| Binary AND | Sets each bit to 1 if both corresponding bits in the operands are 1 |
| Binary OR | Sets each bit to 1 if at least one of the corresponding bits in the operands is 1 |
| Binary XOR | Sets each bit to 1 if only one of the corresponding bits in the operands is 1 |
| Binary Ones Complement | Inverts all the bits of the operand |
| Binary Left Shift | Shifts the bits of the operand to the left by the specified number of positions, filling in with zeros from the right |
| Binary Right Shift | Shifts the bits of the operand to the right by the specified number of positions, filling in with copies of the leftmost bit from the left |
Examples of bitwise operations:
```python
first_num = 75
second_num = 35
result = first_num & second_num # Binary AND
result = first_num | second_num # Binary OR
result = first_num ^ second_num # Binary XOR
result = ~first_num # Binary Ones Complement
result = first_num << 2 # Binary Left Shift
result = first_num >> 2 # Binary Right Shift
```
## Logical Operators
Logical operators are used to perform operations on Boolean values in Python. They grant us decision-making based on multiple conditions and control the flow of a program.
Examples of logical operations:
```python
first_condition = True
second_condition = False
result = first_condition and second_condition # Logical AND
result = first_condition or second_condition # Logical OR
result = not first_condition # Logical NOT
```
## Specific Python Operators
In addition to the above operators, Python has a few specific operators.
### Identity Operators
Identity operators are used to compare the memory locations of two objects in Python. They enable us to find out if two objects are the same object in memory, or if they are different objects with the same values.
Examples of identity operations:
```python
first_obj = [1, 2, 3]
second_obj = [1, 2, 3]
print(first_obj is second_obj) # Output: False
print(first_obj is not second_obj) # Output: True
```
### Membership Operators
Membership operators are used to test if a value or variable is a member of a sequence, such as a list, tuple, or string. With them, we can determine whether an object is present in a sequence.
Examples of membership operations:
```python
numbers = [1, 2, 3, 4, 5]
print(3 in numbers) # Output: True
print(6 not in numbers) # Output: True
```
## Operator Precedence
Operators in Python have a specific order of precedence that determines the order in which operations are performed. The order of precedence is as follows:
1. Exponentiation (**)
2. Complement, unary plus, and minus (e.g. ~x, +x, -x)
3. Multiplication, division, floor division, and modulus (e.g. *, /, //, %)
4. Addition and subtraction (e.g. +, -)
5. Bitwise shift operations (e.g. <<, >>)
6. Bitwise AND (e.g. &)
7. Bitwise XOR (e.g. ^)
8. Bitwise OR (e.g. |)
9. Comparison operators (e.g. ==, !=, >, <, >=, <=)
10. Membership operators (e.g. in, not in)
11. Identity operators (e.g. is, is not)
12. Logical NOT (e.g. not)
13. Logical AND (e.g. and)
14. Logical OR (e.g. or)
Lacking knowledge of the precedence order in Python can affect the outcome of a calculation or comparison. If we're unsure of the order of precedence, it's always a good idea to use parentheses to explicitly define the order of operations.
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CC-MAIN-2023-14/segments/1679296949642.35/warc/CC-MAIN-20230331113819-20230331143819-00396.warc.gz
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webreference.com
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en
| 0.848659
| 2023-03-31T13:46:23
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https://webreference.com/python/basics/operators/
| 0.845982
|
Most circulated Roosevelt dimes are only worth their bullion value. For this list, we are only including five-cent nickels: Shield Nickels, Liberty "V" Nickels, Buffalo Nickels (or Indian Head Nickels), and Jefferson Nickels. One of the more unusual Silver coins was the Jefferson Nickel of 1942 to 1945. Jefferson Nickels were first minted in 1938 and made of 75% Copper and 25% Nickel.
A dime is worth 10 cents and is equal to 2 nickels or 10 pennies. The coin prices and values for 5C Nickels. The dime series, variety, date and mintmarks are found on value charts. The total value of the coins is $3.10. Let x = number of nickels and y = number of dimes.
A dime is worth 10 cents and a nickel is worth 5 cents. The total value of all the coins is $1.50. There are 22 coins in the jar, and the total value of the coins is $1.50. She has three times as many nickels as dimes.
The value of a dime is and the value of a nickel is The total value of all the coins is. We are asked to find the number of dimes and nickels Adalberto has. The person has 20 coins. The total of all of them is 140 cents.
A nickel is worth 5 cents and is equal to 5 pennies. The total value of the coins is $1.40. There are 20 coins in the jar, and the total value of the coins is 1.40. Joe has a collection of nickels and dimes that is worth $5.30.
The United States Mint currently makes Roosevelt dimes for circulation. In 1964, the mint made the last dimes containing 90% silver. All US dimes dated prior to 1965 are 90% silver and follow silver price. Rarity and demand contribute to a wide range in value found among all the different dates.
USA Coin Book has compiled a list of the most valuable US nickels ever known. Our most valuable nickels list includes coins starting in 1866 up to the present (2021) - including rare nickel errors and rare varieties that could still actually be found in pocket change. The list and the prices are current as of 2021.
A step by step process identifies the collector quality dimes worth a higher range. Step 1: Recognize the Different Series of Nickels - US nickels include a variety of series and sub-varieties within series, all important to recognize. Step 2: Date and Mintmark are Identified- Date and mintmark combinations are each valued separately.
The value of a dime coin is worth the same as ten one cent coins. Two nickels have the same value as 1 dime. Less often you can still find 90% silver quarters, as well. Coin dealers usually sell bank rolls or large bags of this “junk silver” grouped together by face value. Common increments are $100 or $1,000 face value.
Dime values today - ***z-mdyear.shtml*** are ***zs-roos-d1.shtml*** each. The least? The more valauble of the dimes are at the top of the list and the less valuable ones are at the bottom, and all the coins in the middle follow that value system.
Kimela T. asked • 12/02/15 A jar contains n nickels and d dimes. The total value of the coins is $1.50. Melanie has $1.80 worth of nickels and dimes. Martha has some nickels and dimes worth $6.25. She has 12 more nickels than dimes.
The child may be encouraged to work out the problem on a piece of paper before entering the solution. We encourage parents and teachers to select the topics according to the needs of the child. We hope that the free math worksheets have been helpful. We welcome your feedback, comments and questions about this site or page.
Student Visa Insurance Get Quotes For visitors, travel, student and other international travel medical Insurance. Embedded content, if any, are copyrights of their respective owners. Note that you will lose points if you ask for hints or clues. Try the given examples, or type in your own problem and check your answers.
Money value (dimes, nickels, pennies) Worksheet. Count the coins and match to its value. Kids count the pennies, nickels, dimes, quarters and half dollars and match to the corresponding value expressed in cents and dollars. This printable batch of simple worksheets helps strengthen counting money skills.
A dime is worth 10 cents. A nickel is worth 5 cents. 1 penny = 1 cent. 1 nickel = 5 cents. The value of a dime is and the value of a nickel is The total value of all the coins is. The total value of the coins is $3.10. The total value of the coins is $1.50. The total value of the coins is $1.40. The total value of the coins is 1.40.
The person has 20 coins. There are 22 coins in the jar. There are 20 coins in the jar. The total of all of them is 140 cents. She has three times as many nickels as dimes. She has 12 more nickels than dimes. How many nickels and dimes are in the jar? How many nickels and how many dimes are in the jar?
The value of all the dimes in dollars:0.10 × 2y = 0.20y. The value of all the nickels in dollars: 0.05 × y = 0.05y. Since the number of dimes are 2 times more than the number of nickels, the number of dimes will be 2y. The number of nickels is y.
The total value of the coins is $4.50. The total value of the coins is $8.40. If the number of dimes was tripled and the number of nickels was increased by 14, the value of the coins would be $8.40. The total value of the coins is $1.80. The total value of the coins is $6.25. The total value of the coins is $5.30.
This money song for kids helps your children learn to identify and know the value of a penny, nickel, dime and quarter. Objective: I know the value of dimes, nickels and pennies. We hope that the kids will also love the fun stuff and puzzles.
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CC-MAIN-2021-25/segments/1623487608856.6/warc/CC-MAIN-20210613131257-20210613161257-00517.warc.gz
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kpi.ua
|
en
| 0.912319
| 2021-06-13T13:22:13
|
https://cad.kpi.ua/wildflower-seedling-vce/nickels-and-dimes-value-76cf99
| 0.758791
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The Tanzalin Method for Easier Integration by Parts
The Tanzalin Method is a technique used to perform certain integrations, providing an alternative to the traditional Integration by Parts method. This approach can be easier to follow and is particularly useful for checking work in examinations.
Example 1: Integration by Parts Method
To compare, let's first examine the traditional Integration by Parts method. We identify u, v, du, and dv as follows:
u = 2x
dv = (3x − 2)^6 dx
du = 2 dx
Integration by Parts gives us:
∫u dv = uv - ∫v du
= (2x)(3x − 2)^6 - ∫(3x − 2)^6 (2 dx)
= (2x)(3x − 2)^6 - 2∫(3x − 2)^6 dx
Now, we find the unknown integral:
∫(3x − 2)^6 dx = (1/3)(3x − 2)^7 + C
Putting it together, we have:
∫(2x)(3x − 2)^6 dx = (2x)(3x − 2)^6 - (2/3)(3x − 2)^7 + C
We can then factor and simplify this to give the final answer.
Example 1: Tanzalin Method
The Tanzalin Method involves setting up a table with successive derivatives of the simplest polynomial term in the first column and the integrals of the second term in the second column.
| Derivatives | Integrals | Sign | Same-color Products |
| --- | --- | --- | --- |
| 2x | (3x − 2)^6 | + | 2x(3x − 2)^6 |
| 2 | (1/3)(3x − 2)^7 | - | -2(1/3)(3x − 2)^7 |
| 0 | (1/3)(3x − 2)^7 | + | 0 |
We multiply the terms with the same color background, and the answer for the integral is the sum of the terms in the final column.
The Tanzalin Method is somewhat less messy than the traditional Integration by Parts method.
Example 2: Tanzalin Method
We'll go straight to the Tanzalin Method for this example.
| Derivatives | Integrals | Sign | Same-color Products |
| --- | --- | --- | --- |
| x | sin x | + | x sin x |
| 1 | -cos x | - | cos x |
| 0 | -sin x | + | 0 |
We multiplied (x) by (-cos x) without changing the sign and then multiplied (1) by (-sin x) with a changed sign.
Adding the final column gives us the answer:
∫x sin x dx = x(-cos x) + cos x + C
Example 3: Tanzalin Method
Using the Tanzalin Method requires four rows in the table this time, since there is one more derivative to find.
| Derivatives | Integrals | Sign | Same-color Products |
| --- | --- | --- | --- |
| x^2 | (3x − 2)^6 | + | x^2(3x − 2)^6 |
| 2x | (1/3)(3x − 2)^7 | - | -2x(1/3)(3x − 2)^7 |
| 2 | (1/9)(3x − 2)^8 | + | 2(1/9)(3x − 2)^8 |
| 0 | (1/9)(3x − 2)^8 | - | 0 |
Our final answer is:
∫x^2(3x − 2)^6 dx = x^2(3x − 2)^6 - (2/3)x(3x − 2)^7 + (2/9)(3x − 2)^8 + C
Example 4: Tanzalin Method
We need to choose ln 4x for the first column, following the Integration by Parts priority recommendations.
| Derivatives | Integrals | Sign | Same-color Products |
| --- | --- | --- | --- |
| ln 4x | x^2 | + | ln 4x(x^2) |
| 1/x | (1/3)x^3 | - | -(1/3)x^2 |
| 0 | (1/3)x^3 | + | 0 |
When do we stop? The derivatives column will continue to grow, as will the integrals column. The Tanzalin Method requires one of the columns to "disappear" (have a value of 0) so we have somewhere to stop.
Our final answer is:
∫ln 4x(x^2) dx = x^2 ln 4x - (1/3)x^3 + C
Conclusion
While the Tanzalin Method only handles integrals involving at least one polynomial expression, it is worth considering as a simpler way of writing Integration by Parts questions. This method provides an alternative approach to integration, making it easier to organize and solve problems.
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CC-MAIN-2019-04/segments/1547583826240.93/warc/CC-MAIN-20190122034213-20190122060213-00295.warc.gz
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intmath.com
|
en
| 0.890537
| 2019-01-22T04:58:18
|
https://www.intmath.com/blog/mathematics/tanzalin-method-for-easier-integration-by-parts-4339
| 0.931644
|
To calculate the geometric mean for a set of numbers, multiply the numbers together and take the nth root of this product. This calculation is only valid when all numbers in the set are nonzero and positive.
For example, consider a set of numbers: 5, 8, 2, 8, and 10. Multiply these numbers together to get a product of 6,400. Since there are five numbers in the set, take the fifth root of 6,400 to find the geometric mean, which is approximately 5.771.
The geometric mean is useful for calculating average rates of return and other investment scenarios where numbers are typically percentages rather than counts. In contrast, the arithmetic mean (which involves adding numbers and dividing by the count) is more suitable for items involving counting rather than multiplication. Key steps for calculating the geometric mean include:
1. Multiplying the numbers in the data set together.
2. Counting the numbers in the set.
3. Taking the nth root of the product, where n is the count of numbers.
By following these steps, you can calculate the geometric mean for any set of nonzero and positive numbers, making it a valuable tool for investment analysis and other applications involving multiplication.
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CC-MAIN-2017-51/segments/1512948517181.32/warc/CC-MAIN-20171212134318-20171212154318-00478.warc.gz
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reference.com
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en
| 0.85321
| 2017-12-12T15:01:32
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https://www.reference.com/math/calculate-geometric-mean-2753eae7a9bfb7c9
| 0.999969
|
To pass in physics board exams, here are some important theory questions of paper-1 that could help you score near 50% plus marks in your 12th physics board exams.
Important questions of physics paper 1 to go with while appearing for class 12 board exams are as follows:
**1st Chapter – Circular Motion**
1. Relation between linear velocity and angular velocity.
2. Difference between centripetal force and centrifugal force.
3. Show that the angle of banking is independent of the mass of the vehicle.
4. Obtain an expression for the time period (T) of a conical pendulum.
**3rd Chapter – Rotational Motion**
1. Derive an expression for the kinetic energy of a rigid rotating body about an axis.
2. Obtain an expression for the torque acting on a rotating body about an axis.
3. State and prove the theorem/principle of parallel axes.
4. State and prove the theorem/principle of perpendicular axes.
5. Expression for the angular momentum of a rotating body.
6. Principle of conservation of angular momentum.
**4th Chapter – Oscillations**
1. Obtain the differential equation of linear S.H.M.
2. Obtain an expression for the period of a simple pendulum.
3. State the laws of a simple pendulum.
**5th Chapter – Elasticity**
All topics in this chapter are important.
**6th Chapter – Surface Tension**
1. Obtain an expression for the rise of a liquid in a capillary tube.
2. Define the angle of contact and its characteristics, including diagrams.
3. What is capillarity, and give some applications of it.
**7th Chapter – Wave Motion**
1. Explain the production of beats and deduce analytically the expression for beats frequency.
2. Explain what is Doppler Effect and its applications.
3. Derive an expression for one-dimensional simple harmonic waves traveling towards the +ve X-axis.
**8th Chapter – Stationary Waves**
1. Explain the formation of stationary waves by analytical method, including nodes and antinodes.
2. State and explain the laws of vibrating strings and end correction.
3. Explain resonance with an example, its merits, and demerits.
4. With a neat diagram, explain the fundamental mode of an air column in a pipe when the pipe is open at both ends and when the pipe is closed at one end.
**9th Chapter – Kinetic Theory of Gases & Radiation**
1. Write the assumptions of the kinetic theory of gases.
2. Explain a perfectly black body.
3. State Kirchhoff’s law of radiation and give its theoretical proof.
4. Explain Newton’s law of cooling and Stefan’s Law.
5. Derive an expression for the pressure exerted by a gas on the basis of the kinetic theory of gases.
6. Deduce Boyle’s law using the expression for pressure exerted by the gas.
These around 30 theory questions can help you score passing marks or near 50-60 percent marks, including your practical exams performance. For example, if you perform well in your practical exams and score around 20-25 out of 30 marks, and then you come across a few of these questions in the theory exam and write proper points as per the requirement of the question, you can score around 10-15 marks out of 70. This can result in a total score of 35, which can help you pass.
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CC-MAIN-2023-14/segments/1679296949694.55/warc/CC-MAIN-20230401001704-20230401031704-00339.warc.gz
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techniyojan.com
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en
| 0.865436
| 2023-04-01T00:25:25
|
https://techniyojan.com/2020/01/how-to-pass-in-physics-board-exams-class-12.html
| 0.496979
|
# STARKs, Part 3: Into the Weeds
STARKs ("Scalable Transparent ARgument of Knowledge") are a technique for creating a proof that \(f(x)=y\) where \(f\) may potentially take a very long time to calculate, but where the proof can be verified very quickly. A STARK is "doubly scalable": for a computation with \(t\) steps, it takes roughly \(O(t \cdot \log{t})\) steps to produce a proof, and it takes ~\(O(\log^2{t})\) steps to verify.
## MIMC
MIMC (see paper) is used as an example because it is simple to understand and interesting enough to be useful in real life. The function can be viewed visually as follows:
MIMC with a very large number of rounds is useful as a verifiable delay function - a function which is difficult to compute, and particularly non-parallelizable to compute, but relatively easy to verify.
## Prime field operations
A convenience class is built to perform prime field operations and polynomial operations over prime fields. The code includes trivial bits such as addition, subtraction, and multiplication, as well as the Extended Euclidean Algorithm for computing modular inverses.
## Fast Fourier Transforms
The FFT only takes \(O(n \cdot log(n))\) time, though it is more restricted in scope; the x coordinates must be a complete set of roots of unity of some order \(N = 2^{k}\).
## Thank Goodness It's FRI-day (that's "Fast Reed-Solomon Interactive Oracle Proofs of Proximity")
A low-degree proof is a (probabilistic) proof that at least some high percentage of a given set of values represent the evaluations of some specific polynomial whose degree is much lower than the number of values given.
## The STARK
The actual meat that puts all of these pieces together is the `def mk_mimc_proof(inp, steps, round_constants)` function, which generates a proof of the execution result of running the MIMC function with the given input for some number of steps.
The extension factor is the extent to which the computational trace is "stretched". The step count multiplied by the extension factor must be at most \(2^{32}\), because there are no roots of unity of order \(2^{k}\) for \(k > 32\).
The computational trace is generated, and then converted into a polynomial. The polynomial is evaluated in a larger set, of successive powers of a root of unity \(g_2\) where \((g_2)^{steps \cdot 8} = 1\).
The round constants of MIMC are converted into a polynomial. Because these round constants loop around very frequently, they form a degree-64 polynomial, and the expression and extension can be computed.
\(C(P(x))\) is calculated, which is \(C(P(x), P(g_1 \cdot x), K(x))\). The goal is that for every \(x\) that is laid down in the computational trace (except for the last step), the next value in the trace is equal to the previous value in the trace cubed, plus the round constant.
There is an algebraic theorem that proves that if \(Q(x)\) is equal to zero at all of these x coordinates, then it is a multiple of the minimal polynomial that is equal to zero at all of these x coordinates: \(Z(x) = (x - x_1) \cdot (x - x_2) \cdot ... \cdot (x - x_n)\).
The quotient \(D(x) = \frac{Q(x)}{Z(x)}\) is provided, and FRI is used to prove that it's an actual polynomial and not a fraction.
The prover wants to prove \(P(1) = input\) and \(P(last\_step) = output\). If \(I(x)\) is the interpolant - the line that crosses the two points \((1, input)\) and \((last\_step, output)\), then \(P(x) - I(x)\) would be equal to zero at those two points.
The Merkle root of \(P\), \(D\), and \(B\) is committed to. A pseudorandom linear combination of \(P\), \(D\), and \(B\) is computed, and an FRI proof is done on that.
The proof consists of a set of Merkle roots, the spot-checked branches, and a low-degree proof of the random linear combination. The largest parts of the proof are the Merkle branches and the FRI proof.
At every position that the prover provides a Merkle proof for, the verifier checks the Merkle proof and checks that \(C(P(x), P(g_1 \cdot x), K(x)) = Z(x) \cdot D(x)\) and \(B(x) \cdot Z_2(x) + I(x) = P(x)\).
The verifier also checks that the linear combination is correct and calls `verify_low_degree_proof` to verify the FRI proof.
The STARK proving overhead for MIMC is remarkably low, because MIMC is almost perfectly "arithmetizable" - its mathematical form is very simple. For "average" computations, which contain less arithmetically clean operations, the overhead is likely much higher, possibly around 10000-50000x.
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cloudflare-ipfs.com
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en
| 0.84683
| 2023-03-24T03:43:14
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https://cloudflare-ipfs.com/ipfs/bafybeigsn4u4nv4uyskxhewakk5m2j2lluzhsbsayp76zh7nbqznrxwm7e/general/2018/07/21/starks_part_3.html
| 0.997855
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## Solving the Equation: (x+1/x-2)^2 + x+1/x-4 - 3(2x-4/x-4)^2 = 0
This equation presents a challenge due to its complex structure. We will break down the steps to solve it:
### 1. Simplification
We can simplify the equation by factoring out common terms and expanding the squares:
(x+1/x-2)^2 + (x+1)/(x-4) - 3(2(x-2)/(x-4))^2 = 0
Expanding the squares gives:
[(x+1)^2/(x-2)^2] + (x+1)/(x-4) - 3[4(x-2)^2/(x-4)^2] = 0
Combining terms with the same denominator, the common denominator is (x-2)^2(x-4)^2:
[(x+1)^2(x-4)^2 + (x+1)(x-2)^2(x-4) - 12(x-2)^2(x-4)] / [(x-2)^2(x-4)^2] = 0
### 2. Solve the Numerator
Since the denominator cannot be zero, we only need to solve the numerator:
(x+1)^2(x-4)^2 + (x+1)(x-2)^2(x-4) - 12(x-2)^2(x-4) = 0
Factoring out common terms gives:
(x-2)^2(x-4)[(x+1)^2 + (x+1)(x-2) - 12] = 0
Simplifying the expression inside the brackets:
(x-2)^2(x-4)[2x^2 + x - 13] = 0
### 3. Solve for x
Now we have a simpler equation to solve. We need to find the values of x that make this equation true:
x - 2 = 0 => x = 2
x - 4 = 0 => x = 4
2x^2 + x - 13 = 0
To solve the quadratic equation 2x^2 + x - 13 = 0, we can use the quadratic formula:
x = [-b ± √(b^2 - 4ac)] / 2a
Where a = 2, b = 1, and c = -13.
Solving for x using the quadratic formula, we get two more solutions:
x = (-1 + √105) / 4
x = (-1 - √105) / 4
### 4. Verification
It's essential to verify if the solutions we found are valid. We need to check if any of the solutions make the original denominator zero.
We find that x = 2 and x = 4 make the denominator zero, so these solutions are extraneous and need to be discarded.
### Conclusion
Therefore, the solutions to the equation (x+1/x-2)^2 + x+1/x-4 - 3(2x-4/x-4)^2 = 0 are:
x = (-1 + √105) / 4
x = (-1 - √105) / 4
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jasonbradley.me
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en
| 0.764842
| 2024-09-08T17:01:41
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https://jasonbradley.me/page/(x%252B1%252Fx-2)%255E2%252Bx%252B1%252Fx-4-3(2x-4%252Fx-4)%255E2%253D0
| 0.999968
|
### Introduction to Gradual Release of Instruction
The gradual release of instruction is a crucial strategy in math instruction, applicable to all topics, including fractions. This approach involves a recursive and cyclical process, rather than a linear one, to layer instruction effectively.
### What is Gradual Release of Instruction?
The traditional gradual release model consists of three stages:
1. **I show the students**: The teacher demonstrates the concept or skill.
2. **We do it together**: The teacher and students work together to practice the concept or skill.
3. **They try it alone**: Students apply the concept or skill independently.
However, this model can be oversimplified. A more effective gradual release plan involves recursive cycles of instruction, practice, and assessment.
### An Alternative Gradual Release Model
In math instruction, a second type of gradual release can be applied. This model focuses on providing students with real-world applications and challenging problems to deepen their understanding. For example:
- **Word problems**: "Sue walked into a bakery to buy some cookies for her family. She noticed that each tray held 24 cookies. If half of the cookies on one of the trays had sprinkles, how many would that be? What if half of the cookies with sprinkles also had chocolate chips–how many would that be?"
- **Mathematical computations**: 3/4 + 4/4 = ?, 5 x 3/8 = ?
- **Equivalent fractions**: Generate a list of 3 fractions equivalent to 4/5.
### Assessing True Understanding
Students may appear to understand fractions by completing practice sheets and identifying given fractions. However, true understanding requires application and real-world context. Teachers must work closely with students, observing, listening, and asking questions to ensure a deep understanding of fractions and their practical applications.
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theteacherstudio.com
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en
| 0.93984
| 2023-03-23T16:55:56
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https://theteacherstudio.com/getting-students-ready-to-learn/
| 0.788911
|
## Introduction to Square Root of 64
The square root of 64 is the number that, when multiplied by itself, equals 64. In other words, this number to the power of 2 equals 64. The square root of 64 can be written as ²√64 or 64^1/2.
## Calculator
Our calculator shows that ²√64 = ±8.
## Second Root of 64
The term "square root of 64" usually refers to the positive number, which is the principal square root. Since the index 2 is even and 64 is greater than 0, 64 has two real square roots: ²√64 (positive) and -²√64 (negative).
## Inverse of Square Root of 64
Extracting the square root is the inverse operation of squaring. The inverse operation of 64 square root is raising the result to the power of 2.
## What is the Square Root of 64?
The square root of 64 is ±8. The parts of the square root symbol are:
- ²√64: square root of 64 symbol
- 2: index
- 64: radicand (the number below the radical sign)
- √: radical symbol or radical only
## Table of nth Roots of 64
The following table provides an overview of the nth roots of 64:
| Index | Radicand | Root | Symbol | Value |
| --- | --- | --- | --- | --- |
| 2 | 64 | Square Root of 64 | ²√64 | ±8 |
| 3 | 64 | Cube Root of 64 | ³√64 | 4 |
| 4 | 64 | Forth Root of 64 | ⁴√64 | ±2.8284271247 |
| 5 | 64 | Fifth Root of 64 | ⁵√64 | 2.29739671 |
| 6 | 64 | Sixth Root of 64 | ⁶√64 | ±2 |
| 7 | 64 | Seventh Root of 64 | ⁷√64 | 1.8114473285 |
| 8 | 64 | Eight Root of 64 | ⁸√64 | ±1.6817928305 |
| 9 | 64 | Nineth Root of 64 | ⁹√64 | 1.587401052 |
| 10 | 64 | Tenth Root of 64 | ¹⁰√64 | ±1.5157165665 |
## Summary
To sum up, the square roots of 64 are ±8, with the positive real value being the principal. Finding the second root of 64 is the inverse operation of rising the result to the power of 2, i.e., (±8)^2 = 64. For more information about roots, visit our page on nth Root.
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industrialdevicesindia.com
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en
| 0.877525
| 2024-09-12T05:38:00
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https://industrialdevicesindia.com/article/what-is-the-square-root-of-64-information-and-calculator/1763
| 0.977974
|
# Resources tagged with: Linear functions
### There are 9 results
Broad Topics > Functions and Graphs > Linear functions
##### Age 11 to 14 Challenge Level:
Collect as many diamonds as possible by drawing three straight lines to understand linear functions.
##### Age 14 to 16 Challenge Level:
Position lines perpendicular to each other and analyze the equations of perpendicular lines. What relationship can be observed between their slopes?
##### Age 11 to 14 Challenge Level:
Investigate how the position of a line affects its equation. Compare the equations of parallel lines and identify any patterns or relationships.
##### Age 11 to 14
Alf Coles discusses creating 'spaces for exploration' in the classroom to facilitate student learning.
##### Age 11 to 14 Challenge Level:
Translate different lines and observe the changes in their equations. Predict and explain the effects of translation on linear equations.
##### Age 11 to 14 Challenge Level:
Reflect different lines in the x or y-axis and analyze the changes in their equations. Predict and explain the effects of reflection on linear equations.
##### Age 14 to 16 Challenge Level:
Given the graph y=4x+7, determine the possible order of four transformations that could have been applied to it.
##### Age 14 to 16 Challenge Level:
Investigate the relationship between the coordinates of a line's endpoints and the number of grid squares it crosses. Is there a consistent pattern or formula?
##### Age 11 to 14 Challenge Level:
Compare different pocket money systems and choose the most preferable one. Consider factors such as weekly allowance, rewards, or penalties.
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maths.org
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en
| 0.870419
| 2020-01-27T21:33:28
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https://nrich.maths.org/public/topic.php?code=59&cl=3&cldcmpid=6982
| 0.977834
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The problem asks for the largest integer for which $n!$ is a factor of the sum $98! + 10,000$. To solve this, we can factor out $n!$ from $98!$, resulting in $\frac{98!}{n!}$. This expression has factors of $n+1, n+2, ..., 98$.
We need to find the largest $n$ for which $n!$ is a factor of $98! + 10,000$. Factoring out $n!$ from $98!$ gives us $\frac{98!}{n!}$, which has factors of $n+1, n+2, ..., 98$. The number $98!$ has $22$ factors of $5$, and $10,000$ has $4$ factors of $5$.
The total number of factors of $5$ in $98! + 10,000$ is $22 + 4 = 26$. Since $n!$ needs to be a factor of $98! + 10,000$, the largest possible value of $n$ is the largest integer for which $n!$ has $26$ or fewer factors of $5$.
This occurs when $n = 22$, since $22!$ has $22$ factors of $5$ from the multiples of $5$ up to $22$, and $23!$ would have $23$ factors of $5$, exceeding the total number of factors of $5$ in $98! + 10,000$.
Therefore, the largest integer $n$ for which $n!$ is a factor of $98! + 10,000$ is $n = 25$, but we must verify if $25!$ is indeed a factor of $98! + 10,000$. Upon closer examination, we see that $25!$ has the necessary factors to divide $98!$, and since $10,000 = 2^4 \cdot 5^4$, $25!$ also has the necessary factors of $2$ and $5$ to divide $10,000$.
Hence, $n = 25$ is indeed the largest integer for which $n!$ is a factor of $98! + 10,000$.
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artofproblemsolving.com
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en
| 0.800055
| 2023-03-25T07:27:59
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https://artofproblemsolving.com/wiki/index.php?title=2017_AMC_8_Problems/Problem_19&oldid=152053
| 0.999297
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## NCERT Solutions for Class 9 Maths Chapter 15 Probability Exercise 15.1
The correct NCERT Solutions for Chapter 15 Probability Exercise 15.1 in Class 9 Maths are provided here to help understand the chapter's basics. These solutions are useful for completing homework and achieving good exam marks. Experts have prepared detailed answers to every question, allowing for easy doubt clearance.
Exercise 15.1 is the sole exercise in this chapter, comprising 13 questions. These questions require preparing frequency distribution tables and calculating the probability of certain events.
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studyrankers.com
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en
| 0.708597
| 2023-03-31T16:23:18
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https://www.studyrankers.com/2020/03/ncert-solutions-for-class-9-maths-chapter-15-exercise-15.1.html
| 0.910289
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The largest subset of [1,2,...,n] which contains no 3-term geometric progression is being discussed. A sequence, now A208746, has been created to track this. For n = 16, omitting 1 and 4 was initially considered, but it was found that 1..16 \ {1,4} still contains 3,6,12 and 9,12,16.
A corrected and extended sequence is:
1,2,3,3,4,5,6,7,7,8,9,10,11,12,13,13,14,14,15,15,16,17,18,19,19,20,21,21,22,23,
24,24,25,26,27,27,28,29,30,31,32,33,34,34,35,36,37,38,38,38,39,39,40,41,42,43,
44,45,46,46,47,48,49,49,50,51,52,52,53,54,55,55,56,57,57,57,58,59,60,61,61,62
This computation used a floating-point IP solver for the packing subproblems. The approach was to enumerate geometric progressions using a nested loop structure and then solve the integer program of maximizing the subset of {1..N} subject to not taking all 3 of any progression.
The original question was posed by Neil Sloane, who also maintains sequence A003002, which gives the size of the largest subset of [1,2,...,n] which contains no 3-term arithmetic progression. The sequence for geometric progressions was initially calculated by hand for n >= 1, resulting in:
1,2,3,3,4,5,6,7,7,8,9,10,11,12,13,14
However, this has been corrected and extended as mentioned above.
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seqfan.eu
|
en
| 0.763072
| 2023-03-26T03:27:48
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http://list.seqfan.eu/pipermail/seqfan/2012-March/060886.html
| 0.883141
|
## Integers and Absolute Value
### Objective
Students will be able to order and compare integers, including the absolute value of integers.
### Lesson Duration
65 minutes
### Launch (10 minutes)
The lesson begins with an opener that allows students to construct viable arguments and critique the reasoning of others, which is mathematical practice 3. The learning target for the day is: "I can compare integers using <, >, and = by identifying their position on a number line. I understand that the absolute value of a number is its distance from zero."
### Explore (50 minutes)
The explore portion of the lesson starts with a video introduction to integers and absolute value. Students fill in their notes during the video and then participate in guided notes with real-world examples of integers, applying mathematical practice 4. A number line is used to demonstrate the progression from -10 to 10, utilizing mathematical practice 5. Comparison problems are discussed, and volunteers explain their answers and reasoning.
The concept of absolute value is introduced, emphasizing that it represents distance and cannot be negative. An example is used to illustrate this concept: if the supermarket is 10 miles north and Grandma's house is 10 miles south, Grandma's house is not -10 miles away, as distance cannot be negative. Students practice problems, including one that requires treating absolute value bars like parentheses in the order of operations, highlighting the importance of precision, which is mathematical practice 6.
### Instructional Strategy
A table challenge may be conducted using XP Math - Math Games Arcade, where tables participate in a task on the smartboard, and the table with the highest score wins. This challenge reminds students of the importance of paying attention and precision in their work.
### Real-World Examples of Absolute Value
Examples include distances between locations, such as the distance between home and school or the distance between two cities. These examples help students understand that absolute value represents a measure of distance without considering direction.
### Similar Lessons and Units
This lesson is part of a larger unit on integers and rational numbers, which includes lessons on adding, subtracting, multiplying, and dividing integers and rational numbers, as well as applying these concepts to real-world problems. The full unit plan includes:
- UNIT 1: Introduction to Mathematical Practices
- UNIT 2: Proportional Reasoning
- UNIT 3: Percents
- UNIT 4: Operations with Rational Numbers
- UNIT 5: Expressions
- UNIT 6: Equations
- UNIT 7: Geometric Figures
- UNIT 8: Geometric Measurement
- UNIT 9: Probability
- UNIT 10: Statistics
- UNIT 11: Culminating Unit: End of Grade Review
Specific lessons in the sequence include:
- LESSON 1: Integers and Absolute Value
- LESSON 2: Modeling Addition
- LESSON 3: Integer Addition Word Problems
- LESSON 4: Adding Integers
- LESSON 5: Multiple Addends
- LESSON 6: Adding Integers Review
- LESSON 7: Adding Integers Test
- LESSON 8: Subtracting Integers
- LESSON 9: Subtracting Integers Practice
- LESSON 10: Addition and Subtraction of Integers - DOMINOES!
- LESSON 11: Adding and Subtracting Integers - Real World Applications
- LESSON 12: Adding and Subtracting Integers - REVIEW!
- LESSON 13: Adding and Subtracting Integers Test
- LESSON 14: Adding and Subtracting Signed Fractions
- LESSON 15: Adding and Subtracting Signed Fractions Fluency Practice
- LESSON 16: Adding and Subtracting Signed Decimals
- LESSON 17: Adding and Subtracting Rational Numbers - Practice Makes Perfect!
- LESSON 18: Adding and Subtracting Rational Numbers - Test
- LESSON 19: Multiplying and Dividing Integers
- LESSON 20: Multiplying and Dividing Rational Numbers
- LESSON 21: Problem Solving with Rational Numbers
- LESSON 22: Fractions to Decimals - Terminate or Repeat?
- LESSON 23: Rational Number Unit Test
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betterlesson.com
|
en
| 0.816985
| 2017-12-16T11:04:57
|
https://betterlesson.com/lesson/443522/integers-and-absolute-value-are-two-steps-forward-and-two-steps-back-the-same-thing?from=consumer_breadcrumb_dropdown_lesson
| 0.989243
|
To determine if the function f(x) = 2x^{3} + 300 + 4 is increasing, decreasing, or not changing at x = -2, we need to find its derivative. First, rewrite the function as f(x) = 2x^{3} + 300x^{0} + 4, which simplifies to f(x) = 2x^{3} + 300 + 4, since x^{0} = 1. However, to apply the power rule for differentiation correctly to the constant term, we treat 300 as 300x^{0}. The derivative of x^{0} is 0, so the derivative of 300 is 0. Thus, we focus on differentiating 2x^{3} and the constant term 4, which also differentiates to 0.
The derivative f'(x) is found by applying the power rule to each term:
- The derivative of 2x^{3} is 2 * 3x^{3-1} = 6x^{2}.
- The derivative of 300 is 0, since the derivative of any constant is 0.
- The derivative of 4 is also 0.
So, f'(x) = 6x^{2}.
To evaluate if the function is increasing or decreasing at x = -2, we substitute x = -2 into f'(x):
f'(-2) = 6(-2)^{2} = 6 * 4 = 24.
However, the original solution provided an incorrect derivative and evaluation. Let's correct that and follow the original function given: f(x) = 2x^{3} + 300 + 4. The correct derivative, considering the function as is, should directly apply the power rule:
- The derivative of 2x^{3} is 6x^{2}.
- The derivatives of 300 and 4 are 0.
Thus, the correct derivative is f'(x) = 6x^{2}. Evaluating at x = -2:
f'(-2) = 6(-2)^{2} = 24.
This indicates the function is increasing at x = -2 because the derivative is positive. The confusion arose from an incorrect manipulation of the function and its derivative in the original solution. The key concept here is applying the power rule correctly to find the derivative and then evaluating it at the specified point to determine the function's behavior.
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expertsmind.com
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en
| 0.833064
| 2017-12-13T17:05:40
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http://www.expertsmind.com/questions/some-interpretations-of-the-derivative-30153374.aspx
| 0.998761
|
## Learning Objectives
By the end of this section, you will be able to:
- Describe the law of conservation of linear momentum.
- Derive an expression for the conservation of momentum.
- Explain conservation of momentum with examples.
- Explain the law of conservation of momentum as it relates to atomic and subatomic particles.
The information presented in this section supports the following AP learning objectives and science practices:
**5.A.2.1** The student is able to define open and closed systems for everyday situations and apply conservation concepts for energy, charge, and linear momentum to those situations.
**5.D.1.4** The student is able to design an experimental test of an application of the principle of the conservation of linear momentum, predict an outcome of the experiment using the principle, analyze data generated by that experiment whose uncertainties are expressed numerically, and evaluate the match between the prediction and the outcome.
**5.D.2.1** The student is able to qualitatively predict, in terms of linear momentum and kinetic energy, how the outcome of a collision between two objects changes depending on whether the collision is elastic or inelastic.
**5.D.2.2** The student is able to plan data collection strategies to test the law of conservation of momentum in a two-object collision that is elastic or inelastic and analyze the resulting data graphically.
**5.D.3.1** The student is able to predict the velocity of the center of mass of a system when there is no interaction outside of the system but there is an interaction within the system.
Momentum is conserved when the net external force on a system is zero. This means that the total momentum of a closed system remains constant over time. The law of conservation of momentum can be expressed mathematically as:
p_tot = p'_tot
where p_tot is the initial total momentum and p'_tot is the final total momentum.
Consider a two-car collision. The momentum of the first car is p1 = m1v1, and the momentum of the second car is p2 = m2v2. After the collision, the momentum of the first car is p'1 = m1v'1, and the momentum of the second car is p'2 = m2v'2. The total momentum before the collision is equal to the total momentum after the collision:
p1 + p2 = p'1 + p'2
Using the definition of impulse, the change in momentum of the first car is given by:
Δp1 = F1Δt
where F1 is the force on the first car and Δt is the time over which the force acts. Similarly, the change in momentum of the second car is given by:
Δp2 = F2Δt
where F2 is the force on the second car. Since the forces are equal and opposite (F1 = -F2), the changes in momentum are also equal and opposite:
Δp1 = -Δp2
The total momentum of the two-car system remains constant:
p1 + p2 = p'1 + p'2
This result can be generalized to any isolated system, with any number of objects. The law of conservation of momentum states that the total momentum of an isolated system remains constant over time.
## Conservation of Momentum Principle
The conservation of momentum principle can be applied to systems as different as a comet striking Earth and a gas containing huge numbers of atoms and molecules. Conservation of momentum is violated only when the net external force is not zero. But another larger system can always be considered in which momentum is conserved by simply including the source of the external force.
## Isolated System
An isolated system is defined as one for which the net external force is zero. In an isolated system, the total momentum remains constant over time.
## Making Connections: Cart Collisions
Consider two air carts with equal mass (m) on a linear track. The first cart moves with a speed v towards the second cart, which is initially at rest. If the collision is elastic, the first cart will stop after the collision, and the second cart will have a final velocity v after the collision. The momentum of the system will be conserved in the collision.
If the two carts stick together after the collision, the final velocity of the two-cart system will be half the initial velocity of the first cart. The kinetic energy of the system will not be conserved in this case.
## Making Connections: Take-Home Investigation—Drop of Tennis Ball and a Basketball
Hold a tennis ball side by side and in contact with a basketball. Drop the balls together. The tennis ball will bounce off the basketball, and the basketball will remain largely stationary. This is because the momentum of the tennis ball is conserved, and the basketball has a much larger mass.
## Making Connections: Take-Home Investigation—Two Tennis Balls in a Ballistic Trajectory
Tie two tennis balls together with a string about a foot long. Hold one ball and let the other hang down, then throw it in a ballistic trajectory. The center of the string will remain stationary, and the two balls will move in opposite directions. This is because the momentum of the system is conserved.
## Applying Science Practices: Verifying the Conservation of Linear Momentum
Design an experiment to verify the conservation of linear momentum in a one-dimensional collision, both elastic and inelastic. Measure the momentum of each object before and after the collision, and compare the results to your prediction.
## Making Connections: Conservation of Momentum and Collision
Conservation of momentum is crucial to our understanding of atomic and subatomic particles. The conservation of momentum principle is used to analyze the masses and other properties of previously undetected particles, such as the nucleus of an atom and the existence of quarks that make up particles of nuclei.
## Subatomic Collisions and Momentum
The conservation of momentum principle is valid when considering systems of particles. Experiments seeking evidence that quarks make up protons scattered high-energy electrons off of protons. The analysis was based partly on the same conservation of momentum principle that works so well on the large scale.
The law of conservation of momentum is a fundamental principle in physics that applies to all systems, from the smallest subatomic particles to the largest galaxies. It states that the total momentum of a closed system remains constant over time, and it is a powerful tool for analyzing and understanding the behavior of physical systems.
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openstax.org
|
en
| 0.811771
| 2024-09-13T15:47:47
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https://openstax.org/books/college-physics-ap-courses/pages/8-3-conservation-of-momentum
| 0.687777
|
**Optimal Control: Direct Solution Methods**
This lecture covers the basics of numerical optimal control, focusing on direct solution methods such as direct single shooting, direct multiple shooting, and collocation with Legendre polynomials. Key concepts include problem formulation, scaling of input and state variables, and the use of toolboxes like CASADI for solving optimal control problems. The lecture also explains time transformation, input rate constraints, and scaling dynamics, providing practical tips for successful problem solving.
**Course Information**
The lecture is part of the doctoral course EE-715: Optimal Control, which introduces optimal control theory, numerical implementation, and problem formulation for applications.
**Related Concepts**
1. Numerical analysis: the study of algorithms using numerical approximation for mathematical analysis problems.
2. Numerical methods for ordinary differential equations: methods for finding numerical approximations to solutions of ordinary differential equations (ODEs).
3. Iterative method: a mathematical procedure using an initial value to generate a sequence of improving approximate solutions.
4. Direct multiple shooting method: a numerical method for solving boundary value problems by dividing the interval into smaller intervals and imposing matching conditions.
5. Numerical methods for partial differential equations: the branch of numerical analysis studying numerical solutions of partial differential equations (PDEs).
**Instructors and Related Lectures**
The lecture is taught by two instructors and is related to 258 other lectures, including:
- Finite Difference Grids (MATH-351)
- Finite Elements: Elasticity and Variational Formulation (MATH-212)
- Nonlinear Problem Solving (MOOC: Numerical Analysis for Engineers)
- Finite Elements: Problem with Limits (MATH-251(b))
- Direct Methods for Linear Systems of Equations (ChE-312)
**Note**: This page is automatically generated and may contain incorrect or outdated information. Please verify the information with EPFL's official sources.
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epfl.ch
|
en
| 0.886774
| 2024-09-11T23:16:55
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https://graphsearch.epfl.ch/en/lecture/0_1816fy63
| 0.999941
|
# How to Calculate Motor Starting Time
Calculating motor starting time can be complex due to various influencing factors. To understand how to calculate this, it's essential to consider the motor characteristic, specifically the torque curve, and how it interacts with the load torque curve.
## Understanding Motor Starting Time
The torque available to accelerate the motor up to speed is the difference between motor torque (CM) and load torque (CL). This difference, denoted as Ca, is crucial for understanding the starting dynamics. The formula for Ca is:
Ca = CM - CL
Where:
- Ca is the torque to accelerate the motor, in N.m
- CM is the motor torque, in N.m
- CL is the load torque, in N.m
Both motor and load torque vary with speed, and the motor torque characteristic depends on the motor's design and construction. Starting methods can also affect the available motor torque and the shape of its curve.
## Deriving the Equation for Starting Time
To derive an equation for the time to accelerate from zero to the running speed, we consider the inertia of both the motor and the load. The equation involves the following parameters:
- t: time to accelerate to running speed, in seconds
- n: motor running speed, in rpm
- C: motor torque, in N.m
- CL: load torque, in N.m
- J: inertia of the motor, in kg.m^2
- JL: inertia of the load, in kg.m^2
The equation for starting time can be complex due to the variability of torque with speed. For precise calculations, especially with complex torque curves or starting arrangements, numerical solutions or software tools are recommended.
## Simplified Approximation for Starting Time
For a more straightforward calculation, simplifications can be introduced:
1. Use an average value of motor torque, considering the inrush torque (CS) and the maximum torque (Cmax).
2. Apply an adjustment factor KL to account for varying load torque due to speed changes.
The load factor (KL) values for different types of loads are:
- Lift: 1
- Fans: 0.33
- Piston: 0.5
- Flywheel: 0
Using these simplifications, the approximate starting time can be calculated with the formula involving the effective acceleration torque (Cacc).
## Example Calculation
For a 90 kW motor driving a fan:
- Motor Rated Speed (n): 1500 rpm
- Motor Full Load Speed: 1486 rpm
- Motor Inertia (J): 1.4 kg.m^2
- Motor Rated Torque: 549 Nm
- Motor Inrush Torque (CS): 1563 Nm
- Motor Maximum Torque (Cmax): 1679 Nm
- Load Inertia (JL): 30 kg.m^2
- Load Torque (CL): 620 Nm
- Load Factor (KL): 0.33
This example demonstrates how to apply the simplified formula to estimate the starting time.
## Conclusion
While accurately calculating motor starting time can be complex, using simplifications and understanding the key factors involved can provide realistic estimates for common starting scenarios. For more precise calculations, especially in critical applications, consulting detailed references or using specialized software is advisable.
## References
- Three-phase asynchronous motors. Generalities and ABB proposals for the coordination of protective devices. ABB, 2008.
For specific load types like centrifugal and positive displacement pumps, the load factor can be approximated as follows:
- Centrifugal pump: similar to a fan, use KL = 0.33
- Positive displacement pump: similar to a piston, use KL = 0.5
These approximations can be used in the simplified formula to estimate the starting time for these types of loads.
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|
myelectrical.com
|
en
| 0.88879
| 2023-03-22T06:24:10
|
https://myelectrical.com/notes/entryid/107/how-to-calculate-motor-starting-time
| 0.875781
|
## Understanding Entropy in Information Theory
Entropy is a measure of uncertainty or randomness in a dataset. It quantifies the level of disorder in the data and helps in decision-making algorithms. In information theory, entropy is related to the reduction of uncertainty. When new knowledge is obtained from data, uncertainty decreases, and information increases.
### A. Information and Uncertainty
Information refers to the reduction of uncertainty. When uncertainty increases, it becomes harder to make predictions. Entropy, at its core, quantifies this uncertainty.
### B. Shannon’s Information Theory and Entropy
Claude Shannon introduced the concept of entropy as a measure of uncertainty. Shannon’s entropy, symbolized as H, quantifies the amount of information contained in a random variable or a probability distribution. Higher entropy implies higher unpredictability.
### C. Calculating Entropy for Discrete and Continuous Probability Distributions
To calculate entropy, we sum up the probabilities of all outcomes, each multiplied by the logarithm of its inverse probability, for discrete distributions. For continuous distributions, we integrate a similar expression.
### D. Interpretation of Entropy Values
Entropy values range from 0 to log(n), where n is the number of distinct outcomes. An entropy of 0 indicates certainty, while maximum entropy indicates maximum uncertainty.
## Entropy as a Measure of Uncertainty in Machine Learning
### A. Entropy as a Metric for Evaluating Decision Trees
Entropy serves as a metric to guide decision tree construction. The information gain achieved by a split is the reduction in entropy. By selecting splits that yield the most significant information gain, decision trees become more accurate.
### B. Information Gain and Its Relationship with Entropy
Information gain measures how much an attribute contributes to reducing overall entropy. By selecting attributes with high information gain, decision trees make better predictions.
### C. Using Entropy to Measure the Purity of a Dataset
Entropy provides a measure of purity, indicating how well-separated classes are. Minimizing entropy leads to higher accuracy and better generalization.
### D. Relationship between Entropy and Gini Impurity
Gini impurity is another criterion for evaluating decision tree splits. Both entropy and Gini impurity aim to minimize uncertainty but can yield different results.
## Applications of Entropy in Machine Learning
### A. Decision Trees and Random Forests
Entropy-based methods help Random Forests select the best attributes for splitting, leading to robust and accurate ensemble models.
### B. Clustering Algorithms
Entropy-based criteria assess the quality of clustering and enhance accuracy. K-means and hierarchical clustering can benefit from entropy-based approaches.
### C. Reinforcement Learning
Entropy plays a crucial role in balancing exploration and exploitation. Entropy regularization encourages exploration, leading to better decision-making.
## Entropy in Deep Learning
### A. Entropy in Neural Network Loss Functions
Cross-entropy loss is a widely-used loss function for classification tasks. It measures the dissimilarity between predicted and actual probabilities.
### B. Regularization Techniques Based on Entropy
Dropout regularization adds noise and randomness, akin to entropy, leading to improved generalization. Maximum entropy regularization aims to maximize the entropy of a model’s predictions.
## Challenges and Considerations
### A. Overfitting and Underfitting
Entropy can lead to overfitting or underfitting. It is essential to address these issues to ensure model performance.
### B. Bias in Data
Data bias influences entropy-based metrics, leading to biased predictions. Addressing bias is crucial for fair and ethical model use.
### C. Handling Imbalanced Datasets
Entropy-based methods can help alleviate the challenges of imbalanced datasets, ensuring fair and accurate model performance.
## FAQs About Machine Learning Entropy
### What is Entropy in Machine Learning?
Entropy is a measure of uncertainty or randomness in a dataset.
### What are the Different Types of Entropy?
There are mainly two types: Shannon entropy and Gini impurity.
### What Does High Entropy Mean?
High entropy indicates a higher level of disorder or uncertainty.
### What is the Entropy of a Model?
The entropy of a model refers to the uncertainty in its predictions.
### How Do You Explain Entropy?
Entropy is a measure of unpredictability or randomness in a dataset.
### What is Entropy Used to Explain?
Entropy explains the amount of disorder or randomness in data.
### What Does the Entropy of 1 Mean?
An entropy value of 1 signifies maximum disorder or uncertainty.
### What is a Good Entropy Value?
A good entropy value depends on the context and algorithm.
### What Does Entropy of 0 Mean?
An entropy value of 0 indicates a perfectly ordered dataset.
### What is Gini and Entropy in Machine Learning?
Gini and entropy are measures of impurity used in decision tree algorithms.
### How is Entropy Calculated?
Entropy is calculated by summing the negative of the probability of each class multiplied by the logarithm of that probability.
## Final Thoughts About Machine Learning Entropy
Machine learning entropy is a powerful concept that measures uncertainty or randomness within a dataset. It plays a crucial role in various machine learning algorithms and helps in building effective models. By embracing entropy, we can make better-informed decisions and improve model performance.
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CC-MAIN-2024-38/segments/1725700651722.42/warc/CC-MAIN-20240917004428-20240917034428-00351.warc.gz
|
kingpassive.com
|
en
| 0.878602
| 2024-09-17T02:04:44
|
https://kingpassive.com/machine-learning-entropy/
| 0.872358
|
The circumference or perimeter of a circle is the distance around the edge of the circle. To calculate the circumference of a circle, use the formula:
C = π × d
* C represents the Circumference.
* d represents the diameter of the circle, which is the distance all the way across the center of the circle.
* π stands for Pi, a number equal to 3.142 rounded to 3 decimal places.
To find the circumference, multiply the diameter of the circle by π.
To find the area of the circle, use the formula:
A = π × r²
* A represents the area.
* r represents the radius of the circle, which is the distance halfway across the center, so it is half the value of the diameter.
Note that the formula for circumference uses the diameter, and the formula for the area uses the radius.
**Example 1**
Find the circumference and area of a circle with a diameter of 38cm. The radius is 19cm.
First, calculate the circumference:
C = π × d
C = π × 38
C = 119.4 cm to 1 decimal place.
Next, calculate the area:
A = π × r²
A = π × 19²
A = 1134 cm² rounded to the nearest whole number.
**Example 2**
Calculate the circumference and area of a circle with a radius of 4.2m. The diameter is 8.4m.
First, calculate the circumference:
C = π × d
C = π × 8.4
C = 26.4 m to 1 decimal place.
Next, calculate the area:
A = π × r²
A = π × 4.2²
A = 55.4 m² rounded to the nearest whole number.
**Example 3**
A bicycle wheel has a diameter of 33cm. To find the distance the bike travels in one turn, calculate the circumference:
C = π × d
C = π × 33
C = 104 cm to the nearest centimeter.
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infobarrel.com
|
en
| 0.847919
| 2016-12-09T23:27:19
|
http://www.infobarrel.com/Calculating_the_area_and_circumference_of_a_circle
| 0.999734
|
Hall's conjecture states that for positive integers x and y, if k = x^3 - y^2 is nonzero, then |k| > x^(1/2-o(1)). This conjecture is a special case of the Masser-Oesterlé "ABC conjecture." Despite limited theoretical progress, significant experimental work has been done, starting with Hall's original paper.
A new algorithm has been developed to find all solutions of |k| << x^(1/2) with x < N in time O(N^(1/2+o(1))). Implementing this algorithm in 64-bit C for N=10^18 and running it for almost a month yielded 10 new cases of 0 < |k| < x^(1/2) in that range. Two of these cases improved on the previous record for x^(1/2) / |k|, with one breaking the old record by a factor of nearly 10.
The following table lists the 24 solutions of 0 < |k| < x^(1/2) with x < 10^18, including x, k, and r = x^(1/2) / |k|. The table does not include y, which is always the integer closest to x^(3/2).
1. |k| = 1641843, x = 5853886516781223, r = 46.60
2. |k| = 30032270, x = 38115991067861271, r = 6.50
3. |k| = 1090, x = 28187351, r = 4.87 (GPZ)
4. |k| = 193234265, x = 810574762403977064, r = 4.66
5. |k| = 17, x = 5234, r = 4.26 (GPZ, P(-3))
6. |k| = 225, x = 720114, r = 3.77 (GPZ)
7. |k| = 24, x = 8158, r = 3.76 (GPZ, P(3))
8. |k| = 307, x = 939787, r = 3.16 (GPZ)
9. |k| = 207, x = 367806, r = 2.93 (GPZ)
10. |k| = 28024, x = 3790689201, r = 2.20 (GPZ)
11. |k| = 117073, x = 65589428378, r = 2.19
12. |k| = 4401169, x = 53197086958290, r = 1.66
13. |k| = 105077952, x = 23415546067124892, r = 1.46
14. |k| = 1, x = 2, r = 1.41
15. |k| = 497218657, x = 471477085999389882, r = 1.38
16. |k| = 14668, x = 384242766, r = 1.34 (GPZ, P(-9))
17. |k| = 14857, x = 390620082, r = 1.33 (GPZ, P(9))
18. |k| = 87002345, x = 12813608766102806, r = 1.30
19. |k| = 2767769, x = 12438517260105, r = 1.27
20. |k| = 8569, x = 110781386, r = 1.23 (GPZ)
21. |k| = 5190544, x = 35495694227489, r = 1.15
22. |k| = 11492, x = 154319269, r = 1.08 (GPZ)
23. |k| = 618, x = 421351, r = 1.05 (GPZ)
24. |k| = 548147655, x = 322001299796379844, r = 1.04 (D)
Notes:
- GPZ: Solutions found by J.Gebel, A.Pethö, and H.G.Zimmer with 1 < |k| < 10^5.
- D: Danilov's infinite family, with r approaching 5^(5/2)/54 = 1.035+.
- P(t): Polynomial family, where t is an integer congruent to 3 mod 6.
- *: Obtained from the record solution by multiplying x, y, k by 4, 8, 64, reducing r by a factor of 32.
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|
harvard.edu
|
en
| 0.787713
| 2024-09-17T02:42:31
|
https://people.math.harvard.edu/~elkies/hall.html
| 0.969047
|
Introduction to Recursion
A binary tree is a tree data structure in which each node has at most two children, referred to as the left child and the right child. The structure of a binary tree can be represented by the following class:
```java
public class Tree {
Object value;
Tree right;
Tree left;
}
```
Not every binary tree is a search tree. Binary search trees have rules about where elements get added, and they will be explored in MP5.
Subtrees of a tree are also considered trees. This property is essential for understanding recursive operations on trees.
Recursion occurs when a problem is defined in terms of itself or its type. In computer science, recursion is a method where the solution to a problem depends on solutions to smaller instances of the same problem.
Recursion vs. Iteration:
- Iteration: repeating the same set of steps over and over again.
- Recursion: breaking a larger problem into smaller problems until they are small enough to solve easily.
For example, consider counting the number of nodes in a tree. This can be done iteratively by visiting every node and incrementing a counter, or recursively by breaking the problem into smaller subproblems and combining the results.
Recursive Node Counting:
1. Break the problem into smaller subproblems.
2. Solve the smallest subproblem.
3. Combine the results.
A recursive function is a function that calls itself. An example of a recursive function is the factorial function:
```java
int factorial(int n) {
if (n == 1) {
return 1;
} else {
return n * factorial(n - 1);
}
}
```
To effectively use recursion, follow these strategies:
1. Know when to stop (base case): return when the smallest subproblem is identified.
2. Make the problem smaller in each step (recursive step): ensure the problem gets smaller to reach the base case.
3. Combine results from recursive calls properly.
The factorial function demonstrates these strategies:
- Base case: n == 1
- Recursive step: decrement n towards 1
- Combine results: multiply current n with the result of the next subproblem
It's essential to reach the base case; otherwise, the problem will not terminate, and the program may crash. The code can fail to reach the base case if it does not properly decrement towards the base case or if the base case is not correctly defined.
When deciding between recursion and iteration, consider clarity as the primary goal. Use the technique that makes the code more understandable, whether it's recursive or iterative. Avoid using recursion solely for brevity or to appear complex.
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CC-MAIN-2020-05/segments/1579250607407.48/warc/CC-MAIN-20200122191620-20200122220620-00021.warc.gz
|
illinois.edu
|
en
| 0.815244
| 2020-01-22T20:44:53
|
https://cs125.cs.illinois.edu/learn/2018_03_12_introduction_to_recursion/
| 0.985443
|
Term 2 Reflection
In Term 2, we covered three units: percents, surface area, and volume. I performed well in surface area and volume, as they involved applying basic arithmetic operations using various formulas. However, I struggled to remember the numerous formulas. A key area for improvement is participating more in class discussions, as consistently noted by my teachers.
The percents unit taught me several concepts, including converting fractions to decimals to percents in any order. I learned that percents are out of 100 and can be represented on 100 grids, with fraction percents shaded on a single square. Mental math and ratio tables can be used to find percents in numbers, and percents can be combined by adding to solve problems.
In the surface area unit, I learned about formulas for finding the surface area of different shapes. For rectangular prisms, the formula is length x width, and they have six faces with equal opposite sides. Triangular prisms have a formula of base x height/2 + length x width, with five faces, two triangles, and three rectangles. Cylinders have multiple formulas, including finding diameter, radius, or circumference, with the easiest formula being (2 π x radius x radius) + (2 x radius x π x height) = Total Surface Area.
The volume unit also introduced various formulas. When given the base and height, the formula is Area of base x height. Without the base, there are specific formulas for each object: v = length x width x height for rectangular prisms, v = base x height (triangle) / 2 x height (prism) for triangular prisms, v = side x side x side for cubes, and v = π x radius x radius x height for cylinders. The key difference between volume and surface area is that volume measures the interior space, while surface area measures the exterior space.
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CC-MAIN-2017-51/segments/1512948521292.23/warc/CC-MAIN-20171213045921-20171213065921-00733.warc.gz
|
blogspot.com
|
en
| 0.732015
| 2017-12-13T05:25:36
|
http://spmath84110.blogspot.com/2011/03/glenesses-term-2-reflection.html
| 0.99746
|
# Kosmin Test Calculator
## Introduction
The Kosmin test is a predictor for middle-distance runners, estimating 800m or 1500m race times using repeated 60-second runs. It is best completed on a track with a timekeeper.
## Test Procedure
The test procedure differs slightly for 800m and 1500m tests.
### 800m Test
The athlete runs flat-out for 60 seconds, and the distance covered is measured. After a 3-minute recovery, the athlete completes another 60-second effort, and the distance is measured. The total distance is used to estimate the 800m race time.
### 1500m Test
The athlete runs flat-out for 60 seconds, and the distance is measured. After a 3-minute recovery, the athlete runs for another 60 seconds, followed by a 2-minute recovery. This is repeated for a total of four 60-second efforts, with recoveries of 3 minutes, 2 minutes, 1 minute, and no recovery after the final effort. The total distance is used to estimate the 1500m race time.
## Benefits
The Kosmin test is specific to the desired distance, making it more likely to predict a realistic race time.
## Using the Calculator
To use the calculator, choose the test distance and sex, enter the total distance covered, and calculate.
## Kosmin Test Calculations
Different formulas are used for 800m and 1500m tests, and for men and women.
| Test | Formula |
| --- | --- |
| 800m Men | 217.77778 - (Total Distance × 0.119556) |
| 800m Women | 1451.46 - (198.54 × Log(Total Distance)) |
| 1500m Men | 500.52609 - (Total Distance × 0.162174) |
| 1500m Women | 500.52609 - (Total Distance × 0.162174) + 10 |
|
CC-MAIN-2024-38/segments/1725700651072.23/warc/CC-MAIN-20240909040201-20240909070201-00829.warc.gz
|
runbundle.com
|
en
| 0.880726
| 2024-09-09T05:08:40
|
https://runbundle.com/tools/race-predictors/kosmin-test
| 0.796136
|
The problem asks to find the number of ordered pairs \(a, b\) of integers such that \(\frac{a + 2}{a + 5} = \frac{b}{4}.\)
To solve this, multiply both sides of the equation by \(a + 5\):
\[a + 2 = b \cdot \frac{a + 5}{4}\]
Then, multiply both sides by 4:
\[4(a + 2) = b(a + 5)\]
Expanding the left side gives:
\[4a + 8 = b(a + 5)\]
This equation implies that \(4a + 8\) is a multiple of \(b\), and \(a + 5\) is a divisor of \(4a + 8\). For integer solutions, \(b\) must be a divisor of 8, which are 1, 2, 4, and 8.
Checking divisibility for each possible value of \(b\):
- If \(b = 1\), \(a + 5\) must divide \(4a + 8\). The only integer solution is \(a = -5\).
- If \(b = 2\), there are no integer solutions because the left side is always even and the right side is always odd.
- If \(b = 4\), \(a + 5\) must divide \(4a + 8\). The only integer solution is \(a = -3\).
- If \(b = 8\), \(a + 5\) must divide \(4a + 8\). The only integer solution is \(a = -1\).
The possible integer solutions for \((a, b)\) are:
- \((-5, 1)\)
- \((-3, 4)\)
- \((-1, 8)\)
There are a total of 3 such ordered pairs.
|
CC-MAIN-2024-38/segments/1725700651344.44/warc/CC-MAIN-20240911052223-20240911082223-00435.warc.gz
|
0calc.com
|
en
| 0.74289
| 2024-09-11T05:50:17
|
https://web2.0calc.com/questions/pls-help-asap_82
| 0.999168
|
### Video Transcript
To solve the simultaneous equations \(x - y = 6\) and \(x^2 - 9xy + y^2 = 36\), we will use the substitution method. This involves rearranging one of the equations to make either \(x\) or \(y\) the subject and then substituting this back into the other equation.
We start with the first equation, \(x - y = 6\), and rearrange it to make \(x\) the subject: \(x = 6 + y\). This is our third equation.
Next, we substitute \(x = 6 + y\) into the second equation, \(x^2 - 9xy + y^2 = 36\), to get \((6 + y)^2 - 9(6 + y)y + y^2 = 36\). Expanding \((6 + y)^2\) gives \(36 + 12y + y^2\).
Substituting back into the equation yields \(36 + 12y + y^2 - 54y + y^2 = 36\). Simplifying, we collect like terms: \(36 + y^2 + 12y - 54y + y^2 = 36\), which simplifies further to \(y^2 - 42y + 36 = 0\) after combining like terms and subtracting 36 from both sides.
However, the correct simplification after expansion should be: \(36 + 12y + y^2 - 54y - 9y^2 + y^2 = 36\), leading to \(-7y^2 - 42y + 36 = 36\). Subtracting 36 from both sides gives \(-7y^2 - 42y = 0\).
Factoring out \(-7y\) from \(-7y^2 - 42y = 0\) gives \(-7y(y + 6) = 0\). Thus, \(y = 0\) or \(y = -6\).
To find \(x\), we substitute \(y = 0\) and \(y = -6\) into \(x = 6 + y\). For \(y = 0\), \(x = 6 + 0 = 6\). For \(y = -6\), \(x = 6 - 6 = 0\).
Therefore, the solutions to the simultaneous equations are \(x = 6\) when \(y = 0\) and \(x = 0\) when \(y = -6\).
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CC-MAIN-2021-25/segments/1623487634616.65/warc/CC-MAIN-20210618013013-20210618043013-00337.warc.gz
|
nagwa.com
|
en
| 0.877406
| 2021-06-18T03:34:20
|
https://www.nagwa.com/en/videos/305140503292/
| 0.997517
|
### Elasticities of Demand
Elasticity measures the responsiveness of the quantity demanded of a product to changes in any of the factors that affect demand. The price elasticity of demand is the percentage change in the quantity of a product demanded divided by the percentage change in the price causing the change in quantity. It indicates the degree of consumer response to variation in price.
The change in price is expressed as a percentage of the average price, and the change in the quantity demanded is expressed as a percentage of the average quantity demanded. This measure is units-free because it is a ratio of two percentage changes, and the percentages cancel each other out. Since a change in price causes the quantity demanded to change in the opposite direction, this ratio is always negative, although economists always ignore the sign and simply use the absolute value.
#### Example 1
A Pizza Hut store can sell 50 pizzas per day at $7 each or 70 pizzas per day at $6 each. The price elasticity is calculated as: [(50 - 70)/60] / [(7 - 6) / 6.5] = -2.17.
#### Types of Demand
Demand can be inelastic, unit elastic, or elastic, and can range from zero to infinity.
- If the elasticity coefficient is greater than 1, demand is **elastic**. A small price change leads to a large change in the quantity demanded.
- When the elasticity coefficient is less than 1, demand is **inelastic**. The more inelastic the demand, the steeper the demand curve.
- When the elasticity coefficient is equal to 1, demand is said to be **unitary elastic**.
#### Example 2
Refer to the graph below. Which of the following is true?
A. Areas C and E are smaller than area A, so demand must be elastic between $10 and $30.
B. Areas C and E are smaller than area A, so demand must be inelastic between $10 and $30.
C. Area F is smaller than areas B and C, so demand must be inelastic between $10 and $30.
Answer: C. Since at $30 the demand is unit elastic, at prices below $30 demand is inelastic. This is because when price rises from $10 to $30, the revenue gained is greater than the revenue lost.
#### Factors that Influence the Elasticity of Demand
The elasticity of demand among products varies substantially. The determinants of price and income elasticity of demand are:
1. **The closeness of substitutes**: The most important determinant is the availability of substitutes. The closer the substitutes for a good or service, the more elastic the demand for it.
2. **The proportion of income spent on the good**: If expenditures on a product are quite small relative to a consumer's budget, the income effect will be small even if there is a substantial increase in the price of the product.
3. **The time elapsed since a price change**: The more time consumers have to adjust to a price change, or the longer a good can be stored without losing its value, the more elastic the demand for that good.
#### Impact on Total Expenditure
Consumers' total expenditure is the same as total revenues from the suppliers' point of view. One of the most important applications of price elasticity is determining how total consumer expenditure on a product changes when the price changes.
- When demand is inelastic, a change in price will cause total expenditures to change in the same direction.
- When demand is elastic, a change in price will cause total expenditures to move in the opposite direction.
- When demand elasticity is unitary, total expenditures will remain unchanged as price changes.
#### Income Elasticity of Demand
The percentage change in the quantity of a product demanded divided by the percentage change in consumer income causing the change in quantity demanded.
- **Normal goods** have positive income elasticity; necessities have low income elasticities (between 0 and 1); luxuries have high income elasticities (greater than 1).
- **Inferior goods** have negative income elasticity; as income expands, the demand for them will decline.
#### Cross-Price Elasticity of Demand
The cross elasticity of demand is a measure of the responsiveness of demand for a good to a change in the price of a substitute or a complement, other factors remaining the same.
- The cross elasticity of demand for a substitute is positive.
- The cross elasticity of demand for a complement is negative.
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|
analystnotes.com
|
en
| 0.905039
| 2023-03-21T18:14:32
|
https://analystnotes.com/cfa-study-notes-calculate-and-interpret-price-income-and-cross-price-elasticities-of-demand-and-describe-factors-that-affect-each-measure.html
| 0.551996
|
Predictive models have become essential for businesses, enabling them to "foresee the future" and make informed decisions. These models can be categorized into regression models (continuous output) and classification models (nominal or binary output). Classification models use algorithms that produce either class outputs or probability outputs.
Algorithms like SVM and KNN create class outputs, where the output is either 0 or 1. In contrast, algorithms like Logistic Regression, Random Forest, and Gradient Boosting produce probability outputs, which can be converted to class outputs by setting a threshold probability.
Evaluating the performance of a machine learning model is crucial to determine its effectiveness on unseen data. This is achieved by using various metrics, including accuracy, precision, recall, F1 score, specificity, and Receiver Operating Characteristics (ROC) curve.
**Model Evaluation Metrics**
1. **Accuracy**: Measures the overall correctness of the model, calculated as (TP+TN)/total.
2. **Precision**: Measures the correctness of positive predictions, calculated as TP/predicted yes.
3. **Recall or Sensitivity**: Measures the ability of the model to detect positive instances, calculated as TP/actual yes.
4. **F1 Score**: The harmonic mean of precision and recall, ranging from 0 to 1, where 1 is perfect precision and recall.
5. **Specificity**: Measures the ability of the model to detect negative instances, calculated as TN/actual no.
6. **ROC Curve**: Plots the true positive rate against the false positive rate, providing a visual representation of the model's performance.
**Additional Metrics**
1. **Log Loss**: Measures the performance of a classification model based on probabilities, with smaller values indicating better performance.
2. **Jaccard Index**: Measures the similarity between two sets, calculated as the size of the intersection divided by the size of the union.
3. **Kolmogorov-Smirnov Chart**: Measures the degree of separation between positive and negative distributions, ranging from 0 to 100.
4. **Gain and Lift Chart**: Evaluates the performance of a classification model by calculating the ratio of results obtained with and without the model.
5. **Gini Coefficient**: Measures the dispersion of data, ranging from 0 to 1, where 0 represents perfect equality and 1 represents perfect inequality.
Understanding these evaluation metrics is essential to assess the performance of machine learning models, especially in cases where the data is imbalanced. By using these metrics, businesses can make informed decisions and improve the overall predictive power of their models.
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CC-MAIN-2023-14/segments/1679296943809.22/warc/CC-MAIN-20230322082826-20230322112826-00188.warc.gz
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datasource.ai
|
en
| 0.842001
| 2023-03-22T09:57:12
|
https://www.datasource.ai/en/data-science-articles/model-evaluation-metrics-in-machine-learning
| 0.839596
|
To represent the decimal 0.375 in place value digits, we write it in its expanded form. The integer part of a decimal is represented by place values such as Ones, Tens, Hundreds, Thousands, Ten Thousands, Hundred Thousands, Millions, and so on. The fractional part of a decimal is represented by place values like one-tenths, one-hundredths, one-thousandths, and so on.
0.375 can be expanded as 3 hundredths and 7 thousandths and 5 ten-thousandths, which equals (3/100) + (7/1000) + (5/10000). This representation helps in understanding the place value of each digit in the decimal number.
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CC-MAIN-2020-05/segments/1579251700988.64/warc/CC-MAIN-20200127143516-20200127173516-00267.warc.gz
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mathsai.com
|
en
| 0.759802
| 2020-01-27T15:27:46
|
https://www.mathsai.com/tools/decimal-number-place-value-calculator.htm
| 0.994253
|
The perimeter of a shape is the sum of all its sides. For a parallelogram, the perimeter can be calculated as P = 2(AB + BC) since opposite sides are congruent. The surface area of a parallelogram is given by S = base x height.
A rectangle has a length (L = AB) and a width (l = BC), and its perimeter can be calculated as P = 2(L + l) or P = AB + BC + CD + DA. The surface area of a rectangle is given by S = length x width.
A square is a special type of rectangle where all sides are equal, and its perimeter can be calculated as P = 4L, where L is the length of a side. The surface area of a square is given by S = AB^2.
For a trapezium, the area can be calculated as A = (Big Base + Small Base) x Height / 2.
A circle has a circumference given by its length, and its area can be calculated using the radius (r).
Regular solids include the regular pyramid, where the lateral area is given by A_lat = (P_b x a_p) / 2.
The volume of a cuboid is given by V = L x l x h, and a cube is a special type of cuboid where all sides are congruent.
For a cylinder, the base area is given by the area of the base circle, and the lateral area, total area, and volume can be calculated using the base radius (r), height (h), and generator edge (g).
The cone has a base area given by the surface of the base circle, and the lateral area, total area, and volume can be calculated using the base radius (r), height (h), and generator (g).
The frustoconic shape has a lateral area, total area, and volume that can be calculated using the big base radius (R), small base radius (r), height (h), and generator (G).
A sphere has an area given by its radius (r), and its volume can be calculated using the radius.
Key formulas:
- Perimeter of a parallelogram: P = 2(AB + BC)
- Surface area of a parallelogram: S = base x height
- Perimeter of a rectangle: P = 2(L + l)
- Surface area of a rectangle: S = length x width
- Perimeter of a square: P = 4L
- Surface area of a square: S = AB^2
- Area of a trapezium: A = (Big Base + Small Base) x Height / 2
- Lateral area of a regular pyramid: A_lat = (P_b x a_p) / 2
- Volume of a cuboid: V = L x l x h
- Volume of a cylinder: V = πr^2h
- Volume of a cone: V = (1/3)πr^2h
- Volume of a sphere: V = (4/3)πr^3
Note: The formulas provided are a selection of key concepts and are not an exhaustive list of all formulas related to the shapes discussed.
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CC-MAIN-2021-25/segments/1623488560777.97/warc/CC-MAIN-20210624233218-20210625023218-00481.warc.gz
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mathematicshelp.org
|
en
| 0.688707
| 2021-06-24T23:37:52
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https://www.mathematicshelp.org/maths/posts/content/35
| 0.994529
|
# Fermi Calculation of Populations and Housing
To determine if it's possible to have a home for every person in the world by the middle of the century, a Fermi calculation can be used. This method, named after Enrico Fermi, involves using rough numbers to estimate the feasibility of a project.
Assuming a total workforce of 20,000,000 workers, which is approximately 1/400 of the world's current population, and dividing them into 200,000 teams of 100 people each, a large-scale construction project can be planned. Each team can build a 10-story building with 20 apartments per story, accommodating 3 people per apartment, resulting in 600 people per building.
With 16,000,000 buildings, each taking up approximately 140 ft by 140 ft of land, the total land required can be calculated. Each building occupies around 20,000 square feet, and multiplying this by 16,000,000 buildings gives a total of 300,000,000 square feet. Since one square mile is approximately 25,000,000 square feet, the total land required is around 12 square miles. This can be spread across the entire world, making each individual section relatively small.
To estimate the cost, a rough calculation can be made. Assuming a certain price per square foot of living space, the cost of one floor can be estimated, then multiplied by 10 to get the cost of all 10 stories, and finally multiplied by 16 million. If each square foot costs $1, the total cost would be approximately $3,000,000,000,000. Although this seems like a significant expense, it's a global-scale construction project, and the cost per person housed would be around $3,000.
Given that each team can build one building in approximately 3 months, in 20 years, one team can build around 80 buildings. With 200,000 teams working simultaneously, all 16,000,000 buildings can be constructed in 20 years. This would provide housing for approximately 10,000,000,000 people, the estimated world population by 2050. If the project starts in 2020, it would be completed by 2040, and the buildings would be full 10 years later.
The key to this plan is that all teams work simultaneously, making it possible to complete the project within the given timeframe. While the cost is a significant factor, the calculation demonstrates that providing housing for every person in the world is theoretically possible, even with rough estimates.
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CC-MAIN-2021-25/segments/1623487655418.58/warc/CC-MAIN-20210620024206-20210620054206-00326.warc.gz
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michaelcurzan.com
|
en
| 0.964195
| 2021-06-20T03:31:07
|
https://www.michaelcurzan.com/post/fermi-calculation-of-populations-and-housing
| 0.675159
|
**How Many Liters Of Propane In A 100 Lb Tank**
A 100 lb propane tank is a common size, but the amount of usable propane it can hold varies due to several factors. One gallon of propane weighs approximately 4 lbs. A standard 100 lb propane tank holds around 23-25 gallons of propane and weighs 170 lbs when full.
The density of propane is 493 gm/ltr at 25°C. Using this density, we can calculate that 1 pound of propane is equivalent to approximately 0.92 liters in volume at 25°C. Therefore, a 100 lb propane tank can hold around 92 liters of propane (100 lbs x 0.92 liters/lb).
The tank's dimensions are typically 18” tall and 12” in diameter, but can vary. The pressure inside a propane tank fluctuates slightly based on the outside temperature, which can affect the amount of propane it can hold.
To give you a better idea, a 100 lb propane tank can last around 7 days and 4 hours when powering a 10,000 btu/h appliance at a temperature of 60 degrees Fahrenheit. The cost to fill a 100 lb propane tank can vary, but on average, it can hold around 23.6 gallons of propane.
It's worth noting that propane tanks come in different sizes, including 20 lb and 100 lb tanks. A 20 lb propane tank is typically 4 feet tall and 18” in diameter. When transporting a propane tank in a car, it's essential to follow safety guidelines and ask your propane retailer if a plug is required.
In summary, a 100 lb propane tank can hold around 92 liters of propane, depending on the temperature and other factors. The tank's size, weight, and btu capacity can vary, but it's a popular choice for heating homes and powering appliances.
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delectablyfree.com
|
en
| 0.888337
| 2023-03-31T12:12:48
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https://delectablyfree.com/blog/how-many-liters-of-propane-in-a-100-lb-tank/
| 0.760039
|
### Introduction to Bungee Jump Simulation
#### Goals and Prerequisites
The goal of this activity is to simulate a bungee jump using a Barbie doll and rubber bands, collecting data to construct a scatterplot, and generating a line of best fit to predict the number of rubber bands needed for a safe jump from a given distance. Prerequisites include understanding uncertainty, functional relationships, and patterns.
#### Background
Students work in groups to formulate a conjecture, design a method, and use data to develop concepts of estimation, slope, and line of best fit. The simulation involves a Barbie doll attached to rubber bands, with the objective of giving Barbie the greatest thrill while ensuring her safety.
#### Key Concepts and Questions
- **Uncertainty**: Variation in data due to repeated measures.
- **Functional Relationships**: Describing how one variable depends on another.
- **Patterns**: Processes of change that repeat in predictable ways.
- **Student Questions**:
1. How many rubber bands are needed for Barbie to safely jump from a height of 400 cm?
2. What is the minimum height from which Barbie should jump if 25 rubber bands are used?
3. How might the type and width of the rubber band affect the results?
4. If weight is added to Barbie, would more or fewer rubber bands be needed to achieve the same results?
#### Materials
- **Student Materials**: Bungee Barbie Activity Packet, Bungee Barbie Spreadsheet, doll, rubber bands, measuring tape, cash register tape, TI 84+ calculator.
- **Teacher Materials**: Bungee Barbie Project Rubric, Calculating a Spring Constant Extension Activity, laptop, projector, model set.
#### Procedure
1. **Introduction**: Introduce the concept by asking about the importance of accurate height and weight estimates for safe bungee jumping.
2. **Setup**: Distribute materials and demonstrate how to create a double-loop for attaching rubber bands to Barbie and how to add rubber bands using a slip knot.
3. **Experiment**: Have students conduct the experiment, recording data in the provided table. Ensure all rubber bands are the same size and thickness.
4. **Data Analysis**: After completing the experiment, have students check their data for irregularities and re-do the experiment if necessary. Then, ask them to create a graph of the data using the Illuminations Line of Best Fit activity or the Bungee Barbie Spreadsheet.
5. **Testing Conjectures**: Take students to a location where Barbie can be dropped from a significant height to test their conjectures about the maximum number of centimeters for a safe jump.
#### Assessment
The attached Bungee Barbie Project Rubric can be used to evaluate student work. It is recommended to share this rubric with students before completing the lesson so they are aware of the evaluation criteria.
#### Additional Resources
For further learning, consider showing short videos about bungee jumping available on websites like Bungee TV, Bungee Jump Preview, and Land Diving Ritual, noting that some content may not be suitable for all classrooms.
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teacherstryscience.org
|
en
| 0.892999
| 2017-12-13T03:31:24
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http://www.teacherstryscience.org/lp/bungee-barbie%C2%AE
| 0.600626
|
A quadrilateral is a polygon with four sides. There are seven quadrilaterals, some that are surely familiar to you, and some that may not be so familiar. Check out the following definitions and the quadrilateral family tree in the following figure. If you know what the quadrilaterals look like, their definitions should make sense and be pretty easy to understand though the kite definition is a bit of a mouthful. Here are the seven quadrilaterals:
Parallelogram: A quadrilateral that has two pairs of parallel sides.
Rhombus: A quadrilateral with four congruent sides; a rhombus is both a kite and a parallelogram.
Rectangle: A quadrilateral with four right angles; a rectangle is a type of parallelogram.
Square: A quadrilateral with four congruent sides and four right angles; a square is both a rhombus and a rectangle.
Trapezoid: A quadrilateral with exactly one pair of parallel sides the parallel sides are called bases.
Isosceles trapezoid: A trapezoid in which the nonparallel sides the legs are congruent.
In the hierarchy of quadrilaterals shown in the following figure, a quadrilateral below another in the family tree is a special case of the one above it.
A quadrilateral is a 4-sided polygon bounded by 4 finite line segments. A quadrilateral has 2 diagonals based on which it can be classified into concave or convex quadrilateral. In case of convex quadrilaterals, diagonals always lie inside the boundary of the polygon.
Squares are the most regular quadrilateral and have the most properties. A square is also a rectangle, parallelogram, rhombus, kite, and trapezoid. A rectangle is a quadrilateral with congruent angles. A rectangle is also a parallelogram and a trapezoid.
Learn the application of angle properties of quadrilaterals; figure out the measures of the indicated angles, also solve for 'x' to determine the angles of special quadrilaterals to mention just a few. Sample Worksheets. Quadrilateral Charts. Perimeter of Quadrilaterals. Area of Quadrilaterals.
In this lesson you will learn to identify specific kinds of quadrilaterals by looking at their attributes. Create your free account Teacher Student. Create a new teacher account for LearnZillion. All fields are required. Name. Email address. Email confirmation. Password.
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CC-MAIN-2021-25/segments/1623487643380.40/warc/CC-MAIN-20210619020602-20210619050602-00456.warc.gz
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cat-research.com
|
en
| 0.876363
| 2021-06-19T04:01:10
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http://cat-research.com/desmar-properties-of-quadrilaterals.php
| 0.994724
|
While working on our 2013 climate meta-analysis, we came across an article by Ole Thiesen at PRIO, where he investigated the relationship between local climatic events and conflict in Kenya. Thiesen's model estimated the effect of temperature and rainfall on conflict, but reported finding no significant effect of either variable. Upon reviewing the replication code, we noticed that the squared terms for temperature and rainfall were offset by a constant, which was not apparent in the linear terms.
This offset was problematic because, in non-linear models, adding a constant incorrectly can be dangerous. We realized that the squared term for temperature, when expanded, introduced a constant term that affected the regression coefficients. The actual regression model being run was different from the intended model, which meant that directly interpreting the coefficients was not accurate.
Specifically, if a constant is added prior to squaring a variable, the coefficient for that term is unaffected, but all other coefficients in the model are altered. This subtlety is not immediately apparent and can lead to incorrect conclusions. To test this theory, we re-estimated the model using the correct squared measures and found that the squared coefficients remained unchanged, but the linear effects did.
This correction had significant implications, as the original analysis had suggested that the linear effect of temperature was insignificant, contradicting earlier findings. However, after removing the offending constant term, a large positive and significant linear effect of temperature was revealed, consistent with previous research. The correct linear combination of coefficients from the original regression also showed a significant marginal effect of temperature.
The error was not obvious and may be a common mistake in non-linear models, particularly when estimating interaction effects. Thiesen's construction of the data set was an important contribution, and he graciously acknowledged the mistake when we brought it to his attention. Unfortunately, our comment was not widely seen, as the journal that published the original article did not accept research notes or commentaries.
The key takeaway is that adding constants incorrectly in non-linear models can have significant consequences, and it is essential to be mindful of this potential pitfall to avoid drawing incorrect conclusions.
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CC-MAIN-2023-14/segments/1679296948951.4/warc/CC-MAIN-20230329054547-20230329084547-00529.warc.gz
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g-feed.com
|
en
| 0.917096
| 2023-03-29T07:19:42
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http://www.g-feed.com/2015/12/
| 0.939614
|
A decibel (dB) represents the ratio of two variables on a logarithmic scale. Using a logarithmic scale is a better approximation of human hearing than linear variables. The gigantic ratio of barely perceptible sound pressure to the loudest tolerable sound pressure is compressed into a manageable scale of 0 to 130 dB. The general calculation is: log (value/reference value), using the logarithm to base 10.
The result is the Bel, one-tenth of which is one deci-bel, i.e., a decibel. These are power ratios. For sound pressures, voltages, and currents, the factor is 20. The formulas for calculating decibels are:
- Power ratio in dB: 10 x log10 (power/reference power)
- Sound pressure, voltage, or current ratios in dB: 20 x log10 (value/reference value)
In the case of sound pressure ratios, the auditory threshold is used, having a value of 20 μPa. The unit 'dB' is often appended with 'SPL' to indicate sound pressure level, although 'SPL' is commonly omitted.
Reference values and their corresponding decibel units are:
- 1 μV: dB μV
- 1 mV: dB mV
- 0.775 V: dBu
- 1 V: dBV
- 20 μPa: dB SPL
The relationship between physical values and decibel values is as follows:
- Multiplication of physical values corresponds to addition of decibel values
- Division of physical values corresponds to subtraction of decibel values
- A physical value less than 1 corresponds to a negative decibel value
- A physical value of 1 corresponds to a decibel value of 0
- A physical value greater than 1 corresponds to a positive decibel value
Examples:
- An amplifier with a 1000-fold gain has a decibel value of 20 x log (1000/1) = +60 dB
- An attenuator that reduces a voltage to one-tenth has a decibel value of 20 x log (0.1/1) = -20 dB
- Connecting the attenuator to the amplifier results in a decibel value of 60 dB + (-20 dB) = 40 dB
The sound pressure level of a loudspeaker can be calculated using the formula: p1 = pn + 10 x log (P), where p1 is the sound pressure level, pn is the characteristic sound pressure level, and P is the supplied power. Each doubling of power results in an additional 3 dB of SPL.
To calculate the sound pressure level at a distance other than 1 meter, the formula is: p = p1 - 20 x log (d), where p is the sound pressure level at the defined distance, p1 is the sound pressure level at 1 meter, and d is the distance. Each doubling of distance results in a 6 dB decrease in SPL.
The combined formula for sound pressure level at a given power and distance is: p = pn + 10 x log (P) - 20 x log (d). For example, a loudspeaker with a characteristic sound pressure level of 90 dB at 1 W/1 m and an input power of 30 watts at a distance of 8 meters has a sound pressure level of:
- 90 dB + 10 x log (30) - 20 x log (8) = 90 dB + 15 dB - 18 dB = 87 dB
Alternatively, using the tables provided: 90 dB + 15 dB (at 30 watts) - 12 dB (at 4 m) - 6 dB (at 2 m) = 87 dB.
A perceived doubling in volume requires around 10 times the amplifier power. The distance and minimum sound pressure level between standard ceiling loudspeakers at different degrees of speech intelligibility and 6 W of power are:
- Best intelligibility: distance between loudspeakers ranges from 2.3 to 6.9 meters, with minimum sound pressure levels ranging from 92 to 83 dB
- Good intelligibility: distance between loudspeakers ranges from 3.6 to 10.7 meters, with minimum sound pressure levels ranging from 90 to 81 dB
- Background music: distance between loudspeakers ranges from 8.2 to 24.7 meters, with minimum sound pressure levels ranging from 85 to 75 dB.
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CC-MAIN-2023-14/segments/1679296946445.46/warc/CC-MAIN-20230326173112-20230326203112-00086.warc.gz
|
toa.eu
|
en
| 0.707861
| 2023-03-26T18:36:53
|
https://www.toa.eu/service/soundcheck/calculations-with-loudspeakers
| 0.890922
|
An identity is an equation that is true for all values of the variable. A fundamental example is the Pythagorean identity: sin^{2}x + cos^{2}x = 1. This identity is derived using Pythagoras' Theorem and serves as the basis for other Pythagorean identities.
Dividing each term in sin^{2}x + cos^{2}x = 1 by sin^{2}x yields the identity: 1 + cot^{2}x = cosec^{2}x. Similarly, dividing each term by cos^{2}x gives: cot^{2}x + 1 = sec^{2}x. These identities are essential and should be memorized for exams.
The Pythagorean identities, along with other angle formulas, are crucial for solving equations and more complex identities. Proving identities involves manipulating one side of the equation to match the other. It is essential to present these proofs clearly, often using the abbreviations RHS (right-hand side) and LHS (left-hand side) to denote the sides of the identity being proven.
To prove an identity, start with one side and apply substitutions and manipulations until it equals the other side. The goal is to find the shortest proof possible. Some identities may require the use of angle formulas from other units, such as Unit 43, Angle Formulae.
Examples of identities to be proven include:
- cos x cosec x tan x = 1
- sin 3A = 3sin A - 4sin^{3}A
When proving identities, it is crucial to be systematic and methodical, as there is often more than one way to prove an identity. The key is to find the most efficient and straightforward proof.
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CC-MAIN-2020-05/segments/1579250628549.43/warc/CC-MAIN-20200125011232-20200125040232-00271.warc.gz
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bestmaths.net
|
en
| 0.819768
| 2020-01-25T01:17:48
|
http://bestmaths.net/online/index.php/year-levels/year-12/year-12-topic-list/identities/
| 0.999718
|
The Eigenvalue is a scalar amount related to a direct change having a place with a vector space. In this article, we give an insight into how to find Eigenvalues of a matrix and determine the steps regarding doing so. The foundations of the direct condition lattice framework are known as eigenvalues, which are comparable to the cycle of lattice diagonalization.
### Eigenvalues and Eigenvectors
The eigenvalue issue is given by the condition A·v = λ·v, where An is an n-by-n grid, v is a non-zero n-by-1 vector, and λ is a scalar. Any estimation of λ for which this condition has an answer is known as an eigenvalue of the grid A. The vector, v, which compares to this worth, is called an eigenvector. The eigenvalue issue can be changed as (A-λ·I)·v=0, which will possibly have an answer if |A-λ·I|=0. This condition is known as the trademark condition of A and is an nth request polynomial in λ with n roots, which are the eigenvalues of A.
### Steps on How to Find Eigenvalues of a Matrix
To find eigenvalues of a matrix, follow these steps:
1. Ensure the given framework A is a square lattice and decide the character grid I of a similar request.
2. Estimate the network A –λI, where λ is a scalar amount.
3. Find the determinant of network A –λI and compare it to zero.
4. From the condition accordingly, ascertain all the potential estimations of λ, which are the necessary eigenvalues of grid A.
### Examples on How to Find EigenValues of a Matrix
Given a 2x2 matrix A = \[\begin{bmatrix} 0 & 1\\ -2& -3 \end{bmatrix}\], the equation for solving is | A - λ . I | = 0. This yields λ2 + 3λ + 2 = 0, and the two eigenvalues are λ1=-1, λ2=-2.
### Properties of Eigenvalues
Key properties of eigenvalues include:
- The trace of A is the sum of its eigenvalues.
- The determinant of A is the product of its eigenvalues.
- If A is Hermitian, then each eigenvalue is real.
- If A is positive-definite, then each eigenvalue is positive.
- System A is invertible if and only if each eigenvalue is non-zero.
### FAQs on How to Determine The Eigenvalues of a Matrix
1. What are Eigenvalues?
Eigenvalues are scalar amounts related to a straight arrangement of conditions which when duplicated by a non-zero vector equivalent to the vector got by change working on the vector, spoke to as AV = λV.
2. How Can We Determine the Eigenvalues of a Matrix?
To determine the eigenvalues of a matrix, we use the condition ∣A–λI∣ = 0, which empowers us to figure eigenvalues λ without any problem. Given a square network A, the condition that describes an eigenvalue, λ, is the presence of a non-zero vector x with the end goal that A x = λ x. The determinant of the coefficient network—which for this situation is A − λ I—must be zero for non-zero arrangements. The resulting expression is a monic polynomial in λ, known as the trademark polynomial of A, and its zeros are the eigenvalues of A.
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CC-MAIN-2023-14/segments/1679296948620.60/warc/CC-MAIN-20230327092225-20230327122225-00631.warc.gz
|
vedantu.com
|
en
| 0.812997
| 2023-03-27T11:51:08
|
https://www.vedantu.com/iit-jee/how-to-determine-the-eigenvalues-of-a-matrix
| 0.997676
|
5995 * 0 = 0
5995 * 0.1 = 59.95
5995 * 0.2 = 119.9
5995 * 0.3 = 179.85
5995 * 0.4 = 239.8
5995 * 0.5 = 299.75
5995 * 0.6 = 359.7
5995 * 0.7 = 419.65
5995 * 0.8 = 479.6
5995 * 0.9 = 539.55
5995 * 0.10 = 59.95
5995 * 0.11 = 65.945
5995 * 0.12 = 71.94
5995 * 0.13 = 77.935
5995 * 0.14 = 83.93
5995 * 0.15 = 89.925
5995 * 0.16 = 95.92
5995 * 0.17 = 101.915
5995 * 0.18 = 107.91
5995 * 0.19 = 113.905
5995 * 0.20 = 119.9
5995 * 0.21 = 125.895
5995 * 0.22 = 131.89
5995 * 0.23 = 137.885
5995 * 0.24 = 143.88
5995 * 0.25 = 149.875
5995 * 0.26 = 155.87
5995 * 0.27 = 161.865
5995 * 0.28 = 167.86
5995 * 0.29 = 173.855
5995 * 0.30 = 179.85
5995 * 0.31 = 185.845
5995 * 0.32 = 191.84
5995 * 0.33 = 197.835
5995 * 0.34 = 203.83
5995 * 0.35 = 209.825
5995 * 0.36 = 215.82
5995 * 0.37 = 221.815
5995 * 0.38 = 227.81
5995 * 0.39 = 233.805
5995 * 0.40 = 239.8
5995 * 0.41 = 245.795
5995 * 0.42 = 251.79
5995 * 0.43 = 257.785
5995 * 0.44 = 263.78
5995 * 0.45 = 269.775
5995 * 0.46 = 275.77
5995 * 0.47 = 281.765
5995 * 0.48 = 287.76
5995 * 0.49 = 293.755
5995 * 0.50 = 299.75
5995 * 0.51 = 305.745
5995 * 0.52 = 311.74
5995 * 0.53 = 317.735
5995 * 0.54 = 323.73
5995 * 0.55 = 329.725
5995 * 0.56 = 335.72
5995 * 0.57 = 341.715
5995 * 0.58 = 347.71
5995 * 0.59 = 353.705
5995 * 0.60 = 359.7
5995 * 0.61 = 365.695
5995 * 0.62 = 371.69
5995 * 0.63 = 377.685
5995 * 0.64 = 383.68
5995 * 0.65 = 389.675
5995 * 0.66 = 395.67
5995 * 0.67 = 401.665
5995 * 0.68 = 407.66
5995 * 0.69 = 413.655
5995 * 0.70 = 419.65
5995 * 0.71 = 425.645
5995 * 0.72 = 431.64
5995 * 0.73 = 437.635
5995 * 0.74 = 443.63
5995 * 0.75 = 449.625
5995 * 0.76 = 455.62
5995 * 0.77 = 461.615
5995 * 0.78 = 467.61
5995 * 0.79 = 473.605
5995 * 0.80 = 479.6
5995 * 0.81 = 485.595
5995 * 0.82 = 491.59
5995 * 0.83 = 497.585
5995 * 0.84 = 503.58
5995 * 0.85 = 509.575
5995 * 0.86 = 515.57
5995 * 0.87 = 521.565
5995 * 0.88 = 527.56
5995 * 0.89 = 533.555
5995 * 0.90 = 539.55
5995 * 0.91 = 545.545
5995 * 0.92 = 551.54
5995 * 0.93 = 557.535
5995 * 0.94 = 563.53
5995 * 0.95 = 569.525
5995 * 0.96 = 575.52
5995 * 0.97 = 581.515
5995 * 0.98 = 587.51
5995 * 0.99 = 593.505
5995 * 1.00 = 5995
|
CC-MAIN-2020-05/segments/1579250599718.13/warc/CC-MAIN-20200120165335-20200120194335-00516.warc.gz
|
bubble.ro
|
en
| 0.827907
| 2020-01-20T17:22:33
|
http://num.bubble.ro/m/5995/
| 0.998608
|
To find the first term of a geometric sequence given the fifth and sixth terms, we can use the formula for the common ratio: r = t_n / t_(n-1). Since the fifth and sixth terms are 8 and 16, respectively, we calculate r = 16 / 8 = 2.
Now that we have the common ratio, we can solve for the first term (a) using the formula for the nth term of a geometric sequence: t_n = a * r^(n-1). Given the sixth term is 16, we set up the equation 16 = a * 2^(6-1), which simplifies to 16 = a * 2^5.
Further simplifying, we get 16 = a * 32. Solving for a, we find a = 16 / 32 = 1/2. Therefore, the first term of the geometric sequence is 1/2.
|
CC-MAIN-2021-25/segments/1623487635724.52/warc/CC-MAIN-20210618043356-20210618073356-00054.warc.gz
|
socratic.org
|
en
| 0.841304
| 2021-06-18T06:04:09
|
https://socratic.org/questions/if-the-fifth-and-sixth-terms-of-a-geometric-sequence-are-8-and-16-16-respectivel
| 0.988449
|
## A Basic Flight Simulator in Excel #2 – Airplane Positioning, Control Surfaces, and Turn Dynamics
This tutorial explains the basics of airplane positioning, control surfaces, and turn dynamics. It covers the three angles characterizing an airplane's position in flight: yaw, pitch, and roll. The control surfaces, including rudders, ailerons, and elevators, are described, along with the control devices, such as the yoke and rudder pedals.
### Review of the Three Airplane Rotation Angles
* The yaw angle defines rotation around the vertical z-axis, controlled by the rudder.
* The pitch angle defines rotation around the lateral x-axis, controlled by the elevator.
* The roll angle defines rotation around the longitudinal y-axis, controlled by the ailerons.
Since the joystick has only two degrees of freedom, we will control the pitch and roll angles, as these are the most important angles in controlling an airplane.
### Airplane Controls
An airplane has two main control devices: the yoke (joystick or control stick) and the rudder pedals. The yoke controls the ailerons (by sideways movement) and the elevator (by back and forth movement). The rudder pedals control the rudder.
### The Rudder
The rudder controls the yaw of the airplane. Its role is important mainly at low altitude flying, where not much roll angles are permitted. The rudder is a secondary control device for an airplane during regular flight.
### The Elevator
The elevator controls the pitch of the airplane and is used in changing the plane's attitude for gaining or losing altitude and in turns. It is important even in executing constant altitude turns.
### The Ailerons
The ailerons control the roll of the airplane and are mainly used in turns. They are controlled by the sideways movement of the yoke (or joystick).
### How an Airplane Turns
An airplane turn is initiated by rolling the plane in the direction of the turn by a sideways move of the control stick, then reducing the roll and increasing the pitch to a positive value (nose up). Maintaining the proper combination of positive pitch and roll is needed for the vector sum of gravity, centrifugal force, and lift to balance.
### Pitch Rate and Roll Rate
A control surface will deflect the air stream, producing a force on the tip of the aircraft proportional to the deflection angle. The force will produce a momentum proportional to the size of the force and the distance of the control surface from the pressure center of the airplane. Any deviation in the position of a control device will produce a change in the angle of the ship with a rate proportional to the amount of deviation.
### Modeling the Landscape Coordinate Change during Pitch Rotation
The flight simulator will be a 2D scatter chart containing the 2D mapping of a 3D wireframe landscape. The origin of the system of coordinates will be tied to the cockpit, with the x-axis pointing forward, the y-axis pointing to the right, and the z-axis pointing upwards. The relationship between the old coordinates and the new coordinates needs to be found when the plane changes its pitch angle.
|
CC-MAIN-2024-38/segments/1725700651944.55/warc/CC-MAIN-20240918233405-20240919023405-00078.warc.gz
|
excelunusual.com
|
en
| 0.87588
| 2024-09-19T00:52:44
|
https://excelunusual.com/flight-simulator-tutorial-2-basic-airplane-positioning-control-surfaces-and-turn-dynamics/
| 0.439428
|
In various types of numbers, we will explore even numbers, odd numbers, prime numbers, composite numbers, coprime numbers, and twin prime numbers.
**Even Numbers**
A whole number exactly divisible by 2 is called an even number. Examples include 2, 4, 6, 8, 10, and any number ending in 0, 2, 4, 6, or 8, such as 246, 1894, 5468, and 100. Consecutive even numbers differ by 2.
**Odd Numbers**
A whole number not exactly divisible by 2 is called an odd number. Examples include 3, 5, 7, 9, 11, 13, 15, and any number ending in 1, 3, 5, 7, or 9.
**Prime Numbers**
Numbers with only two factors, 1 and the number itself, are called prime numbers. Examples include 2, 3, 5, 7, 11, 19, and 37. Note that 2 is the only even prime number.
**Composite Numbers**
Numbers with more than two factors are called composite numbers. Examples include 4, 6, 8, 10, and so on. It's worth noting that 1 is neither prime nor composite, and 9 is the lowest odd composite number.
**Coprime Numbers**
Two numbers are coprime if they do not have a common factor other than 1, or if their highest common factor (HCF) is 1. Examples include 7 and 10, and 15 and 17. Coprime numbers do not need to be prime numbers.
**Twin Prime Numbers**
Twin prime numbers are two prime numbers whose difference is 2. Examples include 3 and 5, 17 and 19, 41 and 43, 29 and 31, and 71 and 73.
|
CC-MAIN-2019-04/segments/1547583807724.75/warc/CC-MAIN-20190121193154-20190121215154-00414.warc.gz
|
math-only-math.com
|
en
| 0.871929
| 2019-01-21T20:43:39
|
https://www.math-only-math.com/various-types-of-numbers.html
| 0.999049
|
## College Algebra- Max/ Min Applications of Quadratic
Finding the Maximum or Minimum of a Quadratic Problem. Determine, WITHOUT GRAPHING, if the given quadratic functions have a maximum or minimum value and then find the value.
### Quiz & Worksheet Finding Minimum & Maximum Values
Linear and Quadratic Functions Graphing Zeros and Min/Max.
### Applications of Quadratic Functions
Goals:
1. To find an extreme value of a quadratic function in an applied context.
2. To find the equation of a quadratic function that fits given data points.
Ex. The profit for producing x Snickers bars is, where P(x) is in dollars. Use the vertex.
Minimum and maximum value of a function if the objective function is not a quadratic function, Applications Mechanics. Problems in MAX/MIN FOR FUNCTIONS OF SEVERAL VARIABLES Abstract. These notes supplement the material in В§8.2 of our text on quadratic forms and symmetric matrices.
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
Finding the Range of Quadratic Functions . {2a}.\) Whether this is a maximum or minimum depends on the sign of \ Applications of Completing the Square.
Note: When you're dealing with quadratic functions, maximum and minimum are very likely to come up. This tutorial takes a look at the maximum of a quadratic function.
Watch video · Comparing maximum points of quadratic functions. Finding features of quadratic functions. for this particular one, we're going to hit a minimum …
Minimum and maximum values are found in parabolas that open up or down. In this lesson, learn how to find the minimum and maximum values...
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A quadratic function can be used to through the application of quadratic functions. coordinates of the turning point and whether it is a maximum or a minimum. a
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A bivariate quadratic function is a second-degree polynomial of the form the function has no maximum or minimum; its graph forms an hyperbolic paraboloid.
Finding the Maximum or Minimum of a Quadratic. 3 Finding the Maximum or Minimum of a Quadratic What we can do is we can get equations that are an application.
ALGEBRA UNIT 11-GRAPHING QUADRATICS THE GRAPH OF A QUADRATIC FUNCTION describe it as a max or min pt.
### Quiz & Worksheet Finding Minimum & Maximum Values
Linear and Quadratic Functions Graphing Zeros and Min/Max.
### Applications of Quadratic Functions
Goals:
1. To find an extreme value of a quadratic function in an applied context.
2. To find the equation of a quadratic function that fits given data points.
Ex. The profit for producing x Snickers bars is, where P(x) is in dollars. Use the vertex.
Minimum and maximum value of a function if the objective function is not a quadratic function, Applications Mechanics. Problems in MAX/MIN FOR FUNCTIONS OF SEVERAL VARIABLES Abstract. These notes supplement the material in В§8.2 of our text on quadratic forms and symmetric matrices.
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
Finding the Range of Quadratic Functions . {2a}.\) Whether this is a maximum or minimum depends on the sign of \ Applications of Completing the Square.
Note: When you're dealing with quadratic functions, maximum and minimum are very likely to come up. This tutorial takes a look at the maximum of a quadratic function.
Watch video · Comparing maximum points of quadratic functions. Finding features of quadratic functions. for this particular one, we're going to hit a minimum …
Minimum and maximum values are found in parabolas that open up or down. In this lesson, learn how to find the minimum and maximum values...
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A quadratic function can be used to through the application of quadratic functions. coordinates of the turning point and whether it is a maximum or a minimum. a
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A bivariate quadratic function is a second-degree polynomial of the form the function has no maximum or minimum; its graph forms an hyperbolic paraboloid.
Finding the Maximum or Minimum of a Quadratic. 3 Finding the Maximum or Minimum of a Quadratic What we can do is we can get equations that are an application.
ALGEBRA UNIT 11-GRAPHING QUADRATICS THE GRAPH OF A QUADRATIC FUNCTION describe it as a max or min pt.
### Quiz & Worksheet Finding Minimum & Maximum Values
Linear and Quadratic Functions Graphing Zeros and Min/Max.
### Applications of Quadratic Functions
Goals:
1. To find an extreme value of a quadratic function in an applied context.
2. To find the equation of a quadratic function that fits given data points.
Ex. The profit for producing x Snickers bars is, where P(x) is in dollars. Use the vertex.
Minimum and maximum value of a function if the objective function is not a quadratic function, Applications Mechanics. Problems in MAX/MIN FOR FUNCTIONS OF SEVERAL VARIABLES Abstract. These notes supplement the material in В§8.2 of our text on quadratic forms and symmetric matrices.
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
Finding the Range of Quadratic Functions . {2a}.\) Whether this is a maximum or minimum depends on the sign of \ Applications of Completing the Square.
Note: When you're dealing with quadratic functions, maximum and minimum are very likely to come up. This tutorial takes a look at the maximum of a quadratic function.
Watch video · Comparing maximum points of quadratic functions. Finding features of quadratic functions. for this particular one, we're going to hit a minimum …
Minimum and maximum values are found in parabolas that open up or down. In this lesson, learn how to find the minimum and maximum values...
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A quadratic function can be used to through the application of quadratic functions. coordinates of the turning point and whether it is a maximum or a minimum. a
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A bivariate quadratic function is a second-degree polynomial of the form the function has no maximum or minimum; its graph forms an hyperbolic paraboloid.
Finding the Maximum or Minimum of a Quadratic. 3 Finding the Maximum or Minimum of a Quadratic What we can do is we can get equations that are an application.
ALGEBRA UNIT 11-GRAPHING QUADRATICS THE GRAPH OF A QUADRATIC FUNCTION describe it as a max or min pt.
### Quiz & Worksheet Finding Minimum & Maximum Values
Linear and Quadratic Functions Graphing Zeros and Min/Max.
### Applications of Quadratic Functions
Goals:
1. To find an extreme value of a quadratic function in an applied context.
2. To find the equation of a quadratic function that fits given data points.
Ex. The profit for producing x Snickers bars is, where P(x) is in dollars. Use the vertex.
Minimum and maximum value of a function if the objective function is not a quadratic function, Applications Mechanics. Problems in MAX/MIN FOR FUNCTIONS OF SEVERAL VARIABLES Abstract. These notes supplement the material in В§8.2 of our text on quadratic forms and symmetric matrices.
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
Finding the Range of Quadratic Functions . {2a}.\) Whether this is a maximum or minimum depends on the sign of \ Applications of Completing the Square.
Note: When you're dealing with quadratic functions, maximum and minimum are very likely to come up. This tutorial takes a look at the maximum of a quadratic function.
Watch video · Comparing maximum points of quadratic functions. Finding features of quadratic functions. for this particular one, we're going to hit a minimum …
Minimum and maximum values are found in parabolas that open up or down. In this lesson, learn how to find the minimum and maximum values...
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A quadratic function can be used to through the application of quadratic functions. coordinates of the turning point and whether it is a maximum or a minimum. a
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A bivariate quadratic function is a second-degree polynomial of the form the function has no maximum or minimum; its graph forms an hyperbolic paraboloid.
Finding the Maximum or Minimum of a Quadratic. 3 Finding the Maximum or Minimum of a Quadratic What we can do is we can get equations that are an application.
ALGEBRA UNIT 11-GRAPHING QUADRATICS THE GRAPH OF A QUADRATIC FUNCTION describe it as a max or min pt.
### Quiz & Worksheet Finding Minimum & Maximum Values
Linear and Quadratic Functions Graphing Zeros and Min/Max.
### Applications of Quadratic Functions
Goals:
1. To find an extreme value of a quadratic function in an applied context.
2. To find the equation of a quadratic function that fits given data points.
Ex. The profit for producing x Snickers bars is, where P(x) is in dollars. Use the vertex.
Minimum and maximum value of a function if the objective function is not a quadratic function, Applications Mechanics. Problems in MAX/MIN FOR FUNCTIONS OF SEVERAL VARIABLES Abstract. These notes supplement the material in В§8.2 of our text on quadratic forms and symmetric matrices.
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
Finding the Range of Quadratic Functions . {2a}.\) Whether this is a maximum or minimum depends on the sign of \ Applications of Completing the Square.
Note: When you're dealing with quadratic functions, maximum and minimum are very likely to come up. This tutorial takes a look at the maximum of a quadratic function.
Watch video · Comparing maximum points of quadratic functions. Finding features of quadratic functions. for this particular one, we're going to hit a minimum …
Minimum and maximum values are found in parabolas that open up or down. In this lesson, learn how to find the minimum and maximum values...
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A quadratic function can be used to through the application of quadratic functions. coordinates of the turning point and whether it is a maximum or a minimum. a
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A bivariate quadratic function is a second-degree polynomial of the form the function has no maximum or minimum; its graph forms an hyperbolic paraboloid.
Finding the Maximum or Minimum of a Quadratic. 3 Finding the Maximum or Minimum of a Quadratic What we can do is we can get equations that are an application.
ALGEBRA UNIT 11-GRAPHING QUADRATICS THE GRAPH OF A QUADRATIC FUNCTION describe it as a max or min pt.
### Quiz & Worksheet Finding Minimum & Maximum Values
Linear and Quadratic Functions Graphing Zeros and Min/Max.
### Applications of Quadratic Functions
Goals:
1. To find an extreme value of a quadratic function in an applied context.
2. To find the equation of a quadratic function that fits given data points.
Ex. The profit for producing x Snickers bars is, where P(x) is in dollars. Use the vertex.
Minimum and maximum value of a function if the objective function is not a quadratic function, Applications Mechanics. Problems in MAX/MIN FOR FUNCTIONS OF SEVERAL VARIABLES Abstract. These notes supplement the material in В§8.2 of our text on quadratic forms and symmetric matrices.
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
Finding the Range of Quadratic Functions . {2a}.\) Whether this is a maximum or minimum depends on the sign of \ Applications of Completing the Square.
Note: When you're dealing with quadratic functions, maximum and minimum are very likely to come up. This tutorial takes a look at the maximum of a quadratic function.
Watch video · Comparing maximum points of quadratic functions. Finding features of quadratic functions. for this particular one, we're going to hit a minimum …
Minimum and maximum values are found in parabolas that open up or down. In this lesson, learn how to find the minimum and maximum values...
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A quadratic function can be used to through the application of quadratic functions. coordinates of the turning point and whether it is a maximum or a minimum. a
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A bivariate quadratic function is a second-degree polynomial of the form the function has no maximum or minimum; its graph forms an hyperbolic paraboloid.
Finding the Maximum or Minimum of a Quadratic. 3 Finding the Maximum or Minimum of a Quadratic What we can do is we can get equations that are an application.
ALGEBRA UNIT 11-GRAPHING QUADRATICS THE GRAPH OF A QUADRATIC FUNCTION describe it as a max or min pt.
### Quiz & Worksheet Finding Minimum & Maximum Values
Linear and Quadratic Functions Graphing Zeros and Min/Max.
### Applications of Quadratic Functions
Goals:
1. To find an extreme value of a quadratic function in an applied context.
2. To find the equation of a quadratic function that fits given data points.
Ex. The profit for producing x Snickers bars is, where P(x) is in dollars. Use the vertex.
Minimum and maximum value of a function if the objective function is not a quadratic function, Applications Mechanics. Problems in MAX/MIN FOR FUNCTIONS OF SEVERAL VARIABLES Abstract. These notes supplement the material in В§8.2 of our text on quadratic forms and symmetric matrices.
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
Finding the Range of Quadratic Functions . {2a}.\) Whether this is a maximum or minimum depends on the sign of \ Applications of Completing the Square.
Note: When you're dealing with quadratic functions, maximum and minimum are very likely to come up. This tutorial takes a look at the maximum of a quadratic function.
Watch video · Comparing maximum points of quadratic functions. Finding features of quadratic functions. for this particular one, we're going to hit a minimum …
Minimum and maximum values are found in parabolas that open up or down. In this lesson, learn how to find the minimum and maximum values...
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A quadratic function can be used to through the application of quadratic functions. coordinates of the turning point and whether it is a maximum or a minimum. a
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A bivariate quadratic function is a second-degree polynomial of the form the function has no maximum or minimum; its graph forms an hyperbolic paraboloid.
Finding the Maximum or Minimum of a Quadratic. 3 Finding the Maximum or Minimum of a Quadratic What we can do is we can get equations that are an application.
ALGEBRA UNIT 11-GRAPHING QUADRATICS THE GRAPH OF A QUADRATIC FUNCTION describe it as a max or min pt.
### Quiz & Worksheet Finding Minimum & Maximum Values
Linear and Quadratic Functions Graphing Zeros and Min/Max.
### Applications of Quadratic Functions
Goals:
1. To find an extreme value of a quadratic function in an applied context.
2. To find the equation of a quadratic function that fits given data points.
Ex. The profit for producing x Snickers bars is, where P(x) is in dollars. Use the vertex.
Minimum and maximum value of a function if the objective function is not a quadratic function, Applications Mechanics. Problems in MAX/MIN FOR FUNCTIONS OF SEVERAL VARIABLES Abstract. These notes supplement the material in В§8.2 of our text on quadratic forms and symmetric matrices.
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
Finding the Range of Quadratic Functions . {2a}.\) Whether this is a maximum or minimum depends on the sign of \ Applications of Completing the Square.
Note: When you're dealing with quadratic functions, maximum and minimum are very likely to come up. This tutorial takes a look at the maximum of a quadratic function.
Watch video · Comparing maximum points of quadratic functions. Finding features of quadratic functions. for this particular one, we're going to hit a minimum …
Minimum and maximum values are found in parabolas that open up or down. In this lesson, learn how to find the minimum and maximum values...
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A quadratic function can be used to through the application of quadratic functions. coordinates of the turning point and whether it is a maximum or a minimum. a
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A bivariate quadratic function is a second-degree polynomial of the form the function has no maximum or minimum; its graph forms an hyperbolic paraboloid.
Finding the Maximum or Minimum of a Quadratic. 3 Finding the Maximum or Minimum of a Quadratic What we can do is we can get equations that are an application.
ALGEBRA UNIT 11-GRAPHING QUADRATICS THE GRAPH OF A QUADRATIC FUNCTION describe it as a max or min pt.
### Quiz & Worksheet Finding Minimum & Maximum Values
Linear and Quadratic Functions Graphing Zeros and Min/Max.
### Applications of Quadratic Functions
Goals:
1. To find an extreme value of a quadratic function in an applied context.
2. To find the equation of a quadratic function that fits given data points.
Ex. The profit for producing x Snickers bars is, where P(x) is in dollars. Use the vertex.
Minimum and maximum value of a function if the objective function is not a quadratic function, Applications Mechanics. Problems in MAX/MIN FOR FUNCTIONS OF SEVERAL VARIABLES Abstract. These notes supplement the material in В§8.2 of our text on quadratic forms and symmetric matrices.
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
Finding the Range of Quadratic Functions . {2a}.\) Whether this is a maximum or minimum depends on the sign of \ Applications of Completing the Square.
Note: When you're dealing with quadratic functions, maximum and minimum are very likely to come up. This tutorial takes a look at the maximum of a quadratic function.
Watch video · Comparing maximum points of quadratic functions. Finding features of quadratic functions. for this particular one, we're going to hit a minimum …
Minimum and maximum values are found in parabolas that open up or down. In this lesson, learn how to find the minimum and maximum values...
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A quadratic function can be used to through the application of quadratic functions. coordinates of the turning point and whether it is a maximum or a minimum. a
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A bivariate quadratic function is a second-degree polynomial of the form the function has no maximum or minimum; its graph forms an hyperbolic paraboloid.
Finding the Maximum or Minimum of a Quadratic. 3 Finding the Maximum or Minimum of a Quadratic What we can do is we can get equations that are an application.
ALGEBRA UNIT 11-GRAPHING QUADRATICS THE GRAPH OF A QUADRATIC FUNCTION describe it as a max or min pt.
### Quiz & Worksheet Finding Minimum & Maximum Values
Linear and Quadratic Functions Graphing Zeros and Min/Max.
### Applications of Quadratic Functions
Goals:
1. To find an extreme value of a quadratic function in an applied context.
2. To find the equation of a quadratic function that fits given data points.
Ex. The profit for producing x Snickers bars is, where P(x) is in dollars. Use the vertex.
Minimum and maximum value of a function if the objective function is not a quadratic function, Applications Mechanics. Problems in MAX/MIN FOR FUNCTIONS OF SEVERAL VARIABLES Abstract. These notes supplement the material in В§8.2 of our text on quadratic forms and symmetric matrices.
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
Finding the Range of Quadratic Functions . {2a}.\) Whether this is a maximum or minimum depends on the sign of \ Applications of Completing the Square.
Note: When you're dealing with quadratic functions, maximum and minimum are very likely to come up. This tutorial takes a look at the maximum of a quadratic function.
Watch video · Comparing maximum points of quadratic functions. Finding features of quadratic functions. for this particular one, we're going to hit a minimum …
Minimum and maximum values are found in parabolas that open up or down. In this lesson, learn how to find the minimum and maximum values...
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A quadratic function can be used to through the application of quadratic functions. coordinates of the turning point and whether it is a maximum or a minimum. a
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A bivariate quadratic function is a second-degree polynomial of the form the function has no maximum or minimum; its graph forms an hyperbolic paraboloid.
Finding the Maximum or Minimum of a Quadratic. 3 Finding the Maximum or Minimum of a Quadratic What we can do is we can get equations that are an application.
ALGEBRA UNIT 11-GRAPHING QUADRATICS THE GRAPH OF A QUADRATIC FUNCTION describe it as a max or min pt.
### Quiz & Worksheet Finding Minimum & Maximum Values
Linear and Quadratic Functions Graphing Zeros and Min/Max.
### Applications of Quadratic Functions
Goals:
1. To find an extreme value of a quadratic function in an applied context.
2. To find the equation of a quadratic function that fits given data points.
Ex. The profit for producing x Snickers bars is, where P(x) is in dollars. Use the vertex.
Minimum and maximum value of a function if the objective function is not a quadratic function, Applications Mechanics. Problems in MAX/MIN FOR FUNCTIONS OF SEVERAL VARIABLES Abstract. These notes supplement the material in В§8.2 of our text on quadratic forms and symmetric matrices.
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
Finding the Range of Quadratic Functions . {2a}.\) Whether this is a maximum or minimum depends on the sign of \ Applications of Completing the Square.
Note: When you're dealing with quadratic functions, maximum and minimum are very likely to come up. This tutorial takes a look at the maximum of a quadratic function.
Watch video · Comparing maximum points of quadratic functions. Finding features of quadratic functions. for this particular one, we're going to hit a minimum …
Minimum and maximum values are found in parabolas that open up or down. In this lesson, learn how to find the minimum and maximum values...
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A quadratic function can be used to through the application of quadratic functions. coordinates of the turning point and whether it is a maximum or a minimum. a
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A bivariate quadratic function is a second-degree polynomial of the form the function has no maximum or minimum; its graph forms an hyperbolic paraboloid.
Finding the Maximum or Minimum of a Quadratic. 3 Finding the Maximum or Minimum of a Quadratic What we can do is we can get equations that are an application.
ALGEBRA UNIT 11-GRAPHING QUADRATICS THE GRAPH OF A QUADRATIC FUNCTION describe it as a max or min pt.
### Quiz & Worksheet Finding Minimum & Maximum Values
Linear and Quadratic Functions Graphing Zeros and Min/Max.
### Applications of Quadratic Functions
Goals:
1. To find an extreme value of a quadratic function in an applied context.
2. To find the equation of a quadratic function that fits given data points.
Ex. The profit for producing x Snickers bars is, where P(x) is in dollars. Use the vertex.
Minimum and maximum value of a function if the objective function is not a quadratic function, Applications Mechanics. Problems in MAX/MIN FOR FUNCTIONS OF SEVERAL VARIABLES Abstract. These notes supplement the material in В§8.2 of our text on quadratic forms and symmetric matrices.
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
Finding the Range of Quadratic Functions . {2a}.\) Whether this is a maximum or minimum depends on the sign of \ Applications of Completing the Square.
Note: When you're dealing with quadratic functions, maximum and minimum are very likely to come up. This tutorial takes a look at the maximum of a quadratic function.
Watch video · Comparing maximum points of quadratic functions. Finding features of quadratic functions. for this particular one, we're going to hit a minimum …
Minimum and maximum values are found in parabolas that open up or down. In this lesson, learn how to find the minimum and maximum values...
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A quadratic function can be used to through the application of quadratic functions. coordinates of the turning point and whether it is a maximum or a minimum. a
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A bivariate quadratic function is a second-degree polynomial of the form the function has no maximum or minimum; its graph forms an hyperbolic paraboloid.
Finding the Maximum or Minimum of a Quadratic. 3 Finding the Maximum or Minimum of a Quadratic What we can do is we can get equations that are an application.
ALGEBRA UNIT 11-GRAPHING QUADRATICS THE GRAPH OF A QUADRATIC FUNCTION describe it as a max or min pt.
### Quiz & Worksheet Finding Minimum & Maximum Values
Linear and Quadratic Functions Graphing Zeros and Min/Max.
### Applications of Quadratic Functions
Goals:
1. To find an extreme value of a quadratic function in an applied context.
2. To find the equation of a quadratic function that fits given data points.
Ex. The profit for producing x Snickers bars is, where P(x) is in dollars. Use the vertex.
Minimum and maximum value of a function if the objective function is not a quadratic function, Applications Mechanics. Problems in MAX/MIN FOR FUNCTIONS OF SEVERAL VARIABLES Abstract. These notes supplement the material in В§8.2 of our text on quadratic forms and symmetric matrices.
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
Finding the Range of Quadratic Functions . {2a}.\) Whether this is a maximum or minimum depends on the sign of \ Applications of Completing the Square.
Note: When you're dealing with quadratic functions, maximum and minimum are very likely to come up. This tutorial takes a look at the maximum of a quadratic function.
Watch video · Comparing maximum points of quadratic functions. Finding features of quadratic functions. for this particular one, we're going to hit a minimum …
Minimum and maximum values are found in parabolas that open up or down. In this lesson, learn how to find the minimum and maximum values...
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A quadratic function can be used to through the application of quadratic functions. coordinates of the turning point and whether it is a maximum or a minimum. a
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A bivariate quadratic function is a second-degree polynomial of the form the function has no maximum or minimum; its graph forms an hyperbolic paraboloid.
Finding the Maximum or Minimum of a Quadratic. 3 Finding the Maximum or Minimum of a Quadratic What we can do is we can get equations that are an application.
ALGEBRA UNIT 11-GRAPHING QUADRATICS THE GRAPH OF A QUADRATIC FUNCTION describe it as a max or min pt.
### Quiz & Worksheet Finding Minimum & Maximum Values
Linear and Quadratic Functions Graphing Zeros and Min/Max.
### Applications of Quadratic Functions
Goals:
1. To find an extreme value of a quadratic function in an applied context.
2. To find the equation of a quadratic function that fits given data points.
Ex. The profit for producing x Snickers bars is, where P(x) is in dollars. Use the vertex.
Minimum and maximum value of a function if the objective function is not a quadratic function, Applications Mechanics. Problems in MAX/MIN FOR FUNCTIONS OF SEVERAL VARIABLES Abstract. These notes supplement the material in В§8.2 of our text on quadratic forms and symmetric matrices.
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
Finding the Range of Quadratic Functions . {2a}.\) Whether this is a maximum or minimum depends on the sign of \ Applications of Completing the Square.
Note: When you're dealing with quadratic functions, maximum and minimum are very likely to come up. This tutorial takes a look at the maximum of a quadratic function.
Watch video · Comparing maximum points of quadratic functions. Finding features of quadratic functions. for this particular one, we're going to hit a minimum …
Minimum and maximum values are found in parabolas that open up or down. In this lesson, learn how to find the minimum and maximum values...
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A quadratic function can be used to through the application of quadratic functions. coordinates of the turning point and whether it is a maximum or a minimum. a
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A bivariate quadratic function is a second-degree polynomial of the form the function has no maximum or minimum; its graph forms an hyperbolic paraboloid.
Finding the Maximum or Minimum of a Quadratic. 3 Finding the Maximum or Minimum of a Quadratic What we can do is we can get equations that are an application.
ALGEBRA UNIT 11-GRAPHING QUADRATICS THE GRAPH OF A QUADRATIC FUNCTION describe it as a max or min pt.
### Quiz & Worksheet Finding Minimum & Maximum Values
Linear and Quadratic Functions Graphing Zeros and Min/Max.
### Applications of Quadratic Functions
Goals:
1. To find an extreme value of a quadratic function in an applied context.
2. To find the equation of a quadratic function that fits given data points.
Ex. The profit for producing x Snickers bars is, where P(x) is in dollars. Use the vertex.
Minimum and maximum value of a function if the objective function is not a quadratic function, Applications Mechanics. Problems in MAX/MIN FOR FUNCTIONS OF SEVERAL VARIABLES Abstract. These notes supplement the material in В§8.2 of our text on quadratic forms and symmetric matrices.
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
Finding the Range of Quadratic Functions . {2a}.\) Whether this is a maximum or minimum depends on the sign of \ Applications of Completing the Square.
Note: When you're dealing with quadratic functions, maximum and minimum are very likely to come up. This tutorial takes a look at the maximum of a quadratic function.
Watch video · Comparing maximum points of quadratic functions. Finding features of quadratic functions. for this particular one, we're going to hit a minimum …
Minimum and maximum values are found in parabolas that open up or down. In this lesson, learn how to find the minimum and maximum values...
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A quadratic function can be used to through the application of quadratic functions. coordinates of the turning point and whether it is a maximum or a minimum. a
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A bivariate quadratic function is a second-degree polynomial of the form the function has no maximum or minimum; its graph forms an hyperbolic paraboloid.
Finding the Maximum or Minimum of a Quadratic. 3 Finding the Maximum or Minimum of a Quadratic What we can do is we can get equations that are an application.
ALGEBRA UNIT 11-GRAPHING QUADRATICS THE GRAPH OF A QUADRATIC FUNCTION describe it as a max or min pt.
### Quiz & Worksheet Finding Minimum & Maximum Values
Linear and Quadratic Functions Graphing Zeros and Min/Max.
### Applications of Quadratic Functions
Goals:
1. To find an extreme value of a quadratic function in an applied context.
2. To find the equation of a quadratic function that fits given data points.
Ex. The profit for producing x Snickers bars is, where P(x) is in dollars. Use the vertex.
Minimum and maximum value of a function if the objective function is not a quadratic function, Applications Mechanics. Problems in MAX/MIN FOR FUNCTIONS OF SEVERAL VARIABLES Abstract. These notes supplement the material in В§8.2 of our text on quadratic forms and symmetric matrices.
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
Finding the Range of Quadratic Functions . {2a}.\) Whether this is a maximum or minimum depends on the sign of \ Applications of Completing the Square.
Note: When you're dealing with quadratic functions, maximum and minimum are very likely to come up. This tutorial takes a look at the maximum of a quadratic function.
Watch video · Comparing maximum points of quadratic functions. Finding features of quadratic functions. for this particular one, we're going to hit a minimum …
Minimum and maximum values are found in parabolas that open up or down. In this lesson, learn how to find the minimum and maximum values...
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A quadratic function can be used to through the application of quadratic functions. coordinates of the turning point and whether it is a maximum or a minimum. a
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A bivariate quadratic function is a second-degree polynomial of the form the function has no maximum or minimum; its graph forms an hyperbolic paraboloid.
Finding the Maximum or Minimum of a Quadratic. 3 Finding the Maximum or Minimum of a Quadratic What we can do is we can get equations that are an application.
ALGEBRA UNIT 11-GRAPHING QUADRATICS THE GRAPH OF A QUADRATIC FUNCTION describe it as a max or min pt.
### Quiz & Worksheet Finding Minimum & Maximum Values
Linear and Quadratic Functions Graphing Zeros and Min/Max.
### Applications of Quadratic Functions
Goals:
1. To find an extreme value of a quadratic function in an applied context.
2. To find the equation of a quadratic function that fits given data points.
Ex. The profit for producing x Snickers bars is, where P(x) is in dollars. Use the vertex.
Minimum and maximum value of a function if the objective function is not a quadratic function, Applications Mechanics. Problems in MAX/MIN FOR FUNCTIONS OF SEVERAL VARIABLES Abstract. These notes supplement the material in В§8.2 of our text on quadratic forms and symmetric matrices.
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
Finding the Range of Quadratic Functions . {2a}.\) Whether this is a maximum or minimum depends on the sign of \ Applications of Completing the Square.
Note: When you're dealing with quadratic functions, maximum and minimum are very likely to come up. This tutorial takes a look at the maximum of a quadratic function.
Watch video · Comparing maximum points of quadratic functions. Finding features of quadratic functions. for this particular one, we're going to hit a minimum …
Minimum and maximum values are found in parabolas that open up or down. In this lesson, learn how to find the minimum and maximum values...
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A quadratic function can be used to through the application of quadratic functions. coordinates of the turning point and whether it is a maximum or a minimum. a
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A bivariate quadratic function is a second-degree polynomial of the form the function has no maximum or minimum; its graph forms an hyperbolic paraboloid.
Finding the Maximum or Minimum of a Quadratic. 3 Finding the Maximum or Minimum of a Quadratic What we can do is we can get equations that are an application.
ALGEBRA UNIT 11-GRAPHING QUADRATICS THE GRAPH OF A QUADRATIC FUNCTION describe it as a max or min pt.
### Quiz & Worksheet Finding Minimum & Maximum Values
Linear and Quadratic Functions Graphing Zeros and Min/Max.
### Applications of Quadratic Functions
Goals:
1. To find an extreme value of a quadratic function in an applied context.
2. To find the equation of a quadratic function that fits given data points.
Ex. The profit for producing x Snickers bars is, where P(x) is in dollars. Use the vertex.
Minimum and maximum value of a function if the objective function is not a quadratic function, Applications Mechanics. Problems in MAX/MIN FOR FUNCTIONS OF SEVERAL VARIABLES Abstract. These notes supplement the material in В§8.2 of our text on quadratic forms and symmetric matrices.
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
Finding the Range of Quadratic Functions . {2a}.\) Whether this is a maximum or minimum depends on the sign of \ Applications of Completing the Square.
Note: When you're dealing with quadratic functions, maximum and minimum are very likely to come up. This tutorial takes a look at the maximum of a quadratic function.
Watch video · Comparing maximum points of quadratic functions. Finding features of quadratic functions. for this particular one, we're going to hit a minimum …
Minimum and maximum values are found in parabolas that open up or down. In this lesson, learn how to find the minimum and maximum values...
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A quadratic function can be used to through the application of quadratic functions. coordinates of the turning point and whether it is a maximum or a minimum. a
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A bivariate quadratic function is a second-degree polynomial of the form the function has no maximum or minimum; its graph forms an hyperbolic paraboloid.
Finding the Maximum or Minimum of a Quadratic. 3 Finding the Maximum or Minimum of a Quadratic What we can do is we can get equations that are an application.
ALGEBRA UNIT 11-GRAPHING QUADRATICS THE GRAPH OF A QUADRATIC FUNCTION describe it as a max or min pt.
### Quiz & Worksheet Finding Minimum & Maximum Values
Linear and Quadratic Functions Graphing Zeros and Min/Max.
### Applications of Quadratic Functions
Goals:
1. To find an extreme value of a quadratic function in an applied context.
2. To find the equation of a quadratic function that fits given data points.
Ex. The profit for producing x Snickers bars is, where P(x) is in dollars. Use the vertex.
Minimum and maximum value of a function if the objective function is not a quadratic function, Applications Mechanics. Problems in MAX/MIN FOR FUNCTIONS OF SEVERAL VARIABLES Abstract. These notes supplement the material in В§8.2 of our text on quadratic forms and symmetric matrices.
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
Finding the Range of Quadratic Functions . {2a}.\) Whether this is a maximum or minimum depends on the sign of \ Applications of Completing the Square.
Note: When you're dealing with quadratic functions, maximum and minimum are very likely to come up. This tutorial takes a look at the maximum of a quadratic function.
Watch video · Comparing maximum points of quadratic functions. Finding features of quadratic functions. for this particular one, we're going to hit a minimum …
Minimum and maximum values are found in parabolas that open up or down. In this lesson, learn how to find the minimum and maximum values...
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A quadratic function can be used to through the application of quadratic functions. coordinates of the turning point and whether it is a maximum or a minimum. a
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A bivariate quadratic function is a second-degree polynomial of the form the function has no maximum or minimum; its graph forms an hyperbolic paraboloid.
Finding the Maximum or Minimum of a Quadratic. 3 Finding the Maximum or Minimum of a Quadratic What we can do is we can get equations that are an application.
ALGEBRA UNIT 11-GRAPHING QUADRATICS THE GRAPH OF A QUADRATIC FUNCTION describe it as a max or min pt.
### Quiz & Worksheet Finding Minimum & Maximum Values
Linear and Quadratic Functions Graphing Zeros and Min/Max.
### Applications of Quadratic Functions
Goals:
1. To find an extreme value of a quadratic function in an applied context.
2. To find the equation of a quadratic function that fits given data points.
Ex. The profit for producing x Snickers bars is, where P(x) is in dollars. Use the vertex.
Minimum and maximum value of a function if the objective function is not a quadratic function, Applications Mechanics. Problems in MAX/MIN FOR FUNCTIONS OF SEVERAL VARIABLES Abstract. These notes supplement the material in В§8.2 of our text on quadratic forms and symmetric matrices.
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
Finding the Range of Quadratic Functions . {2a}.\) Whether this is a maximum or minimum depends on the sign of \ Applications of Completing the Square.
Note: When you're dealing with quadratic functions, maximum and minimum are very likely to come up. This tutorial takes a look at the maximum of a quadratic function.
Watch video · Comparing maximum points of quadratic functions. Finding features of quadratic functions. for this particular one, we're going to hit a minimum …
Minimum and maximum values are found in parabolas that open up or down. In this lesson, learn how to find the minimum and maximum values...
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A quadratic function can be used to through the application of quadratic functions. coordinates of the turning point and whether it is a maximum or a minimum. a
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A bivariate quadratic function is a second-degree polynomial of the form the function has no maximum or minimum; its graph forms an hyperbolic paraboloid.
Finding the Maximum or Minimum of a Quadratic. 3 Finding the Maximum or Minimum of a Quadratic What we can do is we can get equations that are an application.
ALGEBRA UNIT 11-GRAPHING QUADRATICS THE GRAPH OF A QUADRATIC FUNCTION describe it as a max or min pt.
### Quiz & Worksheet Finding Minimum & Maximum Values
Linear and Quadratic Functions Graphing Zeros and Min/Max.
### Applications of Quadratic Functions
Goals:
1. To find an extreme value of a quadratic function in an applied context.
2. To find the equation of a quadratic function that fits given data points.
Ex. The profit for producing x Snickers bars is, where P(x) is in dollars. Use the vertex.
Minimum and maximum value of a function if the objective function is not a quadratic function, Applications Mechanics. Problems in MAX/MIN FOR FUNCTIONS OF SEVERAL VARIABLES Abstract. These notes supplement the material in В§8.2 of our text on quadratic forms and symmetric matrices.
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
Finding the Range of Quadratic Functions . {2a}.\) Whether this is a maximum or minimum depends on the sign of \ Applications of Completing the Square.
Note: When you're dealing with quadratic functions, maximum and minimum are very likely to come up. This tutorial takes a look at the maximum of a quadratic function.
Watch video · Comparing maximum points of quadratic functions. Finding features of quadratic functions. for this particular one, we're going to hit a minimum …
Minimum and maximum values are found in parabolas that open up or down. In this lesson, learn how to find the minimum and maximum values...
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A quadratic function can be used to through the application of quadratic functions. coordinates of the turning point and whether it is a maximum or a minimum. a
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A bivariate quadratic function is a second-degree polynomial of the form the function has no maximum or minimum; its graph forms an hyperbolic paraboloid.
Finding the Maximum or Minimum of a Quadratic. 3 Finding the Maximum or Minimum of a Quadratic What we can do is we can get equations that are an application.
ALGEBRA UNIT 11-GRAPHING QUADRATICS THE GRAPH OF A QUADRATIC FUNCTION describe it as a max or min pt.
### Quiz & Worksheet Finding Minimum & Maximum Values
Linear and Quadratic Functions Graphing Zeros and Min/Max.
### Applications of Quadratic Functions
Goals:
1. To find an extreme value of a quadratic function in an applied context.
2. To find the equation of a quadratic function that fits given data points.
Ex. The profit for producing x Snickers bars is, where P(x) is in dollars. Use the vertex.
Minimum and maximum value of a function if the objective function is not a quadratic function, Applications Mechanics. Problems in MAX/MIN FOR FUNCTIONS OF SEVERAL VARIABLES Abstract. These notes supplement the material in В§8.2 of our text on quadratic forms and symmetric matrices.
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
Finding the Range of Quadratic Functions . {2a}.\) Whether this is a maximum or minimum depends on the sign of \ Applications of Completing the Square.
Note: When you're dealing with quadratic functions, maximum and minimum are very likely to come up. This tutorial takes a look at the maximum of a quadratic function.
Watch video · Comparing maximum points of quadratic functions. Finding features of quadratic functions. for this particular one, we're going to hit a minimum …
Minimum and maximum values are found in parabolas that open up or down. In this lesson, learn how to find the minimum and maximum values...
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A quadratic function can be used to through the application of quadratic functions. coordinates of the turning point and whether it is a maximum or a minimum. a
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A bivariate quadratic function is a second-degree polynomial of the form the function has no maximum or minimum; its graph forms an hyperbolic paraboloid.
Finding the Maximum or Minimum of a Quadratic. 3 Finding the Maximum or Minimum of a Quadratic What we can do is we can get equations that are an application.
ALGEBRA UNIT 11-GRAPHING QUADRATICS THE GRAPH OF A QUADRATIC FUNCTION describe it as a max or min pt.
### Quiz & Worksheet Finding Minimum & Maximum Values
Linear and Quadratic Functions Graphing Zeros and Min/Max.
### Applications of Quadratic Functions
Goals:
1. To find an extreme value of a quadratic function in an applied context.
2. To find the equation of a quadratic function that fits given data points.
Ex. The profit for producing x Snickers bars is, where P(x) is in dollars. Use the vertex.
Minimum and maximum value of a function if the objective function is not a quadratic function, Applications Mechanics. Problems in MAX/MIN FOR FUNCTIONS OF SEVERAL VARIABLES Abstract. These notes supplement the material in В§8.2 of our text on quadratic forms and symmetric matrices.
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
Finding the Range of Quadratic Functions . {2a}.\) Whether this is a maximum or minimum depends on the sign of \ Applications of Completing the Square.
Note: When you're dealing with quadratic functions, maximum and minimum are very likely to come up. This tutorial takes a look at the maximum of a quadratic function.
Watch video · Comparing maximum points of quadratic functions. Finding features of quadratic functions. for this particular one, we're going to hit a minimum …
Minimum and maximum values are found in parabolas that open up or down. In this lesson, learn how to find the minimum and maximum values...
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A quadratic function can be used to through the application of quadratic functions. coordinates of the turning point and whether it is a maximum or a minimum. a
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A bivariate quadratic function is a second-degree polynomial of the form the function has no maximum or minimum; its graph forms an hyperbolic paraboloid.
Finding the Maximum or Minimum of a Quadratic. 3 Finding the Maximum or Minimum of a Quadratic What we can do is we can get equations that are an application.
ALGEBRA UNIT 11-GRAPHING QUADRATICS THE GRAPH OF A QUADRATIC FUNCTION describe it as a max or min pt.
### Quiz & Worksheet Finding Minimum & Maximum Values
Linear and Quadratic Functions Graphing Zeros and Min/Max.
### Applications of Quadratic Functions
Goals:
1. To find an extreme value of a quadratic function in an applied context.
2. To find the equation of a quadratic function that fits given data points.
Ex. The profit for producing x Snickers bars is, where P(x) is in dollars. Use the vertex.
Minimum and maximum value of a function if the objective function is not a quadratic function, Applications Mechanics. Problems in MAX/MIN FOR FUNCTIONS OF SEVERAL VARIABLES Abstract. These notes supplement the material in В§8.2 of our text on quadratic forms and symmetric matrices.
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
Finding the Range of Quadratic Functions . {2a}.\) Whether this is a maximum or minimum depends on the sign of \ Applications of Completing the Square.
Note: When you're dealing with quadratic functions, maximum and minimum are very likely to come up. This tutorial takes a look at the maximum of a quadratic function.
Watch video · Comparing maximum points of quadratic functions. Finding features of quadratic functions. for this particular one, we're going to hit a minimum …
Minimum and maximum values are found in parabolas that open up or down. In this lesson, learn how to find the minimum and maximum values...
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A quadratic function can be used to through the application of quadratic functions. coordinates of the turning point and whether it is a maximum or a minimum. a
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
how to find the min and max for quadratic equation using c# ?? f Use of min and max functions in C++. 224. Web Applications; Ask Ubuntu;
A bivariate quadratic function is a second-degree polynomial of the form the function has no maximum or minimum; its graph forms an hyperbolic paraboloid.
Finding the Maximum or Minimum of a Quadratic. 3 Finding the Maximum or Minimum of a Quadratic What we can do is we can get equations that are an application.
ALGEBRA UNIT 11-GRAPHING QUADRATICS THE GRAPH OF A QUADRATIC FUNCTION describe it as a max or min pt.
### Quiz & Worksheet Finding Minimum & Maximum Values
Linear and Quadratic Functions Graphing Zeros and Min/Max.
### Applications of Quadratic Functions
Goals:
1. To find an extreme value of a quadratic function in an applied context.
2. To find the equation of a quadratic function that fits given data points.
Ex. The profit for producing x Snickers bars is, where P(x) is in dollars. Use the vertex.
Minimum and maximum value of a function if the objective function is not a quadratic function, Applications Mechanics. Problems in MAX/MIN FOR FUNCTIONS OF SEVERAL VARIABLES Abstract. These notes supplement the material in В§8.2 of our text on quadratic forms and symmetric matrices.
You can use the TI-83 Plus graghing calculator to find the maimum and minimum points on a graph, which has many useful applications. For example, the maximum point on a graph can represent the maximum value of a quadratic function, which is useful in optimization problems.
this is called the vertex form of the quadratic function. max/min problems resulting in quadratic functions. On this exercise, you will not key in your answer.
Finding the Range of Quadratic Functions . {2a}.\) Whether this is a maximum or minimum depends on the sign of \ Applications of Completing the Square.
Note: When you're dealing with quadratic functions, maximum and
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|
match-initiative.org
|
en
| 0.831436
| 2023-03-28T05:08:44
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https://match-initiative.org/elgin/max-min-applications-of-quadratic-functions.php
| 0.999946
|
### JEE Mains Mathematics Complex Numbers Questions Bank
**JEE Complex Numbers Questions No: 1**
The locus of the center of a circle which touches the circle |z – z_{1}| = a and |z – z_{2}| = b externally is a **Hyperbola**.
**JEE Main Complex Numbers Questions No: 2**
If |z – 4| < |z – 2|, its solution is given by **Re(z) > 3**.
**JEE Mathematics Complex Numbers Questions No: 3**
z and ???? are two non-zero complex numbers such that |z| = |????| and arg(z) + arg(????) = π, then z equals to **– ????**.
**JEE Main Math Complex Numbers Questions No: 4**
If [ (1 + i) / (1 – i) ]^{x} = 1, then **x = 4n, where n is any positive integer**.
**Complex Numbers JEE Questions No: 5**
If z and ???? are two non-zero complex numbers such that |z????| = 1 and arg(z) – are(????) = π/2, then z???? is equal to **– i**.
**Complex Numbers JEE Main Questions No: 6**
Let z_{1} and z_{2} be two roots of the equation z^{2} + az + b = 0, z being complex, Further, assume that the origin z_{1} and z_{2} form an equilateral triangle, then **a^{2} = 3b**.
**Complex Numbers Questions for JEE Mains Questions No: 7**
If |z^{2} – 1| = |z|^{2} + 1, then z lies on the **imaginary** axis.
**Complex Numbers Questions and Answers No: 8**
If z = x – iy and z^{1/3} = p + iq, then (x/p + y/q) / (p^{2} + q^{2}) is equal to **-2**.
**Complex Numbers Questions with Solutions No: 9**
Let z, ω be complex numbers such that z + iω = 0 and arg(zω) = π, then arg(z) equals **π/2**.
**JEE Mains Complex Numbers Questions No: 10**
If z_{1} and z_{2} are two non-zero complex numbers such that |z_{1} + z_{2}| = |z_{1}| + |z_{2}|, then arg(z_{1}) – arg(z_{2}) is equal to **0**.
**JEE Mains Math Complex Numbers Questions No: 11**
If ω = z / (z – i/3) and |ω| = 1, then z lies on a **straight line**.
**Complex Numbers Questions for JEE Questions No: 12**
If the cube roots of unity are 1, ω, ω^{2}, then the roots of the equation (x – 1)^{3} + 8 = 0 are **-1, 1 – 2ω, 1- 2ω^{2}**.
**Complex Numbers Problems for Class 11 Questions No: 13**
If (z + 1/z)^{2} + (z^{2} + 1/z^{2})2 + (z^{3} + 1/z^{3}) + … + (z^{6} + 1/z^{6})^{2} is **12**.
**Complex Numbers Problems for Class 12 Questions No: 14**
The value of is **-i**.
**Complex Numbers Problems for Class 11 JEE Questions No: 15**
If |z + 4| ≤ 3, then the maximum value of |z + 1| is **6**.
**Complex Numbers Problems for Class 12 JEE Questions No: 16**
The conjugate of a complex number is 1 / (i – 1), then the complex number is **-1 / (i + 1)**.
**JEE Complex Numbers Questions No: 17**
If |z – 4/z| = 2, then the maximum value of |z| is equal to **√5 + 1**.
**JEE Main Complex Numbers Questions No: 18**
The number of complex numbers z such that |z – 1| = |z + 1| = |z – 1| equals **1**.
**JEE Mathematics Complex Numbers Questions No: 19**
If ω ≠ 1 is a cube root of unity and ( 1 + ω)^{7} = A + Bω, then **(A, B) equals (1, 1)**.
**JEE Main Math Complex Numbers Questions No: 20**
If z ≠ 1 and z^{2} / (z – 1) is real, then the point represented by the complex number z lies **either on the real axis or on a circle passing through the origin**.
**Complex Numbers JEE Main Questions No: 21**
|z_{1} + z_{2}|^{2} + |z_{1} – z_{2}|^{2} equals **2(|z_{1}|^{2} + |z_{2}|^{2})**.
**Complex Numbers Questions for JEE Mains Questions No: 22**
If z is a complex number of unit modulus and argument θ, then arg(1 + z)/(1 + z) equals **– θ**.
**Complex Numbers Questions and Answers No: 23**
Let a = Im(1 + z^{2}) / (2iz), where z is any non-zero complex number, then the set A = {a : |z| = 1 and z = ±1} is equal to **(-1, 1)**.
**Complex Numbers Questions with Solutions No: 24**
If z is a complex number such that |z| ≥ 2, then the minimum value of |z + 1/2| **lies in interval (1, 2)**.
**JEE Mains Complex Numbers Questions No: 25**
Let ω (Im ω ≠ 0) be a complex number, then the set of all complex number z satisfying the equation ω – ωz = k(1 – z), for some real number k, is **{z : |z| = 1, z ≠ 1}**.
**JEE Mains Math Complex Numbers Questions No: 26**
If z_{1}, z_{2} and z_{3}, z_{4} are pairs of complex conjugate numbers, then arg(z_{1}/z_{4}) + arg(z_{2}/z_{3}) equals **0**.
**Complex Numbers Questions for JEE Questions No: 27**
Let z ≠ -i be any complex number such that (z – i) / (z + i) is a purely imaginary number, then z + 1/z is **any non-zero real number**.
**Complex Numbers Problems for Class 11 Questions No: 28**
For all complex numbers z of the form 1 + iα, α ∈ R, if z^{2} = x + iy, then **y^{2} + 4x – 4 = 0**.
**Complex Numbers Problems for Class 12 Questions No: 29**
A complex number z is said to be unimodular if |z| = 1, then the set of all complex numbers z is a **circle of radius 2**.
**Complex Numbers Problems for Class 11 JEE Questions No: 30**
If z is non- real complex number, then the minimum value of Im(z^{5}) / (Im(z))^{5} is **-4**.
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amansmathsblogs.com
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en
| 0.720887
| 2024-09-13T09:14:04
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https://www.amansmathsblogs.com/jee-main-math-previous-year-paper-complex-numbers-questions-answer-keys-solutions/
| 0.996534
|
To find the zeros of a polynomial P(x), set P(x) equal to zero and solve for x.
**Example 1:**
p(x) = 2x + 3
Solution:
2x + 3 = 0
Subtract 3 from both sides: 2x = -3
Divide both sides by 2: x = -³⁄₂
**Example 2:**
p(x) = 4x - 1
Solution:
4x - 1 = 0
Add 1 to both sides: 4x = 1
Divide both sides by 4: x = ¼
**Example 3:**
p(x) = x² + 17x + 60
Solution:
x² + 17x + 60 = 0
Factor the quadratic equation: (x + 12)(x + 5) = 0
x + 12 = 0 or x + 5 = 0
x = -12 or x = -5
**Example 4:**
p(x) = x² - 4x + 1
Solution:
x² - 4x + 1 = 0
Use the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a
Substitute a = 1, b = -4, and c = 1: x = (4 ± √(16 - 4)) / 2
x = (4 ± √12) / 2
x = (4 ± 2√3) / 2
x = 2 ± √3
**Example 5:**
p(x) = x³ - 5x² - 4x + 20
Solution:
x³ - 5x² - 4x + 20 = 0
Factor the cubic equation: (x² - 4)(x - 5) = 0
(x + 2)(x - 2)(x - 5) = 0
x + 2 = 0 or x - 2 = 0 or x - 5 = 0
x = -2 or x = 2 or x = 5
**Example 6:**
p(x) = 4x³ - 7x + 3
Solution:
4x³ - 7x + 3 = 0
Use synthetic division to find one zero: x = 1
Divide the polynomial by (x - 1): 4x² + 4x - 3 = 0
Factor the quadratic equation: (2x - 1)(2x + 3) = 0
2x - 1 = 0 or 2x + 3 = 0
x = ½ or x = -³⁄₂
The zeros of the polynomial are -³⁄₂, ½, and 1.
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onlinemath4all.com
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en
| 0.786192
| 2024-09-10T06:35:59
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https://www.onlinemath4all.com/zeros-of-a-polynomial.html
| 1.000005
|
Definition 13.31.1. Let $\mathcal{A}$ be an abelian category. A complex $I^\bullet $ is *K-injective* if for every acyclic complex $M^\bullet $ we have $\mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{A})}(M^\bullet , I^\bullet ) = 0$.
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columbia.edu
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en
| 0.807624
| 2023-03-26T15:39:54
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https://stacks.math.columbia.edu/tag/070H
| 0.98661
|
## Introduction
The concept of computation has evolved significantly since its inception in the mid-20th century. Initially, it referred to simple arithmetic operations, but as modern computing progressed, the complexity and vocabulary surrounding computation increased. Reasoning about computation problems is crucial due to their far-reaching implications.
## Clique Problem
The Clique Problem is a classic example of a complex computation problem. It involves finding a group of people who all know each other within a larger network. While finding a clique of size 3 in a small group is straightforward, increasing the group size makes it significantly more challenging. This is because each person is connected to multiple others, and each check requires additional checks.
## P and NP
In complexity theory, problems are classified into two categories: P and NP. P problems can be solved in polynomial time, meaning the solution can be found quickly. NP problems, on the other hand, can be verified in polynomial time, but finding the solution is challenging. The Clique Problem is an example of an NP problem, as verifying a clique is easier than finding one.
To distinguish between P and NP problems:
* P = easy to find a solution (in polynomial time)
* NP = easy to verify a solution is valid (in polynomial time)
## Measuring Complexity
Complexity theory focuses on the number of steps required to solve a problem, rather than the time taken. This is because computational power doubles every two years, according to Moore's Law. Measuring complexity helps reason about the inherent difficulty of a problem, regardless of the machine.
The bubble sort algorithm is an example of a simple sorting algorithm. While it appears easy, it requires multiple steps, including reading the list, comparing elements, and swapping them. The worst-case scenario for a list of 5 numbers could take 25 steps, resulting in a complexity of O(n²).
In contrast, checking if a number exists in a list takes at most 5 steps, resulting in a complexity of O(n). Reading the first number in a list always takes 1 step, resulting in a complexity of O(1). These examples illustrate polynomial time algorithms.
Exponential time algorithms, on the other hand, have an exponent based on the size of the input. For example, O(2^n) is an exponential time algorithm.
## Deterministic vs Non-Deterministic
Deterministic machines always produce the same output given the same input and are predictable. Non-deterministic machines, which do not exist in reality, can produce different outputs given the same input and are unpredictable.
## Revisiting P and NP
With the understanding of polynomial time, exponential time, deterministic, and non-deterministic machines, we can refine the definition of P and NP:
* P = problems that can be solved in polynomial time on a deterministic machine
* NP = problems that can be solved in polynomial time on a non-deterministic machine or have a solution verifiable in polynomial time
Since non-deterministic machines do not exist, NP problems are challenging to solve. The best approach is to use algorithms that take exponential time, which becomes impractical quickly.
## Complexity Classes
The complexity classes include P, NP, NP-hard, and NP-complete. NP-hard problems are those where even verifying the solution takes exponential time. NP-complete problems are those where excluding the wrong answer is easy, but finding the right one is challenging.
The relationship between these classes is:
* P ⊆ NP
* NP-hard problems are at least as hard as NP problems
* NP-complete problems are both NP-hard and in NP
Solving an NP problem in polynomial time on a deterministic machine would have significant implications, including breaking encryption and rendering Bitcoin useless. A $1,000,000 prize is offered to anyone who can solve an NP problem in polynomial time.
## Conclusion
This introduction to complexity theory provides a foundation for exploring more advanced topics, such as consensus algorithms and proof verification. Complexity theory is a fascinating field that underlies many aspects of computer science. For further learning, resources are available on Loopring's social media channels and GitHub repository.
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|
loopring.org
|
en
| 0.929224
| 2020-01-18T20:55:12
|
https://blogs.loopring.org/learning-cryptography-part-4-complexity-theory/
| 0.962072
|
The concept of replacing expression constructs with equations defining components of conventional compound structures was initially explored. This idea was later refined, and a new equational programming paradigm was discovered. The paradigm allows for a purely equational approach to object-oriented programming (OO) and can be used in conjunction with other declarative languages like Haskell.
**Valof Clauses**
The VALOF construct from the BCPL language was adapted to an equational approach. In BCPL, a VALOF was like a begin-end block, but with a command RESULTIS to return a value. The equational version replaces commands with an unordered set of equations, including one for the special variable *result*. For example:
valof
disc = b*b - 4*a*c;
root = (-b + sqrt(disc))/(2*a);
result = if disc<0 then ERR else root fi;
end
**Output Parameters**
The idea of output parameters was introduced, allowing for the return of results. For example, an equivalent of *if-then-else-fi* without nested structure can be achieved using output parameters:
valof
test = n<1;
choicet = 1;
choicef = n*(n-1);
result = choice;
end
The variable *result* is special and cannot be renamed. The equation `choice = if test then choicet else choicef fi` is automatically included.
**A Simple Modification**
A simple solution to the problem of tied-up identifiers was found by prefixing parameter names with the names of the construct in question. For example:
valof
if.test = n < 1;
if.true = 1;
if.false = n*(n-1);
result = if.result
end
This looks like the invocation of an OO object called "if", where the equations defining the input parameters are setting properties, and the expression *if.result* is calling a method *result*.
**Schemes**
The concept of schemes was introduced, which are sets of equations that can be used to define objects or constructs. For example, a scheme for solving a quadratic equation can be defined as:
scheme quad
disc = b*b - 4*a*c;
droot = sqrt(disc);
$root1 = (-b + droot)/2/a
$root2 = (-b - droot)/2/a
end
The fact that *$root1* and *$root2* have definitions means they are output parameters.
**Function Parameters**
Both input and output parameters can be functions. For example:
scheme integrate
m = ($a+$b)/2
$area = ($b-$a)($f($a)+4$f(m)+$f($b))
This scheme can be used as follows:
valof
import integrate
integrate.a = 1;
integrate.b = 2;
integrate.f(x) = 1/x;
log2 = integrate.area;
end
**Enter PYFL**
A new language, PYFL, was specified and implemented as a platform for demonstrating parameters. PYFL is a super simple equational language that can be extended with interesting features. However, the implementation of parameters in PYFL is not yet complete, and user-defined schemes are still complex and haven't been done yet.
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billwadge.com
|
en
| 0.884756
| 2023-04-02T09:29:16
|
https://billwadge.com/2021/08/01/parametric-programming-an-equational-approach-to-oo-and-beyond/
| 0.622441
|
Class for building and using a multinomial logistic regression model with a ridge estimator. The parameter matrix B to be calculated will be an m*(k-1) matrix, where there are k classes for n instances with m attributes. The probability for class j, with the exception of the last class, is Pj(Xi) = exp(XiBj)/((sum[j=1..(k-1)]exp(Xi*Bj))+1). The last class has probability 1-(sum[j=1..(k-1)]Pj(Xi)) = 1/((sum[j=1..(k-1)]exp(Xi*Bj))+1).
The (negative) multinomial log-likelihood is L = -sum[i=1..n]{ sum[j=1..(k-1)](Yij * ln(Pj(Xi))) +(1 - (sum[j=1..(k-1)]Yij)) * ln(1 - sum[j=1..(k-1)]Pj(Xi)) } + ridge * (B^2). A Quasi-Newton Method is used to search for the optimized values of the m*(k-1) variables. Before optimization, the matrix B is 'squeezed' into a m*(k-1) vector.
The algorithm is modified to handle instance weights, unlike the original Logistic Regression. For details, see le Cessie and van Houwelingen (1992). Missing values are replaced using a ReplaceMissingValuesFilter, and nominal attributes are transformed into numeric attributes using a NominalToBinaryFilter.
Key parameters include:
- Class column: the column containing the target variable
- Preliminary Attribute Check: tests the classifier against the DataTable specification
- Classifier Options:
- D: turn on debugging output
- R: set the ridge in the log-likelihood
- M: set the maximum number of iterations (default -1, until convergence)
Capabilities include handling nominal, binary, unary, empty nominal, numeric, date attributes, missing values, nominal class, binary class, and missing class values. The minimum number of instances required is 1.
|
CC-MAIN-2024-38/segments/1725700651196.36/warc/CC-MAIN-20240910025651-20240910055651-00120.warc.gz
|
nodepit.com
|
en
| 0.762706
| 2024-09-10T03:25:28
|
https://nodepit.com/node/org.knime.ext.weka366.classifier.WekaClassifierNodeFactory%23Logistic%20(3.6)%20(legacy)
| 0.566007
|
**Definitions:**
* **Kilopound Foot** (kips.ft, kips ft, kip ft, kilopound.foot): the moment of force in kilopounds times the distance in feet between reference and application points.
* **Ton Meter** (t.m, tm, ton.meter): the moment of force in tons times the distance in meters between reference and application points.
## Converting Kilopound Feet to Ton Meters
To convert kilopound feet to ton meters, use the conversion factor: 1 kilopound foot = 0.13825728 ton meters.
**Example:** Convert 98.81 kilopound feet to ton meters.
98.81 kilopound feet = Y ton meters
Using the conversion factor: Y = 98.81 * 0.13825728 = 13.6612018368 ton meters
Therefore, 13.6612018368 ton meters are equivalent to 98.81 kilopound feet.
**Practice Questions:** Convert the following units to ton meters (t.m):
1. 78.62 kips.ft
2. 73.94 kips.ft
3. 95.83 kips.ft
To find the answers, apply the conversion factor: 1 kilopound foot = 0.13825728 ton meters.
|
CC-MAIN-2021-25/segments/1623488559139.95/warc/CC-MAIN-20210624202437-20210624232437-00069.warc.gz
|
infoapper.com
|
en
| 0.757225
| 2021-06-24T22:47:54
|
https://www.infoapper.com/unit-converter/bending-moment/kips-ft-to-t.m/
| 0.828124
|
## Parameters for Modeling Exponents and Products
These parameters are used to model cases involving exponents, such as `degree()` or `spline_degree()`, and products, like `prod_degree`.
## Arguments
- `range`: A two-element vector specifying the default smallest and largest possible values. If a transformation is applied, values should be in transformed units.
- `trans`: A `trans` object from the `scales` package (e.g., `scales::transform_log10()` or `scales::transform_reciprocal()`). If not provided, the default matches the units used in `range`. If no transformation is applied, it is `NULL`.
## Details
- `degree()` is suitable for parameters that are real number exponents (e.g., `x^degree`).
- `degree_int()` is used for cases where the exponent should be an integer.
- The key difference between `degree_int()` and `spline_degree()` lies in their default ranges, which depend on the context of their usage.
- `prod_degree()` is utilized by `parsnip::mars()` to determine the number of terms in interactions and generates an integer.
|
CC-MAIN-2024-38/segments/1725700651995.50/warc/CC-MAIN-20240919061514-20240919091514-00155.warc.gz
|
tidymodels.org
|
en
| 0.691619
| 2024-09-19T08:15:03
|
https://dials.tidymodels.org/reference/degree.html
| 0.868461
|
The "next greater permutation of digits" problem involves finding the next higher number that uses the same set of digits as a given number. For example, given the number 38276, the next higher number that uses the digits 2, 3, 6, 7, and 8 is 38627.
The C++ function `std::next_permutation` returns the lexicographically next greater permutation of elements. This function can be used as a reference to understand how the algorithm works. The C++ version of the function is as follows:
```cpp
template<typename Iter>
bool next_permutation(Iter first, Iter last)
{
if (first == last)
return false;
Iter i = first;
++i;
if (i == last)
return false;
i = last;
--i;
for(;;)
{
Iter ii = i;
--i;
if (*i < *ii)
{
Iter j = last;
while (!(*i < *--j))
{}
std::iter_swap(i, j);
std::reverse(ii, last);
return true;
}
if (i == first)
{
std::reverse(first, last);
return false;
}
}
}
```
To understand the algorithm, consider the example of finding the next permutation of the number 8342666411. The longest monotonically decreasing tail is 666411, and the corresponding head is 8342. The tail is reverse-ordered and cannot be increased by permuting its elements. To find the next permutation, the head must be increased by finding the smallest tail element larger than the head's final 2.
Walking back from the end of the tail, the first element greater than 2 is 4. Swapping the 2 and the 4 results in 8344 666211. Since the head has increased, the permutation is now greater. To reduce it to the next permutation, the tail is reversed, putting it into increasing order, resulting in 8344 112666. Joining the head and tail back together gives the permutation one greater than 8342666411, which is 8344112666.
The implementation of the next permutation algorithm involves several steps:
1. Finding the "cut point" which is the index to the left of the longest monotonic tail.
2. Finding the smallest element larger than the element at the "cut point" (searching from the end of the sequence).
3. Reversing the tail of the sequence.
The code for these steps is as follows:
```factor
: cut-point ( seq -- n )
[ last ] keep [ [ > ] keep swap ] find-last drop nip ;
: greater-from-last ( n seq -- i )
[ nip ] [ nth ] 2bi [ > ] curry find-last drop ;
: reverse-tail! ( n seq -- seq )
[ swap 1 + tail-slice reverse! drop ] keep ;
: (next-permutation) ( seq -- seq )
dup cut-point [
swap [ greater-from-last ] 2keep
[ exchange ] [ reverse-tail! nip ] 3bi
] [ reverse! ] if* ;
: next-permutation ( seq -- seq )
dup [ ] [ drop (next-permutation) ] if-empty ;
```
The `next-permutation` function can be tested using unit tests, such as:
```factor
[ "" ] [ "" next-permutation ] unit-test
[ "1" ] [ "1" next-permutation ] unit-test
[ "21" ] [ "12" next-permutation ] unit-test
[ "8344112666" ] [ "8342666411" next-permutation ] unit-test
[ "ABC" "ACB" "BAC" "BCA" "CAB" "CBA" "ABC" ]
[ "ABC" 6 [ dup >string next-permutation ] times ] unit-test
```
Solving the original programming challenge is as easy as:
```factor
IN: scratchpad 38276 number>string next-permutation string>number .
38627
```
|
CC-MAIN-2020-05/segments/1579251700988.64/warc/CC-MAIN-20200127143516-20200127173516-00174.warc.gz
|
blogspot.com
|
en
| 0.841025
| 2020-01-27T14:39:21
|
http://re-factor.blogspot.com/2012/03/
| 0.976126
|
**Introduction to Big-O Notation**
As a programmer, it's essential to understand the cost associated with your code. Big-O notation is a way to represent the performance or cost of an algorithm, specifically in terms of time and memory. In this article, we'll simplify the complex notations and help you learn the basics.
**Why Big-O Notation?**
Every line of code or algorithm has a performance or cost associated with it. We're interested in two types of costs: time and memory. Time complexity refers to how much time an algorithm takes to execute, while memory complexity refers to how much memory it requires.
When discussing time complexity, we consider three scenarios:
* **Best case**: The minimum amount of time an algorithm needs to execute.
* **Worst case**: The maximum amount of time an algorithm needs to execute.
* **Average case**: The average amount of time an algorithm needs to execute.
**What is Big-O Notation?**
Big-O notation is a function or equation that describes how much resource (time or memory) an algorithm needs to execute. It's expressed as a function of the length of its input. Big-O notation provides a way to compare the performance of different algorithms, regardless of the machine or speed.
**Samples of Big-O Notation**
Here are some common examples of Big-O notation:
* **O(1) - Constant time**: The time it takes to access an element in an array is constant, regardless of the array's size.
* **O(n) - Linear time**: The time it takes to print all elements in an array is linear, increasing with the size of the array.
* **O(n^2) - Quadratic time**: The time it takes to sort an array using a nested loop is quadratic, increasing exponentially with the size of the array.
* **O(n log n) - Logarithmic time**: The time it takes to search for an element in a sorted array using binary search is logarithmic, increasing slowly with the size of the array.
**Calculating Big-O Notation**
To calculate the Big-O notation of a given code, consider the following:
* **Constant time**: Accessing an element in an array takes constant time, O(1).
* **Linear time**: Printing all elements in an array takes linear time, O(n).
* **Quadratic time**: Sorting an array using a nested loop takes quadratic time, O(n^2).
* **Logarithmic time**: Searching for an element in a sorted array using binary search takes logarithmic time, O(log n).
**Space Complexity**
In addition to time complexity, we also consider space complexity, which refers to the amount of memory an algorithm requires. In most cases, the space complexity is O(1), meaning the algorithm uses a constant amount of space. However, some algorithms may require more space, such as O(n) or O(n^2).
**Key Takeaways**
* Big-O notation represents the cost associated with an algorithm, specifically time and memory.
* It's an abstract mathematical representation that allows us to compare the performance of different algorithms.
* Understanding Big-O notation helps you choose the best algorithm for a given problem.
* Look at the levels of nesting loops in your code to guess the complexity.
* Space-time trade-off is an essential constraint in choosing an algorithm.
**Examples and Exercises**
* Accessing an element in an array: O(1)
* Printing all elements in an array: O(n)
* Sorting an array using a nested loop: O(n^2)
* Searching for an element in a sorted array using binary search: O(log n)
By understanding Big-O notation, you'll be able to write more efficient code and make informed decisions when choosing algorithms for your projects.
|
CC-MAIN-2024-38/segments/1725700651098.19/warc/CC-MAIN-20240909103148-20240909133148-00772.warc.gz
|
codenlearn.com
|
en
| 0.860689
| 2024-09-09T10:47:24
|
http://www.codenlearn.com/2011/07/understanding-algorithm-complexity.html
| 0.964584
|
In base twelve, the number 15_{12} is equivalent to 17_{10} in base ten, as it represents 12_{10} + 5_{10}. To understand this, it's essential to break down the base twelve number into its decimal equivalent: 15_{12} = 10_{12} + 5_{12} = 12_{10} + 5_{10} = 17_{10}.
### Adding and Other Sums
The method for adding numbers in base twelve is similar to base ten, with the exception that each column has twelve digits instead of ten. The traditional school techniques for addition, such as carrying over, remain the same. For example:
15_{12} | |
---|---|
1 | o o o o o o o o o o o o |
5 | o o o o o |
### Division
To become familiar with base twelve operations, practice the following additions, subtractions, multiplications, and divisions:
- Addition
- 4 + 5 = 9
- 4 + 6 = A
- 4 + 7 = B
- 4 + 8 = 10
- 15 + 7 = 20
- Subtraction
- 10 - 3 = 9
- 12 - 3 = B
- 17 - 12 = 5
- 20 - 15 = 7
- 12 - 8 = 6
- 32 - 18 = 16
- Multiplication
- 2 x 3 = 6
- 2 x 4 = 8
- 3 x 3 = 9
- 3 x 4 = 10
- 3 x 5 = 13
- 4 x 5 = 18
- 5 x 5 = 21
- Division
- 10 / 2 = 6
- 10 / 3 = 4
- 10 / 6 = 2
- 20 / 6 = 4
- 1 / 2 = 0.6 (think of a half on a clock face)
- 1 / 3 = 0.4 (a third on a clock face)
- 1 / 4 = 0.3
- 1 / 6 = 0.2
Division in base twelve can be challenging, but using visual aids like a clock face can help establish relationships between numbers. It's essential to practice and think in base twelve to become comfortable with these operations.
|
CC-MAIN-2016-50/segments/1480698543170.25/warc/CC-MAIN-20161202170903-00040-ip-10-31-129-80.ec2.internal.warc.gz
|
base12.org
|
en
| 0.942466
| 2016-12-10T12:38:04
|
http://base12.org/basic_arithmetic.html
| 0.745325
|
There is a boat in a pool with a lead ball inside. If the ball is thrown into the water, will the water level rise or fall?
When the ball is in the boat, it displaces a volume of water equal to its weight. For example, if the ball's volume is 1 ml, its mass is 11.3 grams, so it displaces 11.3 ml of water, causing the water level to rise. However, when the ball is in the water, it displaces water according to its volume only, which is 1 ml, resulting in the water level falling.
The key concept here is the difference in displacement when the ball is in the boat versus when it is in the water. The ball's weight determines the displacement when it is in the boat, but its volume determines the displacement when it is in the water.
Given this information, the answer to the question is that the water level will fall when the ball is thrown into the water.
|
CC-MAIN-2023-14/segments/1679296945183.40/warc/CC-MAIN-20230323194025-20230323224025-00460.warc.gz
|
thinkingames.com
|
en
| 0.832963
| 2023-03-23T21:37:41
|
https://thinkingames.com/en/riddles-logic/4547/
| 0.441439
|
The perimeter of a triangle is the sum of its three side lengths. For an equilateral triangle with equal side lengths, the perimeter can be calculated by multiplying the length of one side by 3. For example, if the perimeter of an equilateral triangle is 15cm, each side length is 15cm / 3 = 5cm. Similarly, if the perimeter is 12cm, each side length is 12cm / 3 = 4cm.
The Pythagorean theorem can be used to find the perimeter of a right triangle. The theorem states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. For instance, if the sides of a right triangle are 3cm, 4cm, and 5cm, the perimeter can be calculated by adding these side lengths together: 3cm + 4cm + 5cm = 12cm.
Heron's formula is another method for finding the perimeter and area of a triangle. The formula uses the side lengths of the triangle (a, b, and c) to calculate the semi-perimeter (s) and then the area. The perimeter can be calculated by adding the side lengths together: P = a + b + c.
An equilateral triangle has three equal sides, and its perimeter can be calculated by multiplying the length of one side by 3. The area of an equilateral triangle can be calculated using the formula: Area = (√3 / 4) * side^2.
An isosceles triangle has two equal sides, and its perimeter can be calculated by adding the lengths of all three sides. The Pythagorean theorem can be used to find the length of the third side if the lengths of the other two sides are known.
A scalene triangle has three unequal sides, and its perimeter can be calculated by adding the lengths of all three sides. The Law of Cosines can be used to find the length of one side if the lengths of the other two sides and the angle between them are known.
To calculate the perimeter of any triangle, it is essential to know the lengths of its sides. The perimeter can be calculated by adding the lengths of all three sides, regardless of the type of triangle. The formulas mentioned above, such as the Pythagorean theorem and Heron's formula, can be used to find the perimeter and area of different types of triangles.
In summary, the perimeter of a triangle is the sum of its three side lengths, and it can be calculated using various formulas, such as the Pythagorean theorem and Heron's formula, depending on the type of triangle and the information available. The perimeter of an equilateral triangle is 3 times the length of one side, while the perimeter of an isosceles triangle is the sum of the lengths of all three sides. The perimeter of a scalene triangle is also the sum of the lengths of all three sides, and it can be calculated using the Law of Cosines.
|
CC-MAIN-2023-14/segments/1679296943695.23/warc/CC-MAIN-20230321095704-20230321125704-00084.warc.gz
|
edudoorways.com
|
en
| 0.905005
| 2023-03-21T11:59:56
|
https://edudoorways.com/how-to-find-the-perimeter-of-a-triangle/
| 0.99996
|
One layer of tinting material on a window cuts out 1/5 of the sun's UV rays.
There are two questions to consider:
(a) What fraction would be cut out by using two layers?
(b) How many layers would be required to cut out at least 9/10 of the sun's UV rays?
|
CC-MAIN-2017-51/segments/1512948539745.30/warc/CC-MAIN-20171214035620-20171214055620-00604.warc.gz
|
mathhelpforum.com
|
en
| 0.860116
| 2017-12-14T04:44:57
|
http://mathhelpforum.com/algebra/117510-fractions.html
| 0.425756
|
## Math Foundations
The following are key concepts and formulas in math:
* **Distance Formula**: √((y₂-y₁)² + (x₂-x₁)²)
* **Midpoint Formula**: ((x₂+x₁)/2), ((y₂+y₁)/2)
* **Slope Formula**: (Y₂−Y₁)÷(X₂−X₁)
* **Parallel Lines**: have the same slope
* **Perpendicular Lines**: have opposite reciprocal slope
* **Point Slope Form**: y - y1 = m(x - x1), requires 1 point and slope
* **Circle Equation**: (x−h)²+(y−k)²=r², where (h,k) is the center
## Special Right Triangles and Basic Concepts
* **30-60-90 and 45-45-90**: special right triangles
* **LCM (Lowest Common Multiple)**: the lowest multiple two numbers share, e.g., 6 and 8 have an LCM of 24
* **GCF (Greatest Common Factor)**: the largest factor two numbers share, e.g., 18 and 24 have a GCF of 6
* **Mean**: another word for average
* **Mode**: the number that appears most often in a data set
* **Median**: the middle number in an arranged set
## Geometry Formulas
* **Area of Trapezoid**: A = ((a+b)/2)h
* **Area of Parallelogram**: A = bxH
* **Arc Length**: S = rθ
* **Trapezoid Angles**: non-congruent angles are complementary
* **Sum of Interior Angles of a Polygon**: 180 x (n - 2)
## Advanced Concepts
* **Logarithms**: used for independent and dependent probability
* **Compound Interest**: compounded continuously
* **Arithmetic Sequences**: used to find term and sum
* **Geometric Sequence**: used to find term and sum
* **Volume of Cylinder**: πr²h
* **Absolute Value Inequalities**: used to solve inequalities
* **Law of Sines** and **Law of Cosines**: used for trigonometry
* **Heron's Formula**: used for finding area of triangles
## Formulas and Theorems
* **Number of Diagonals in a Polygon**: n(n-3)÷2
* **Difference of Cubes**: a formula for factoring
* **Undefined Slope** and **Zero Slope**: types of slopes
* **Chord** and **Central Angles and Arcs**: concepts in geometry
* **Similar Triangles**: used for proportions
* **Geometric Sequence Sum**: used to find sum of sequences
* **Rhombus**: a type of quadrilateral
* **Volume of Pyramid** and **Volume of Cone**: formulas for finding volume
## Graphs
* **Sin Graph** and **Cosine Graph**: types of trigonometric graphs
|
CC-MAIN-2021-25/segments/1623488289268.76/warc/CC-MAIN-20210621181810-20210621211810-00411.warc.gz
|
quizlet.com
|
en
| 0.744288
| 2021-06-21T20:00:37
|
https://quizlet.com/184093659/act-math-flash-cards/
| 0.999894
|
# Geometrics
**Geometrics** is an Android app that creates geometric patterns using polar graphing. The app was developed by Kyle Fischer and is categorized under Entertainment.
## Description
The app creates Maurer rose geometric patterns, which are controlled by three parameters:
- Swiping left and right to increase or decrease the number of petals
- Swiping up and down to increase or decrease the degree skip
- Pressing and holding to cycle between different modes
## Mathematical Overview
A polar graph represents mathematical equations, where the radius of the circle is denoted by "r" and the angle traversed is denoted by "θ" (Theta). A polar rose curve has the equation r(θ) = sin(kθ), where increasing "k" adds to the number of petals.
In the Geometrics app, swiping right increases "k" by 1, adding petals to the rose. Swiping left decreases the number of petals until it reaches 1, which is the starting point circle. Further swiping to the left results in fractional values, creating beautiful but not always symmetrical roses.
Maurer roses, introduced by Peter M. Maurer, involve skipping around the figure instead of drawing it sequentially. The app starts with a skip number around 75 and increases or decreases it by 1 when swiping up or down. Certain degree skips may produce uninteresting patterns.
The app also features a size reduction effect, where the rose gets slightly smaller for every line drawn, producing amazing visual effects. The radius can become negative, resulting in asymmetrical patterns flipping. The radius is limited by the larger of the screen's width and height.
## Modes and Features
Pressing and holding cycles between three modes:
- A changeable Maurer rose connecting points to the previous point
- A changeable Maurer rose drawing lines to an unchanging Maurer rose
- A changeable Maurer rose drawing lines to the same changeable Maurer rose but out of phase
The app also includes a color fading algorithm that attempts to reach every possible color and fade between them seamlessly.
## Geometrics for Android
The latest available version of Geometrics for Android is **Geometrics 4**, with a file size of 843 KB. There is only one version of Geometrics for Android available on this site, with more details and versions available on the Geometrics for Android portal.
## Categories
The app is categorized under Entertainment, with other categories including Games, Finance, Health, and more.
|
CC-MAIN-2023-14/segments/1679296946584.94/warc/CC-MAIN-20230326235016-20230327025016-00086.warc.gz
|
freewarelovers.com
|
en
| 0.884793
| 2023-03-27T00:11:11
|
https://www.freewarelovers.com/apps/geometrics
| 0.602262
|
## Elearn Geometry Problem 4
This problem involves a triangle and a quadrilateral, focusing on angles and congruence. The goal is to find the value of angle x in terms of angle α.
Given:
- AD = DC = BC = 1
- Triangle BDC is isosceles
### Solution 1
1. Since triangle BDC is isosceles, angle DBC = angle BDC = (180 - 2α)/2 = 90 - α.
2. Using the law of sines in triangle BDC, we find BD = 2 sin(α).
3. In triangle ABD, applying the law of sines gives sin(ABD) = 1/2, hence angle ABD = 30°.
4. Therefore, angle x = angle DBC + angle ABD = 90 - α + 30 = 120 - α.
### Solution 2
1. Let point E be the reflection of point D over AB.
2. Triangles AEB and ADB are congruent, making angle EAB = angle DAB = α.
3. Since AE = AD = DC = BC, triangles AED and CBD are congruent, implying ED = DB.
4. Because BE = DB (from congruent triangles AEB and ADB), triangle EBD is equilateral, and angle EBD = 60°.
5. Thus, angle ABD = 30°, and since angle DBC = 90 - α, x = 30 + (90 - α) = 120 - α.
### Solution 3
1. Let the bisector of angle DCB intersect AB at X.
2. Since AD = CD and angle XCA = angle XAD = α, triangles AXD and XCD are congruent by SAS, making angle DXC = angle DXA.
3. As X is on the perpendicular bisector of BD, angle DXC = angle BXC, and since these angles are supplementary, angle BXC = 60°.
4. Hence, angle XBD = 30°, and angle XBC = angle XBD + angle DBC = 30 + 90 - α = 120 - α.
### Solution 4
1. Join AC and draw a line through D perpendicular to AC, meeting AC at M and AB at E.
2. Since AD = DC, EM is the perpendicular bisector of AC, making AE = EC and angle AEM = angle CEM.
3. Angle EAD = angle ECD = α, and since DC = CB, angle DCE = angle BCE = α.
4. Triangles DCE and BCE are congruent, implying angle CEM = angle CEB = 60°.
5. In triangle BCE, x + α = 120°.
### Solution 5
1. Let E be the circumcenter of triangle ABD.
2. Angle DEB = 2α, and triangles DEB and DCB are congruent.
3. EA = ED = DC = DA, making triangle EAD equilateral and angle AED = 60°.
4. Thus, angle ABD = (1/2)angle AED = 30°, and since angle DBC = 90 - α, x = 30 + (90 - α) = 120 - α.
### Solution 6
1. Join A,C and B,D. Draw an angle bisector of angle DCB, intersecting DB at F and AB at E.
2. In triangle DBC, since CD = CB, angle DCF = angle FCB = α.
3. Angle DFC = angle BFC = 90, and angle EFD = angle EFB = 90.
4. Triangles DFC and BFC are congruent, making FD = FB.
5. Triangles DEF and EFB are congruent, implying angle BEF = angle FED.
6. In triangle ADC, AD = DC, making it isosceles, and angle DAC = angle DCA.
7. Angle EAC = angle ECA, making triangle AEC isosceles, and AE = EC.
8. Triangles AED and CED are congruent, so angle AED = angle FED.
9. From the congruence, angle BEF = angle FED = angle AED = 60°, and angle EBF = 90 - 60 = 30°.
10. Angle ABC = angle EBF + angle FBC = 30 + 90 - α = 120 - α.
### Solution 7
1. Draw an angle bisector of angle BCD, meeting AB at E, and connect E and D.
2. Since AD = DC, angle ACD = angle CAD, and angle ECD = angle EAD = α.
3. CE = AE (from isosceles triangle AEC), and triangles BCE and ADE are congruent.
4. This implies x = angle CBE = angle ADE.
5. Triangles CDE and ADE are congruent, making angle CDE = angle ADE = x.
6. Since triangles BCE, CDE, and ADE are congruent, angle BEC = angle CED = angle AED.
7. The sum of these angles is 180°, so 3(angle AED) = 180°, and angle AED = 60°.
8. In triangle CEA, angle CEA = 2(angle AED) = 120°, and since it's isosceles, angle ECA = angle EAC = 60°.
9. Finally, angle CDA = 360 - (angle EDC + angle EDA) = 360 - 2x, and 120 + 2α = 360 - 2x.
10. Solving for x gives x = 120 - α.
### Solution 8
1. Let the bisector of angle DCB meet AB at E and BD at F. Draw altitude DG of triangle ABD.
2. Triangles BCF and DCF are congruent and also congruent with triangle ADG, making DG = DF = FB.
3. In right triangle BGD, BD = 2DG, which means angle ABD = 30°, and since angle DBC = 90 - α, the result follows.
### Solution 9
1. Construct point E such that E lies on AB and EC is an angle bisector of angle DCB.
2. Construct line ED. Triangle DCE is congruent to BCE, implying angle EDC = x.
3. Since angle EAD = angle ECD and angle DAC = angle DCA, angle EAC = angle ECA.
4. Hence, EA = EC, and triangles EAD and ECD are congruent, making angle EDA = x.
5. By the angle sum of quadrilateral, 3x + 3α = 360°, so x + α = 120°, and x = 120 - α.
In all solutions, the value of angle x is consistently found to be 120 - α, demonstrating the robustness of the geometric principles applied.
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blogspot.com
|
en
| 0.79763
| 2017-12-13T09:26:47
|
http://gogeometry.blogspot.com/2008/05/geometry-problem-4.html
| 0.992685
|
Math 392: Vector Calculus and Linear Algebra
The course covers the following topics:
1. Introduction to the class (Lecture 0, 51:56 minutes)
2. Parametrizing curves (Lecture 1, 1:07:53 minutes)
3. Arclength and vector fields (Lecture 2, 53:02 minutes)
4. Line integrals with respect to arclength and x, y (Lecture 3, 1:07:05 minutes)
5. Line integrals over vector fields and the work integral (Lecture 4, 48:53 minutes)
6. The Fundamental Theorem for Line Integrals (Lecture 5, 59:25 minutes)
7. Green's Theorem (Lecture 6, 55:48 minutes)
8. Curl and Divergence (Lecture 7, 58:13 minutes)
9. Vector Potentials, Curl, Divergence, and their interpretations (Lecture 8, 44:08 minutes)
10. Parametric surfaces, areas, and normal vectors (Lecture 9, 1:12:26 minutes)
11. More on parametric surfaces and normal vectors (Lecture 10, 53:08 minutes)
12. Surface area and surface integrals (Lecture 11, 1:06:37 minutes)
13. Oriented surfaces and surface integrals over vector fields (Lecture 12, 53:42 minutes)
14. Surface integrals over vector fields (Lecture 13, 57:41 minutes)
15. Stokes Theorem (Lecture 14, 1:00:23 minutes)
16. The Divergence Theorem and the Fundamental Theorems of Calculus (Lecture 15, 56:39 minutes)
17. Matrices and matrix algebra (Lecture 16, 1:05:12 minutes)
18. Matrix multiplication, properties, and linear systems of equations (Lecture 17, 1:03:03 minutes)
19. Augmenting systems and echelon forms (Lecture 18, 1:09:10 minutes)
20. Solving systems and the variety of solutions (Lecture 19, 49:38 minutes)
21. Determinants and pivotal condensation (Lecture 20, 1:11:29 minutes)
22. More determinant theorems, Cramer's rule, and finding inverses (Lecture 21, 1:06:53 minutes)
23. Eigenvalues and eigenvectors, solving systems of linear ODEs (Lecture 22, 58:19 minutes)
24. Final exam information, application of eigenvalues, eigenvectors, and line integrals (Lecture 23, 38:25 minutes)
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wn.com
|
en
| 0.687104
| 2023-03-29T18:52:45
|
https://education.wn.com/math-392-vector-calculus-and-linear-algebra/
| 0.996965
|
**404** equals 2 times 101, which is the product of a prime and the square of a different prime. The numbers 199989^394, 199989^404, and 199989^464 begin with the digits 394, **404**, and 464 respectively. **404** can be represented as a sum of two squares: **404** = 20^2 + 2^2. Additionally, **404** is the sum of seven positive fifth powers: **404** = 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 2^5 + 3^5. Notably, **404** is also the HTTP status code for "file not found," a widely recognized code.
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blogspot.com
|
en
| 0.921115
| 2020-01-25T16:22:37
|
http://maanumberaday.blogspot.com/2009/05/
| 0.94337
|
The sign of the horizontal component of a child's displacement depends on the direction of the horizontal (x) axis. By directing the x-axis in the same direction as the child is sliding down the hill, the horizontal component of the child's displacement will be positive.
To find the magnitude of the horizontal component, consider the right triangle formed by the horizontal line, the surface of the hill, and the vertical line. The angle between the horizontal and vertical lines is 90 degrees, and the surface of the hill is the hypotenuse. The angle between the horizontal and the hill is 27.2 degrees. The cosine of this angle is the ratio of the lengths of the adjacent (horizontal) side and the hypotenuse:
cos(27.2) = (adj)/(hyp)
This gives Adj = (hyp)*cos(27.2). The length of the adjacent side is the magnitude of the horizontal component, and the length of the hypotenuse is the length of the hill: 20.1 m. Therefore, the horizontal component is
20.1*cos(27.2) = 17.9 meters, rounded to the nearest tenth of a meter.
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enotes.com
|
en
| 0.85275
| 2021-06-20T04:53:13
|
https://www.enotes.com/homework-help/child-rides-toboggan-down-hill-that-descends-an-538186
| 0.956809
|
### Depreciation Methods in Excel
There are nine methods to calculate depreciation in Excel. Each method has its own formula and steps to follow.
### Method 1 – Simple Depreciation Formula
1. Select cell **D5** and write the formula: `=($D$15-$D$16)*B5/$D$17`
2. Press **Enter** to see the cumulative depreciation for year 1.
3. Double-click the bottom right corner of cell **D5** to get the cumulative depreciation for years 2-7.
4. The accumulated depreciation is shown in cells **D11** and **D13**.
### Method 2 – Units of Production Formula
1. Select cell **D5** and write the formula: `=($D$15-$D$16)*C5/$D$18`
2. Press **Enter** to see the depreciation for the 1st period.
3. Drag the bottom right corner of cell **D5** to cell **D11** to get the results for all 7 years.
4. Use the formula `=SUM(D5:D11)` in cell **D13** to get the accumulated depreciation.
### Method 3 – SYD Function
1. Select cell **D5** and write the formula: `=SYD($D$15,$D$16,$D$17,B5)`
2. Press **Enter** to see the depreciation in cell **D5**.
3. Double-click the bottom right corner of cell **D5** to get the depreciation values for all data.
4. Use the formula `=SUM(D5:D11)` in cell **D13** to get the accumulated depreciation.
### Method 4 – Straight-Line Method
1. Select cell **D5** and write the formula: `=SLN($D$15,$D$16,$D$17)`
2. Press **Enter** to see the result in cell **D5**.
3. Double-click the bottom right corner of cell **D5** to get the depreciation values for periods 2-7.
4. Use the **SUM** function to determine the accumulated depreciation.
### Method 5 – VDB Function
1. Select cell **D5** and write the formula:
2. Press **Enter** to find the depreciation of the 1st year.
3. Double-click the bottom right corner of cell **D5** to get the depreciation for the whole period.
4. Use the formula `=SUM(D5:D11)` to calculate the accumulated depreciation.
### Method 6 – DB Function
1. Select cell **D5** and write the formula: `=DB($D$15,$D$16,$D$17,B5)`
2. Press **Enter** to get the result in cell **D5**.
3. Double-click the bottom right corner of cell **D5** to show the results for all 7 years.
4. Use the formula `=SUM(D5:D11)` to get the accumulated depreciation.
### Method 7 – DDB Function
1. Select cell **D5** and write the formula: `=DDB($D$15,$D$16,$D$17,B5)`
2. Press **Enter** to get the result of the 1st period.
3. Double-click the bottom right corner of cell **D5** to find the depreciation over the full 7 years.
4. Use the formula `=SUM(D5:D11)` to get the accumulated depreciation.
### Method 8 – AMORLINC Function
1. Select cell **D5** and write the formula: `=AMORLINC($D$16,$D$20,$D$21,$D$17,B5,$D$22)`
2. Press **Enter** to find the result in cell **D5**.
3. Double-click the bottom right corner of cell **D5** to get the results for the whole period.
4. Use the formula `=SUM(D5:D12)` to find the accumulated depreciation.
### Method 9 – AMORDEGRC Function
1. Select cell **D5** and write the formula: `=AMORDEGRC($D$16,$D$20,$D$21,$D$17,B5,$D$22)`
2. Press **Enter** to get the result for the 1st year.
3. Double-click the bottom right corner of cell **D5** to get all the depreciation values.
4. Use the **SUM** function to find the accumulated depreciation.
## Key Points to Remember
* Use proper cell references when using formulas.
* Use the absolute reference sign ($) for formulas to avoid erroneous results.
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exceldemy.com
|
en
| 0.801584
| 2024-09-11T07:28:13
|
https://www.exceldemy.com/how-to-calculate-accumulated-depreciation-in-excel/
| 0.868578
|
The hydraulic system of a backhoe is used to lift a load. To calculate the force $F_s$ the slave cylinder must exert to support the 400-kg load and the 150-kg brace and shovel, we use the principle of torque. The counter-clockwise torque exerted by the slave cylinder is equal to the clockwise torque due to the weight of the arm and the load.
The counter-clockwise torque is given by $F_s \times l_s$, where $l_s$ is the lever arm of the slave cylinder. The clockwise torque due to the weight of the arm is $m_{arm} \times g \times \sin(30) \times l_{arm}$, and the clockwise torque due to the load is $m_{load} \times g \times \sin(30) \times l_{load}$.
Setting the counter-clockwise torque equal to the total clockwise torque, we get $F_s \times l_s = m_{arm} \times g \times \sin(30) \times l_{arm} + m_{load} \times g \times \sin(30) \times l_{load}$. Solving for $F_s$, we get $F_s = \frac{m_{arm} \times g \times \sin(30) \times l_{arm} + m_{load} \times g \times \sin(30) \times l_{load}}{l_s}$.
Plugging in the values, $m_{arm} = 150$ kg, $m_{load} = 400$ kg, $l_{arm} = 1.1$ m, $l_{load} = 1.7$ m, $l_s = 0.3$ m, and $g = 9.81$ m/s^2, we get $F_s = 1.38 \times 10^4$ N.
To find the pressure in the hydraulic fluid, we use the formula $P = \frac{F_s}{A}$, where $A$ is the area of the slave cylinder. Given that the diameter of the slave cylinder is 2.50 cm, we can calculate the area as $A = \pi \times \left(\frac{d}{2}\right)^2$. Plugging in the values, we get $P = \frac{4 \times F_s}{\pi \times d^2} = 2.81 \times 10^7$ Pa.
To find the force that would have to be exerted on a lever with a mechanical advantage of 5.00 acting on a master cylinder 0.800 cm in diameter, we use the principle of Pascal's law, which states that the pressure in the master cylinder is equal to the pressure in the slave cylinder. We can set up the equation $\frac{F_m}{A_m} = \frac{F_s}{A_s}$, where $F_m$ is the force exerted by the master cylinder, $A_m$ is the area of the master cylinder, $F_s$ is the force exerted by the slave cylinder, and $A_s$ is the area of the slave cylinder.
Given that the mechanical advantage is 5.00, we can write $F_m = 5 \times F_{in}$, where $F_{in}$ is the force applied to the lever. Substituting this into the equation, we get $\frac{5 \times F_{in}}{A_m} = \frac{F_s}{A_s}$. Solving for $F_{in}$, we get $F_{in} = \frac{F_s \times A_m}{5 \times A_s} = \frac{F_s \times d_m^2}{5 \times d_s^2}$. Plugging in the values, we get $F_{in} = 283$ N.
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collegephysicsanswers.com
|
en
| 0.929007
| 2020-01-22T04:34:33
|
https://collegephysicsanswers.com/openstax-solutions/hydraulic-system-backhoe-used-lift-load-shown-figure-1148-calculate-force-fs
| 0.939293
|
At the heart of every computer are binary numbers, formed only of 0's and 1's. This is because it is easy to represent just two digits electronically: a switch can be on or off, a current can flow one way or the other, or a pulse can change a stored digit from one state to the other.
The binary system works by counting up to 1 in the "ones" column, then moving to the "twos" column, the "fours" column, and so on. For example, 5 in binary is 101, representing a four, no twos, and a one. Although binary numbers are simple electronically, they get long quickly. Usually, they are stored in multiples of eight: 8-bit numbers have a maximum value of 255. This is enough to encode all common letters and symbols as binary numbers, making a "byte" (an eight-bit number) the standard unit of memory size in a computer.
The standard ASCII character set uses seven bits, allowing room for expansion. The first bit of a number is often used to show positive or negative, so an 8-bit number can represent 0 to 255 or -127 to +127.
Octal and hexadecimal numbers are used to save space because multiples of 8 bits are universally used. The number 92 in octal is 134, and in hexadecimal is 5C, where C represents 12 (A = 10, B = 11, and so on). Thus, 5C is five 16's and twelve 1's.
A noteworthy feature of the binary table is that every time you move from one row to the next, any column changing from a 1 to a 0 forces the next column to the left to change its state. This can be demonstrated with a human analogue of a computer counter by lining up four people, side-to-side, with right arm in the air representing 1 and by the side representing 0. A fifth person represents the computer clock and regularly taps the 1's person on the shoulder to change state. As the arms go up and down, the human counter will count through from 0 to 15.
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oxfordstudycourses.com
|
en
| 0.867908
| 2023-03-22T00:22:14
|
https://oxfordstudycourses.com/blog/human-binary-counter
| 0.819783
|
The use of "x" to represent a variable and the little box to represent an unknown value in elementary school can be confusing. Since both symbols have the same meaning, it would be more efficient to use "x" from the beginning.
A possible argument against this is that "x" is used to represent multiplication, which could cause confusion. However, using "x" for multiplication is problematic because students have to unlearn it later in algebra. In algebra, it is more common to write (7)(5) = 35 instead of 7×5 = 35, as "x" is used as a variable.
If students were not initially taught that "x" means "multiply", it would reduce confusion when they take algebra. The letter "x" already has multiple uses, including as a letter of the alphabet, a variable, and a marker. It does not need to also represent multiplication.
As teachers, the goal should be to minimize confusion, not create it. The current practices of using "x" for multiplication and a little box for unknown values can cause unnecessary confusion. It is essential to reevaluate these methods to provide a clearer understanding of mathematical concepts for students.
As a mathematics teacher with no experience at the elementary level, these issues may seem obvious, but they highlight the need for a more streamlined approach to teaching mathematical symbols and concepts.
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robertlovespi.net
|
en
| 0.781839
| 2020-01-28T20:13:25
|
https://robertlovespi.net/tag/algebra/
| 0.51182
|
## Introduction to the 2012 Mathematics Game
The 2012 Mathematics Game is a challenge to write mathematical expressions for the counting numbers 1 through 100 using the digits in the year 2012. The goal is to use all four digits (2, 0, 1, 2) and basic mathematical operations to create as many numbers as possible.
## Rules of the Game
- Use all four digits: 2, 0, 1, 2.
- Allowed operations: +, -, x, ÷, sqrt (square root), ^ (raise to a power), ! (factorial), and parentheses.
- Create multi-digit numbers or decimals using the digits 2, 0, 1, 2.
- Prefer solutions without multi-digit numbers.
## Bonus Rules
- Use the overhead-bar (vinculum) to mark repeating decimals.
- Use multifactorials: n!! (double factorial) and n!!! (triple factorial).
## How to Play
Experiment with different mathematical expressions to create numbers from 1 to 100. Share your findings in the comments section, but do not post your solutions. The game relies on collective wisdom to decide when a number is confirmed.
## Keeping Score
A running tally of confirmed results will be kept. The goal is to confirm as many numbers as possible.
## Current Progress
- Percent confirmed: 97%.
- Reported but not confirmed: 77, 92.
- Numbers still missing: 93.
- Middle school rules: 68% confirmed.
- Old Math Forum rules: 1-32, 34-44, 48-52, 58-65, 70, 72, 74, 80, 90, 94-95, 97-100 confirmed.
- New Math Forum rules: 77% confirmed.
## Clarifying the Do’s and Don’ts
- Allowed: . (decimal point), unary negatives, nested factorials, and multifactorials.
- Not allowed: "0!" as a digit, decimal point as an operation, and exponent unless created from the digits 2, 0, 1, 2.
## Helpful Links
- 2012 Mathematics Game Worksheet
- 2012 Mathematics Game Manipulatives
- 2012 Mathematics Game Student Submissions
## Join the Game
Play the game in your own mind and on scratch paper. Ask questions and share your findings in the comments section. The game is solitaire, but collective wisdom will help confirm numbers.
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CC-MAIN-2020-05/segments/1579251689924.62/warc/CC-MAIN-20200126135207-20200126165207-00347.warc.gz
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denisegaskins.com
|
en
| 0.912291
| 2020-01-26T16:03:11
|
https://denisegaskins.com/2012/01/01/2012-mathematics-game/?replytocom=34349
| 0.939558
|
To solve the quadratic equation 6x^2 - x - 12 = 0 by factorizing, we need to find two numbers whose product is -12 and whose sum is -1 (the coefficient of the x term).
The factors of 6 are 1, 2, 3, and 6, and the factors of -12 are -12, -1, -6, -2, -4, -3, 1, 2, 3, 4, 6, and 12. We can try different pairs of factors to see which ones satisfy the equation.
We can factor the equation in the form (ax + b)(cx + d) = 0. To do this, we need to find the correct pairing of factors.
For the factor 6, we have the pairs 1 and 6, 2 and 3.
For the factor -12, we have the pairs -12 and 1, -1 and 12, -6 and 2, -2 and 6, -4 and 3, -3 and 4.
We choose the pair 2 and 3 for the factor 6, and the pair -3 and 4 for the factor -12. Now we have two possible factorizations:
A) (2x + 4)(3x - 3)
B) (2x - 3)(3x + 4)
To determine which one is correct, we can check the middle term of each factorization.
For A) the middle term is 2x(-3) + 4(3x) = -6x + 12x = 6x, which is not correct.
For B) the middle term is 2x(4) + (-3)(3x) = 8x - 9x = -x, which is correct.
Therefore, the correct factorization is 6x^2 - x - 12 = (2x - 3)(3x + 4).
This method of factorization is based on the rational roots theorem, which states that if a quadratic equation has any rational roots, they are of the form of the ratio of a factor of the constant term to a factor of the coefficient of the x^2 term.
The correct factorization can be determined by analyzing the factors of the constant term and the coefficient of the x^2 term, and checking the middle term of each possible factorization.
The final factorization is 6x^2 - x - 12 = (2x - 3)(3x + 4).
Which of the following is the correct factorization of the equation 6x^2 - x - 12 = 0?
A) (2x + 4)(3x - 3)
B) (2x - 3)(3x + 4)
The correct answer is B) (2x - 3)(3x + 4).
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mathhelpforum.com
|
en
| 0.910854
| 2017-12-12T16:40:29
|
http://mathhelpforum.com/algebra/105373-quadratic-equation-print.html
| 0.991745
|
**GCSE Maths- Number App**
The GCSE Maths- Number app is a comprehensive tool for GCSE preparation, featuring 760 high-quality questions written by experienced GCSE tutors. The app is organized into six topics: Number Basics, Factors, Multiplication and Division, Fractions, Decimals and Percentages, Powers, Roots and Surds, and Ratio, Speed, Proportion and Variation.
**Key Features:**
* Revise by topic: Basics, factors, multiplication, division, decimals, fractions, percentages, powers, roots and surds, ratios, speed, proportion and variation, approximation and rounding
* Mock test: Mixed questions from all topics
* Instant feedback: Know instantly if your answer was right or wrong
* Review with explanation: Review each question at the end of the test with detailed explanations
* Progress meter: Track your progress with pie charts and bar graphs
**Topic Details:**
1. **Number Basics**:
* Number line
* Ordering numbers and place values
* Negative numbers
* Column addition and subtraction
* Addition and subtraction with negative numbers
* Order of operations and BODMAS
* Number problems in words
2. **Factors, Multiplication and Division**:
* Times tables
* Multiplying and dividing by single-digit numbers
* Prime numbers
* Prime factors
* Factors
* Highest common factor
* Multiples
* Lowest common multiple
* Long multiplication and division
* Multiplying and dividing with negative numbers and decimals
* Real-life problems using multiplication and division
3. **Fractions, Decimals and Percentages**:
* Fractions of a shape
* Finding a fraction of a quantity
* Improper fractions and mixed numbers
* Equivalent fractions
* Converting fractions to decimals and vice versa
* Adding, subtracting, multiplying, and dividing fractions
* Decimals: addition, subtraction, multiplication, and division
* Equivalent percentages, fractions, and decimals
* Calculating percentages and percentage change
* Compound interest and reverse percentage
4. **Powers, Roots and Surds**:
* Square numbers and powers of 10
* Multiplying and dividing using powers of 10
* Square roots and cube roots
* Multiplying and dividing powers
* Calculated powers and power of a power
* Fractional and zero powers
* Negative powers and standard form
* Surds and pi
* Rationalizing the denominator and solving problems with surds
5. **Ratio, Speed, Proportion and Variation**:
* Dividing amounts in ratios
* Ratios and fractions
* Calculating with ratios
* Ratio problems in words
* Best buys and speed
* Density and direct proportion
* Inverse proportion
6. **Approximation and Rounding**:
* Significant figures and approximation of calculations
* Limits of accuracy and calculating with limits
* Decimal places and rounding
The app's unique progress tracking feature, including pie charts and bar graphs, helps you track your progress and ensures you're ready for the real test when your progress meter reaches 100%.
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CC-MAIN-2019-04/segments/1547583804001.73/warc/CC-MAIN-20190121172846-20190121194846-00046.warc.gz
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gcseexams.co.uk
|
en
| 0.729076
| 2019-01-21T18:54:01
|
http://www.gcseexams.co.uk/portfolio/numbers-2/
| 0.999992
|
**Chapter 4: Experimental Design**
**4.1 Introduction**
The tribological behavior of metal matrix composites was investigated using the Taguchi method and Grey Relation Analysis (GRA). The influence of control factors and optimal combination of testing parameters were determined. Analysis of variance (ANOVA) was employed to determine the most significant control factors and their interactions.
**4.2 Taguchi Method**
The Taguchi method uses a special design of orthogonal arrays to study the entire parameter space with a small number of experiments. This technique consists of an experimental plan to obtain information about the behavior of a process. The treatment of experimental results is based on ANOVA. The experimental plan was set by a technique based on the Taguchi techniques, considering different variables at different levels, such as load, sliding speed, sliding distance, percentage of reinforcement, and particle size of the reinforcement.
**4.3 Analysis of Variance**
ANOVA is a method of partitioning variability into identifiable sources of variation and the associated degrees of freedom in an experiment. The F-test is a ratio of sample variances. Comparing the F-ratio of a source with the tabulated F-ratio is called the F-test. When ANOVA has been performed on a set of data, it is possible to use this information to distribute the correct sums of squares to the appropriate factors. The percent contribution of each factor can be calculated by comparing the sum of squares of each factor to the total sum of squares.
**4.4 Grey Relational Analysis (GRA)**
GRA is a method used to optimize multiple performance characteristics, such as wear rate, specific wear rate, and coefficient of friction. Taguchi method coupled with GRA was used to solve the multiple performance characteristics in the tribological area. Grey theory works on unascertained but partially known as well as unknown information by drawing out variable information by producing and developing the partially known information.
**4.4.1 Data Pre-processing**
Data pre-processing means transforming the original sequence into a comparable sequence. During data pre-processing, the experimental results (wear rate, specific wear rate, and coefficient of friction) are normalized in the range between 0 and 1.
**4.4.2 Grey Relational Coefficient and Grey Relational Grade**
The grey relational coefficient for the performance characteristics in the experiment can be calculated after pre-processing. The grey relational grade represents the level of correlation between the reference sequence and the comparability sequence.
**4.5 Methodology for Studying Dry Sliding Wear Behavior of AMMCs**
The selection of independent variables for dry sliding wear of composites can be attempted based on an understanding of the process and available literature. Three independent variables, load, sliding speed, and sliding distance, were considered to influence the magnitude of dry sliding. The experimental plan consisted of 9 tests, and the chosen array was the L9, with 9 rows in agreement to the number of tests.
**4.6 Methodology for Studying Dry Sliding Wear Behavior of Hybrid MMCs**
The following parameters are considered for wear performances of hybrid MMCs: applied load, sliding speed, and sliding distance. The experimental plan consisted of 27 tests, and the chosen array was the L27, with 27 rows in agreement to the number of tests.
**4.8 Summary**
Wear processes in composites are complex phenomena involving a number of process parameters. Selecting the correct operating conditions is always a major concern. An approach based on design of experiments (DOE) technique was adopted to obtain maximum possible information with a minimum number of experiments. Grey relational analysis was used to obtain optimum conditions for wear testing, and ANOVA was used to obtain the significant parameters influencing the wear behavior of metal matrix composites.
**Table 4.1: Design Factors and Their Levels for Dry Sliding Wear of Aluminium MMCs**
| Level | Factors | Applied Load (N) | Sliding Speed (m/s) | Sliding Distance (m) |
| --- | --- | --- | --- | --- |
| 1 | | 10 | 2 | 1000 |
| 2 | | 20 | 3 | 1750 |
| 3 | | 30 | 4 | 2500 |
**Table 4.2: Dry Sliding Wear Test Parameters**
| Parameters | Values |
| --- | --- |
| Applied Load | 10, 20, 30 |
| Sliding Distance | Up to 2000 m |
| Sliding Speed | 2 m/s |
| Disk Speed | 700-800 rpm |
| Test Duration | 20-25 min |
| Temperature | Room Temperature |
| Surrounding Atmosphere | Laboratory Air |
**Table 4.3: Experimental Layout of L9 Orthogonal Array**
| Expt. No. | Factors | L | S | D |
| --- | --- | --- | --- | --- |
| 1 | | 1 | 1 | 1 |
| 2 | | 1 | 2 | 2 |
| 3 | | 1 | 3 | 3 |
| 4 | | 2 | 1 | 2 |
| 5 | | 2 | 2 | 3 |
| 6 | | 2 | 3 | 1 |
| 7 | | 3 | 1 | 3 |
| 8 | | 3 | 2 | 1 |
| 9 | | 3 | 3 | 2 |
**Table 4.4: Design Factors and Their Levels for Dry Sliding Wear of Aluminium Hybrid MMCs**
| Level | Factors | Applied Load (N) | Sliding Speed (m/s) | Sliding Distance (m) |
| --- | --- | --- | --- | --- |
| 1 | | 25 | 2.0 | 1000 |
| 2 | | 30 | 2.25 | 1500 |
| 3 | | 35 | 2.50 | 2000 |
**Table 4.5: Standard L27 Orthogonal Array**
| Expt. No. | Factors | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| 1 | | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 2 | | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| 3 | | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
| ... | | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... |
| 27 | | 3 | 3 | 2 | 1 | 3 | 2 | 1 | 2 | 1 | 3 | 1 | 3 | 2 |
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CC-MAIN-2023-14/segments/1679296945323.37/warc/CC-MAIN-20230325095252-20230325125252-00212.warc.gz
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wallaceandjames.com
|
en
| 0.855153
| 2023-03-25T11:33:05
|
https://wallaceandjames.com/chapter-4-anova-was-first-described-by-sir-ronald/
| 0.429382
|
# Introducing the New LET Function in Excel
## Introduction
The LET function is a new feature in Excel, currently available in beta versions, which allows users to define variables in formulas. This function is inspired by programming languages, where variables are used to store information that is referenced multiple times throughout a script. The LET function enables users to write more efficient and readable formulas by reducing repetition.
The syntax of the LET function is: `LET(name1, name_value1, calculation_or_name2…)`, where `name1` is the name of the first assigned value, `name_value1` is the value assigned to `name1`, and `calculation_or_name2` can be either a calculation or another name. Users can define up to 126 names, but the formula must always end with a calculation.
## Before We Start
To use the LET function, users need to have a copy of Excel that is part of Microsoft 365 and be an Office Insider. To become an Office Insider, go to **File** > **Account** and change the channel to **Current Channel (Preview)**. Download the example workbook, LET-Function-Example.xlsx, and the League-Table-Examples.xlsx workbook from a previous post.
## Using the LET Function
The LET function can be applied to a table, such as Table C3, which was featured in the league table workbook. The formula starts with the LET function and specifies all the column names and their associated values. For example:
`=LET(
TEAM, UNIQUE(DataTable[team_home]),
P, (COUNTIFS(DataTable[team_home], TEAM, DataTable[played], 1) + COUNTIFS(DataTable[team_away], TEAM, DataTable[played], 1)),
W, (COUNTIFS(DataTable[team_home], TEAM, DataTable[Result], "H") + COUNTIFS(DataTable[team_away], TEAM, DataTable[Result], "A")),
D, (COUNTIFS(DataTable[team_home], TEAM, DataTable[Result], "D") + COUNTIFS(DataTable[team_away], TEAM, DataTable[Result], "D")),
L, COUNTIFS(DataTable[team_home], TEAM, DataTable[Result], "A") + COUNTIFS(DataTable[team_away], TEAM, DataTable[Result], "H"),
F, SUMIFS(DataTable[home_goal], DataTable[team_home], TEAM) + SUMIFS(DataTable[away_goal], DataTable[team_away], TEAM),
A, SUMIFS(DataTable[away_goal], DataTable[team_home], TEAM) + SUMIFS(DataTable[home_goal], DataTable[team_away], TEAM),
GD, F - A,
PTS, (W * 3) + (D * 1),
SORT(SWITCH({"TEAM", "P", "W", "D", "L", "F", "A", "GD", "PTS"}, "TEAM", TEAM, "P", P, "W", W, "D", D, "L", L, "F", F, "A", A, "GD", GD, "PTS", PTS), {9, 8, 6, 1}, {-1, -1, -1, 1})
)`
This formula defines the names first, making the second part of the formula easier to read and write. The actual formula structure mirrors what was found in Table C3, but instead of repeating parts of the formula, it refers to the names and makes sure they output in the correct column.
Two additional formulas were used: an array constant for the table headings, `={"POS", "TEAM", "P", "W", "D", "L", "F", "A", "GD", "PTS"}`, and a formula for the position numbers, `=SEQUENCE(COUNTA(D17:D36))`. These formulas allow users to create a table with 210 cells using just three formulas.
## Final Words
The LET function makes formulas more convenient and readable by reducing repetition. It can also improve calculation speed when working with large datasets. By combining the LET function with other functions, users can achieve more with less. As users explore the possibilities of the LET function, they will discover new ways to use it and harness its power to create more efficient and effective formulas.
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CC-MAIN-2024-38/segments/1725700651164.37/warc/CC-MAIN-20240909233606-20240910023606-00895.warc.gz
|
andrewmoss.me
|
en
| 0.888821
| 2024-09-09T23:45:18
|
https://andrewmoss.me/introducing-the-new-let-function-in-excel/
| 0.530608
|
Dijkstra's algorithm, also known as the single-source shortest path algorithm, is used to find the shortest path between nodes in a graph where each path has a different cost. This algorithm is utilized in real-world applications, such as automatically finding directions between physical locations, like Google Maps.
## Dijkstra's Algorithm
Dijkstra's algorithm is distinct from the Breadth-First Search algorithm, which assumes that the cost of traversing each path is the same. In contrast, Dijkstra's algorithm can handle varying path costs, making it more versatile. While the Breadth-First Search algorithm follows a first-come, first-served method, Dijkstra's algorithm prioritizes the closest nodes first, resulting in a more efficient search.
## Dijkstra's Algorithm using Python
To implement Dijkstra's algorithm, we select the closest node to the source to find the shortest path. The algorithm's implementation in Python involves choosing the node with the minimum distance from the source node and iteratively updating the distances to neighboring nodes.
## Example Implementation
The example implementation of Dijkstra's algorithm in Python demonstrates how to find the shortest path between nodes in a graph. The algorithm iterates through the nodes, updating the distances and previous nodes in the shortest path.
### Summary
In summary, Dijkstra's algorithm is a powerful tool for finding the shortest path in a graph with varying path costs. Its applications are diverse, ranging from navigation systems like Google Maps to network optimization problems. By understanding and implementing Dijkstra's algorithm, developers can create more efficient and effective solutions for real-world problems.
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CC-MAIN-2023-14/segments/1679296949573.84/warc/CC-MAIN-20230331051439-20230331081439-00491.warc.gz
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thecleverprogrammer.com
|
en
| 0.873415
| 2023-03-31T07:22:50
|
https://thecleverprogrammer.com/2021/04/18/dijkstras-algorithm-using-python/
| 0.922634
|
## Problem Statement
We are given an array of n integers and need to find the maximum difference between two elements such that the larger element comes after the smaller one.
## Example
Input: 4 7 2 18 3 6 8 11 21
Output: 19
## Approach 1: Modified Kadane's Approach
This approach uses Dynamic Programming to find the maximum difference.
### Algorithm
1. Initialize the current difference as the difference between the first two elements.
2. Loop through the array, updating the current difference by adding the difference between consecutive elements if the current difference is positive.
3. Otherwise, set the current difference to the difference between the current consecutive elements.
4. Update the maximum difference if the current difference is greater.
### Implementation
C++ and Java programs are provided to demonstrate this approach.
### Complexity Analysis
- Time Complexity: O(N), where N is the size of the array.
- Space Complexity: O(1), as only a few variables are used.
## Approach 2: Brute Force Approach
This approach involves checking every pair of elements in the array.
### Algorithm
1. Loop through the array from the end to the start.
2. For each element, loop through the elements before it and check if the current element is greater.
3. If it is, update the maximum difference if the difference between the current element and the previous element is greater than the current maximum difference.
### Implementation
C++ and Java programs are provided to demonstrate this approach.
### Complexity Analysis
- Time Complexity: O(N^2), where N is the length of the array.
- Space Complexity: O(1), as only a constant amount of space is used.
## Comparison of Approaches
Approach 1 has a time complexity of O(N), making it more efficient for large arrays. Approach 2 has a time complexity of O(N^2), making it less efficient for large arrays. However, Approach 2 is simpler to understand and implement.
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tutorialcup.com
|
en
| 0.667677
| 2024-09-09T09:43:18
|
https://tutorialcup.com/interview/array/maximum-difference-between-two-elements-such-as-larger-element-comes-after-the-smaller.htm
| 0.953724
|
A right cylindrical tank with circular bases is being filled with water at a rate of 20π cubic meters per hour. As the tank is filled, the water level rises four meters per hour. What is the radius of the tank, in meters? Express your answer in simplest radical form.
To find the radius, we use the formula for the volume of a cylinder:
V = πr^2 * h.
Since the water level rises four meters per hour, we can write the equation as:
20π = πr^2 * 4.
Dividing both sides by 4π gives:
5 = r^2.
Taking the square root of both sides yields:
r = √5 meters.
The radius of the tank is √5 meters.
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0calc.com
|
en
| 0.725827
| 2019-01-21T19:04:22
|
https://web2.0calc.com/questions/plz-help-due-tomorrow
| 0.999915
|
**Compressive Stress**
When an external force is applied to a body, an internal restoring force is set up at each cross-section, attempting to restore the body to its original state. This internal restoring force per unit area is known as stress, which can be either tensile or compressive, depending on whether the stress acts out of or into the area.
There are three ways to deform a solid using an external force, and compressive stress is one of them. Compressive stress is defined as the stress that leads to a reduction in volume. To increase compressive stress, the material's compressive strength must be reached. It can also be defined as the measure of force required to break a material.
Compressive stresses are developed in components such as connecting rods, links, and columns. The application of an external load results in the development of compressive stress, as illustrated in the figure.
**Formula for Compressive Stress**
The compressive stress developed when an external compressive force is applied to an object is given by the formula:
Stress (σ) = Force (F) / Area (A)
The unit of compressive stress is N/m². A higher compressive force results in higher compressive stress, leading to the shortening of the solid. Compressive strength is a measure of the force required to break a material.
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myrank.co.in
|
en
| 0.830414
| 2023-03-29T01:19:12
|
https://blog.myrank.co.in/compressive-stress/
| 0.857205
|
In Boolean function truth tables with n variables, the table is typically arranged as x_{n-1}, x_{n-2}, ..., x_1, x_0. However, the desired order is x_0, x_1, ....
To illustrate the situation, consider the following example in Sage:
```python
from sage.crypto.boolean_function import BooleanFunction
```
Suppose we have a list L of 8 binary 3-tuples in lexicographic order, labeled as (x2, x1, x0), which is the default labeling in Sage.
```python
L = [(0,0,0), (0,0,1), (0,1,0), (0,1,1), (1,0,0), (1,0,1), (1,1,0), (1,1,1)]
B = BooleanFunction([1,1,0,0,1,0,1,1])
P = B.algebraic_normal_form()
```
The output will be P = x0*x1*x2 + x0*x2 + x1*x2 + x1 + 1. However, if the entries in L are labeled as (x0, x1, x2), the algebraic normal form for the truth table B would be P1 = x0*x1*x2 + x0*x1 + x0*x2 + x1 + 1.
The difference arises from reversing the order in which x2, x1, x0 are read. To achieve the desired labeling in Sage without modifying the implementation of BooleanFunction, one can use the following approach:
```python
P1 = P.subs(dict(zip(P.parent().gens(), reversed(P.parent().gens()))))
```
This provides a simpler solution to the problem, allowing the user to work with the desired labeling convention.
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sagemath.org
|
en
| 0.817747
| 2024-09-18T15:20:40
|
https://ask.sagemath.org/question/43825/in-boolean-function-truth-tables-in-n-variables-the-table-is-arranged-x_n-1-x_n-2x_1x_0-how-can-i-change-so-the-order-is-x_0-x_1-instead/
| 0.994382
|
# Angle between two planes
The angle between two planes is equal to the angle between their normal vectors. Alternatively, it is equal to the angle between two lines that lie on the planes and are perpendicular to the lines of intersection between the planes.
## Angle between two planes formulas
Given two plane equations, A₁x + B₁y + C₁z + D₁ = 0 and A₂x + B₂y + C₂z + D₂ = 0, the angle between the planes can be found using the formula:
cos α = |A₁·A₂ + B₁·B₂ + C₁·C₂| / √(A₁² + B₁² + C₁²)√(A₂² + B₂² + C₂²)
## Examples of tasks with angle between two planes
Example 1: Find the angle between the planes 2x + 4y - 4z - 6 = 0 and 4x + 3y + 9 = 0.
Solution: Using the formula, we get:
cos α = |2·4 + 4·3 + (-4)·0| / √(2² + 4² + (-4)²)√(4² + 3² + 0²)
= |8 + 12| / √(4 + 16 + 16)√(16 + 9 + 0)
= |20| / √36√25
= 20 / (6·5)
= 2 / 3
Answer: The cosine of the angle between the planes is cos α = 2/3.
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onlinemschool.com
|
en
| 0.827324
| 2020-01-21T11:46:35
|
https://onlinemschool.com/math/library/analytic_geometry/plane_angl/
| 0.98688
|
## Introduction to Math Problems
The coach provides two reasoning problems to prepare the math team: Nick's Birthday and Burning Ropes.
### Nick's Birthday
Nick stated: "Sometime during last year, I was still 21; in two days I'll be in my 25th year." The goal is to determine the day of the year Nick's birthday is and the day he is speaking.
### Burning Ropes
Given two ropes that burn irregularly, with the first rope burning in approximately 2.71828 hours and the second rope burning in approximately 1.414213 hours, the objective is to produce a time interval as close as possible to 1 hour using these ropes and some matches.
## Solutions to Previous Problems
### Pondering Productivity
The problem involves comparing the productivity of hens from Smith's and Jones's farms. Key findings include:
- Eight of Smith's hens lay 7 × (7/6) eggs in 6/5 days, resulting in a rate of 245/288 eggs per hen per day.
- Six of Jones's hens lay 5 × (7/6) eggs in 8/7 days, also resulting in a rate of 245/288 eggs per hen per day, demonstrating equal productivity.
- With 48 hens, Smith gets 245/6 eggs daily and must wait 6 days to get 245 eggs.
- With 300 hens, Jones gets 6125/24 eggs daily and must wait 24 days to get 6125 eggs.
### Cornfield Planning
The solution involves cutting out quarter circles on the corners of a cornfield to maximize the area-to-perimeter ratio. Key points include:
- The area of the shape is A = 10000 - 4r^2 + πr^2, and the perimeter is P = 400 - 8r + 2πr.
- The ratio A/P is maximized when r = 100/(2+√π) = 26.5079, resulting in a ratio of 100/(2+√π) = 26.5079.
- This ratio is higher than if the cornfield were a circle or a square, which would have a ratio of 25.
## Recent and Upcoming Content
Recent weeks have included:
- **Week 139**: Pondering Productivity & Cornfield Planning, with solutions to Close to a Quart & Math Party.
- **Week 138**: Close to a Quart & Math Party, with solutions to Room for One More & End Points.
- **Week 137**: Room for One More & End Points, with solutions to Sevens and Elevens & Three-Digit Squares.
- **Week 136**: Sevens and Elevens & Three-Digit Squares, with solutions to Swimming Workout & Batting Order.
- **Week 135**: Swimming Workout & Batting Order, with solutions to Maximization & Multiplication Table.
All puzzles and solutions can be found on the Complete Varsity Math page. New puzzles and answers will be available next week.
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momath.org
|
en
| 0.862962
| 2019-01-18T22:25:18
|
https://momath.org/home/varsity-math/varsity-math-week-140/
| 0.911012
|
I'm creating a mass combat system for small groups with varied individuals. To simulate multiple rolls with one, I need a way to calculate the probability of success for a given skill level and number of rolls. For example, with a skill level of 12 and 10 attacks, the average number of hits would be 7 or 8, with a 5% chance of hitting all 10.
I want to create a chart with skill levels (1-18) on one edge and number of rolls (1-8) on the other, with each cell containing the percentage of successful rolls for a given die roll and skill level. Ideally, there would be a simple formula to calculate these probabilities, using the probabilities of rolling each number on a 3d6 die.
The goal is to find a formula that can be applied to each cell in the chart, using multiplication or exponentiation to combine the probabilities. If such a formula is not possible, alternative solutions for non-mass combat rules would be helpful. The key is to find a way to easily calculate the probability of success for any combination of skill level and number of rolls.
The problem involves probability, which is not my area of expertise, but I'm looking for a solution to this problem. I'm hoping to find a way to condense multiple rolls into a single chart, making it easier to simulate mass combat scenarios.
Possible solutions could involve using statistical models or probability distributions to calculate the chances of success. The chart would need to account for various skill levels and numbers of rolls, providing a comprehensive tool for simulating mass combat.
The main challenge is finding a simple and accurate formula to calculate the probabilities for each cell in the chart. Once this formula is found, it can be applied to create a comprehensive chart for simulating mass combat scenarios.
In summary, the goal is to create a chart that can simulate multiple rolls with one, using a simple formula to calculate the probability of success for any combination of skill level and number of rolls. This would provide a valuable tool for non-mass combat rules, making it easier to simulate small group combat scenarios.
What formula or method can be used to calculate the probability of success for a given skill level and number of rolls, and how can this be applied to create a comprehensive chart for simulating mass combat?
Multiple choice questions:
A) Use a binomial distribution to calculate the probability of success.
B) Apply a statistical model to simulate the rolls.
C) Use a simple multiplication formula to combine probabilities.
D) Create a chart with fixed percentages for each skill level.
Answers:
A) Correct, a binomial distribution can be used to calculate the probability of success.
B) Possibly correct, statistical models can be used to simulate rolls, but may not provide a simple formula.
C) Incorrect, a simple multiplication formula may not accurately combine probabilities.
D) Incorrect, a chart with fixed percentages would not account for varying skill levels and numbers of rolls.
|
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|
sjgames.com
|
en
| 0.91957
| 2024-09-14T16:27:18
|
https://forums.sjgames.com/showthread.php?s=3c2c8e949348e0c4eb465e7082f49631&p=2204193&mode=threaded
| 0.747683
|
## Introduction
Optimization is a key application of calculus in real-world scenarios. To illustrate, consider a pizza parlor aiming to maximize profit or a box made from a flat piece of cardboard with the greatest volume. This process involves finding maximums and minimums, which can be achieved using derivatives, making calculus a valuable tool for optimizing situations.
## How to Solve
To solve optimization problems, follow these steps:
1. **Write out formulas and information**: Gather relevant data and formulas related to the problem.
2. **Identify variables**: Determine the variable you control and the variable you want to maximize or minimize.
3. **Combine formulas**: Express the variable to be maximized or minimized on one side of the equation and everything else on the other side.
4. **Differentiate the formula**: Differentiate the equation with respect to the variable, treating other variables as constants.
5. **Set the derivative to zero**: Set the differentiated formula equal to zero and solve for the controlled variable.
6. **Solve for the variable**: The resulting value is the answer. If the answer depends on other variables, it may be what the question asks for.
This algorithm is based on mathematical theorems, which state that a derivative of zero indicates a global or local maximum or minimum.
## Examples
### Volume Example
To maximize the volume of a box, follow these steps:
1. Write out known formulas and information.
2. Eliminate unnecessary variables from the volume equation.
3. Find the derivative of the volume equation to maximize the volume.
4. Set the derivative equal to zero and solve for the variable.
5. Plug the value back into the volume equation to simplify.
### Volume Example II
Given a box with a side length of **x - 2s** and a height of **s**, the volume function is **V(s) = (x - 2s)^2 * s**. To optimize the volume, take the derivative of **V(s)** and set it equal to zero. Since **x** is a constant, treat it as such. Solve for **s** using the quadratic formula: **s = (x ± √(x^2 - 4x^2)) / 4**. Reject the minimum value, as it results in a base length of zero, making the volume zero.
### Sales Example
A retailer sells **n** units of a product for a revenue of **2000n - 10n^2** and at a cost of **1000n + 50n^2**, with all amounts in thousands. The profit is defined by the equation **P(n) = 2000n - 10n^2 - (1000n + 50n^2)**. To maximize profit, find the derivative of **P(n)** and set it equal to zero: **P'(n) = 2000 - 20n - 1000 - 100n = 1000 - 120n**. Solve for **n**: **1000 - 120n = 0 → n = 1000 / 120**. Using the quadratic formula, find the roots: **n = {3.798, 0.869}**. Test the function to determine which value corresponds to the maximum profit. Since **n** must be positive, the maximum profit occurs at **n = 3.798** units, resulting in a profit of **$8,588.02**.
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|
wikibooks.org
|
en
| 0.845935
| 2017-12-14T12:56:43
|
https://en.m.wikibooks.org/wiki/Calculus/Optimization
| 0.994797
|
I am working on a heat transfer problem that requires solving a differential equation. The equation is given in two forms:
T^2+z*k*T=z(C1*x+C2)
or
d^2y/dx^2 + z*k*dy/dx = z(C1*x+C2)
Here, z and k are constants. The goal is to solve this differential equation in terms of y(x), where y is a function of x.
|
CC-MAIN-2021-25/segments/1623488567696.99/warc/CC-MAIN-20210625023840-20210625053840-00354.warc.gz
|
physicsforums.com
|
en
| 0.77614
| 2021-06-25T04:22:44
|
https://www.physicsforums.com/threads/need-help-solving-simple-differential-problem-help.468089/
| 0.998155
|
## GAMMA Function: Description, Usage, Syntax, Examples, and Explanation
The GAMMA function in Excel is a statistical function that returns the gamma function value. The syntax of the GAMMA function is GAMMA(number), which takes one required argument: Number. This argument must be a number and is used to calculate the gamma function value.
The GAMMA function uses the following equation: Г(N+1) = N * Г(N), where N is the input number. The function returns a number based on this equation. The GAMMA function is useful in various statistical and mathematical calculations, particularly in the fields of engineering, physics, and data analysis.
The syntax of the GAMMA function is straightforward, with the number as the only required argument. By using the GAMMA function, users can easily calculate the gamma function value for a given number, making it a convenient tool for complex calculations.
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xlsoffice.com
|
en
| 0.703499
| 2023-03-31T06:27:46
|
https://www.xlsoffice.com/tag/gamma-function/
| 0.998634
|
## Problem 12: Triangle, Cevian, Angles, Congruence
The problem involves a triangle ABC with a point E on AC such that BC = CE. Joining B to E, we observe that AE = CD and angle AEB = 115 = angle BDC, which implies BE = BD. This leads to the conclusion that triangles ABE and CBD are SAS congruent, resulting in x = 50.
An alternative approach denotes angle ABD as α = 115º - x. Assuming α > 65º, we find AD > AB, and since AD = BC, BC > AB. In triangle ABC, this yields x > 50º, or 115º - α > 50º, implying α < 65º, which contradicts the assumption α > 65º. Therefore, α = 65º and x = 50º.
Another solution involves taking E as the reflection of C in BD, resulting in DE = CD. An angle chase yields ∠ADE = 50, making triangles ADE and BCD congruent (SAS). Consequently, AE = BD and ∠DAE = 15, leading to the congruence of triangles ABE and BAD (SSS) with ∠BAE = ∠ABD. This ultimately gives x + 15 = 115 - x, or x = 50.
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blogspot.com
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en
| 0.791183
| 2017-12-12T15:52:27
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http://gogeometry.blogspot.com/2009/02/problem-12-triangle-cevian-angles.html
| 0.988839
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## Algebra 1 (FL B.E.S.T.)
### Course: Algebra 1 (FL B.E.S.T.) > Unit 5
Lesson 1: Solutions of two-variable inequalities - Testing solutions to systems of inequalities
CCSS.Math:
The system of inequalities is given by y≥2x+1 and x>1. To determine if the ordered pair (2,5) is a solution, we need to check if it satisfies both inequalities.
### Checking the Solution
For the first inequality, y≥2x+1, substitute x=2 and y=5:
5 ≥ 2(2) + 1
5 ≥ 4 + 1
5 ≥ 5
This is true, so the first inequality is satisfied.
For the second inequality, x>1, substitute x=2:
2 > 1
This is true, so the second inequality is satisfied.
Since the ordered pair (2,5) satisfies both inequalities, it is a solution to the system.
### Key Concepts
* Each inequality has its own solution set.
* The solution set of both inequalities must satisfy both inequalities.
* If the two shaded areas in the graph overlap, any point in the dual shaded area satisfies both equations.
* A system of inequalities can have multiple solutions, but each solution must satisfy both inequalities.
### Frequently Asked Questions
* Can a solution set satisfy only one inequality?
No, a solution set must satisfy both inequalities in a system of inequalities.
* How would you shade a graph with this solution?
To shade a graph, identify the regions that satisfy each inequality and overlap them to find the solution set.
* Can we have a system with one inequality and one equation?
While it's technically possible, it's not a common scenario and may not be relevant in most cases.
* How would you check a solution without the y-value?
Even if there's no y-value, you can still fill in the given x-value and check if it works out.
* Are there any practice problems for testing solutions for a system of inequalities?
Yes, practice problems can be found in previous videos or by clicking on "Practice this concept" above the video.
### Multiple Choice Questions
* What is required for an ordered pair to be a solution to a system of inequalities?
A) It must satisfy only one inequality.
B) It must satisfy both inequalities.
C) It must be in one of the shaded regions.
D) It must be outside the shaded regions.
Answer: B) It must satisfy both inequalities.
* If a point is only in one shaded region and not in both, is it a solution to the system?
A) Yes, it is a solution.
B) No, it is not a solution.
C) Maybe, it depends on the system.
D) It's not possible to determine.
Answer: B) No, it is not a solution.
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khanacademy.org
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en
| 0.944162
| 2023-03-23T18:27:54
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https://www.khanacademy.org/math/algebra-1-fl-best/x91c6a5a4a9698230:inequalities-graphs-systems/x91c6a5a4a9698230:solutions-of-two-variable-inequalities/v/testing-solutions-for-a-system-of-inequalities
| 0.978783
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## Definition of Continuity
A function f(x) is said to be continuous at a point x = a, in its domain if the following three conditions are satisfied:
1. f(a) exists (i.e., the value of f(a) is finite)
2. Limxa f(x) exists (i.e., the right-hand limit = left-hand limit, and both are finite)
3. Limxa f(x) = f(a)
## Continuity at a Point
A function f is continuous at x = c if f(c) is defined and the limit of f(x) as x approaches c equals f(c). In other words, a function is continuous at a point if the function’s value at that point is the same as the limit at that point.
## Continuity in Mathematics
Continuity, in mathematics, is a rigorous formulation of the intuitive concept of a function that varies with no abrupt breaks or jumps. A function is a relationship in which every value of an independent variable (say x) is associated with a value of a dependent variable (say y).
## Explaining Continuity
In calculus, a function is continuous at x = a if – and only if – all three of the following conditions are met:
1. The function is defined at x = a; that is, f(a) equals a real number.
2. The limit of the function as x approaches a exists.
3. The limit of the function as x approaches a is equal to the function value at x = a.
## Principle of Continuity
The continuity principle, or continuity equation, is a principle of fluid mechanics. It states that what flows into a defined volume in a defined time, minus what flows out of that volume in that time, must accumulate in that volume.
## Continuity Theory Examples
Examples of Continuity Theory include an elderly individual continuing to run for exercise but doing so in a less strenuous manner, and middle-aged people staying in contact with friends from their childhood or university years.
## Derivation of Continuity Equation
The continuity equation is defined as the product of the cross-sectional area of the pipe and the velocity of the fluid at any given point along the pipe being constant. The derivation involves the following steps:
1. Δx1 = v1Δt
2. V = A1 Δx1 = A1 v1 Δt
3. Δm1 = Density × Volume = ρ1A1v1Δt
4. Δm1/Δt = ρ1A1v1
## Significance of Continuity Equation
The continuity equation is important for describing the movement of fluids as they pass from a tube of greater diameter to one of smaller diameter. It is critical to keep in mind that the fluid has to be of constant density and be incompressible.
## Continuity Equation
The continuity equation (Eq. 4.1) is the statement of conservation of mass in the pipeline: mass in minus mass out equals change of mass. The equation is ∂(ρvA)/∂x, which represents “mass flow in minus mass flow out” of a slice of the pipeline cross-section.
## Mass Continuity Equation
In fluid dynamics, the continuity equation states that the rate at which mass enters a system is equal to the rate at which mass leaves the system plus the accumulation of mass within the system.
## Assumptions in Fluid Flow Problems
The most common assumption while dealing with fluid flow problems using the continuity equation is that the flow is steady.
## Continuity Equation of Flow
The continuity equation is simply a mathematical expression of the principle of conservation of mass. For a control volume that has a single inlet and a single outlet, the principle of conservation of mass states that, for steady-state flow, the mass flow rate into the volume must equal the mass flow rate out.
## Equation for Fluid Flow Problems
The continuity equation must be perfunctorily satisfied while dealing with fluid flow problems.
## Unit for Discharge
The units that are typically used to express discharge in streams or rivers include m³/s (cubic meters per second), ft³/s (cubic feet per second or cfs), and/or acre-feet per day.
## Formula for Discharge
Discharge = V x D x W, where V is the velocity, D is the depth, and W is the width. If length is measured in feet and time in seconds, Discharge has units of feet³/sec or cubic feet per second (cfs).
## Calculating Specific Discharge
The actual flow velocity v may be calculated with the following formula: v = Q/(A*f) = q/n, where n is the porosity, and q is the specific discharge.
## Calculating Water Discharge
The flow rate of a stream is equal to the flow velocity (speed) multiplied by the cross-sectional area of the flow. The equation Q = AV (Q = discharge rate, A = area, V = velocity) is sometimes known as the discharge equation.
## Discharge in Fluid Mechanics
Discharge (also called flow rate) is the amount of fluid passing a section of a stream in unit time. If v is the mean velocity and A is the cross-sectional area, the discharge Q is defined by Q = Av, which is known as volume flow rate.
## Velocity Formula
Velocity formula = displacement ÷ time. Time = taken to cover the distance.
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moorejustinmusic.com
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en
| 0.93185
| 2021-06-19T19:32:28
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https://moorejustinmusic.com/other-papers/what-is-the-3-part-definition-of-continuity/
| 0.995657
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Fractions and decimal numbers are two different ways of representing numbers less than one. Any number below one can be represented with either a fraction or a decimal. There are specific mathematic equations that allow you to figure out what the equivalent of a fraction would be in decimal form, and vice versa.
**Understanding Fractions and Decimals**
A fraction consists of three parts: the numerator (top part), the slash, and the denominator (bottom part). The denominator represents how many equal parts there are in the whole. For example, a pizza cut into 8 pieces has a denominator of 8. The numerator represents a part, or parts, of the whole. One slice of the whole pizza would be represented by the numerator "1".
Decimals do not use a slash to indicate what part of the whole they represent. Instead, the decimal point signifies that the numbers are below one. With a decimal, the whole is considered to be based on 10, 100, 1000, etc. Decimals are often read in a way that demonstrates their similarity to fractions. For example, 0.05 would be read aloud as "five hundredths," the same as 5/100.
**Converting Fractions to Decimals Using Division**
To convert a fraction to a decimal, think of the fraction as a math problem. Read the fraction as if it were a division problem, with the number on the top being divided by the number on the bottom. For example, the fraction 2/3 can also be stated as 2 divided by 3. Divide the numerator by the denominator using a calculator, long division, or mental math. Double-check your math by multiplying the decimal equivalent by the denominator to get the numerator.
**Converting Fractions with "Power of 10" Denominators**
Another way to convert a fraction into a decimal is to use a "power of 10" denominator. A "power of 10" denominator is a denominator consisting of any positive number that can be multiplied to make a multiple of 10, such as 10 or 100. To convert, multiply the fraction by another fraction with a denominator that creates a multiple of 10 when multiplied together. The top of this second fraction will be the same as its denominator, making it equal to one. For example, to convert 1/5 to a fraction with a denominator of 10, multiply it by 2/2, resulting in 2/10.
**Converting "Power of 10" Fractions to Decimals**
To convert a "power of 10" fraction to a decimal, take the numerator and rewrite it with a decimal point at the end. Count the number of zeros in the denominator and move the decimal point to the left by that number of spaces. For instance, the fraction 2/10 has one zero in the denominator, so the decimal equivalent is 0.2.
**Memorizing Important Decimal Equivalents of Fractions**
To memorize decimal equivalents of fractions, convert common fractions to decimals by dividing the numerator by the denominator. Some basic fraction to decimal conversions to know by heart are 1/4 = 0.25, 1/2 = 0.5, and 3/4 = 0.75. Create flashcards with the fraction on one side and its decimal equivalent on the other to practice and memorize.
**Community Q&A**
Q: How do you turn a mixed number into a decimal?
A: First, convert the mixed number into an improper fraction. Multiply the whole number by the denominator, then add the result to the numerator. Divide the new numerator by the denominator to get the decimal.
Q: How can I convert a fraction to a percent?
A: First, convert the fraction into a decimal. Multiply the result by 100 to get a percent. Move the decimal point to the right 2 places.
Q: Can you explain things in simpler terms?
A: To convert a fraction into a decimal, divide the top number by the bottom number.
Q: What about with denominators that don't divide into 10, 100, or 1000, such as 7?
A: You still divide the denominator into the numerator, no matter what the numbers are. You may wind up with a quotient consisting of a decimal number.
Q: What is the point of this?
A: There are many situations in which operating with a decimal number is easier than operating with a fraction.
Q: Can you explain ratios and unit rates?
A: A ratio is a comparison between two numbers. A "rate" is a ratio that compares quantities expressed in different units. A "unit rate" is a rate in which the second number in the ratio is 1.
Q: How do I turn a fraction into a mixed number?
A: Turn an improper fraction into a mixed number by dividing the numerator by the denominator. The quotient becomes the integer of the mixed number, and the remainder is the numerator of the mixed number.
Q: What happens if I can’t use a calculator, but the number doesn’t go into 100 equally, e.g. 7?
A: You would have to do the division manually, carried out to as many decimal places as needed.
Q: How can I convert 0.625 to a fraction?
A: 0.625 = 625/1000, which reduces to 5/8.
Q: How do I turn 2.072 into a fraction?
A: 2.072 is the same as the mixed number 2 72/1,000, which reduces to 2 9/125.
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wikihow.com
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en
| 0.720551
| 2020-01-28T19:22:15
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https://m.wikihow.com/Convert-Fractions-to-Decimals
| 0.99963
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Function notation is an alternative way to express the y-value of a function. When graphing, the y-axis can be labeled as f(x). Key concepts related to function notation include:
Basic concepts:
- Solving linear equations using multiplication and division
- Solving two-step linear equations, such as $ax + b = c$ and ${x \over a} + b = c$
- Solving linear equations using the distributive property, such as $a(x + b) = c$
- Solving linear equations with variables on both sides
Related concepts:
- Function notation (Advanced)
- Operations with functions
Examples of function notation include:
a) ${f(\heartsuit)}$
b) ${f(\theta)}$
c) ${f(3)}$
d) ${f(1)}$
e) ${f(3x)}$
f) ${f(x)}$
g) ${f(3x4)}$
h) ${3f(x)}$
i) ${f(x)3}$
Evaluating function notation:
a) $f(3)$
b) $f(8)$
c) $f(2/5)$
Function notation can also be used with various mathematical operations, such as:
a) ${\sqrt{x}+5}$
b) ${\sqrt{x+5}}$
c) ${\sqrt{2x3}}$
d) ${8\sqrt{x}}$
e) ${8\sqrt{2x3}}$
f) $4\sqrt{x^{5}+9}$
Multiple choice questions and answers have been refined for clarity and concision.
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studypug.com
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en
| 0.677446
| 2019-01-23T07:44:29
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https://www.studypug.com/algebra-help/function-notations
| 0.999994
|
# Unit 1 Congruence, Structure, and Proof
## Lesson 1
### Learning Focus
Identify the defining features of translation, rotation, reflection, and dilation transformations. Use function notation to describe transformations.
### Lesson Summary
This lesson reviews the characteristics of rigid transformations (translation, rotation, reflection) that preserve angle and distance measurements, and dilation transformations that produce similar figures. It also examines notation for describing transformations symbolically, illustrating that geometric transformations are functions mapping input points to unique output points.
## Lesson 2
### Learning Focus
Justify triangle congruence criteria using reasoning based on rigid transformations.
### Lesson Summary
This lesson reviews triangle congruence criteria that guarantee congruence without requiring all corresponding sides and angles to be congruent. It justifies each criterion using rigid transformations and explains what it means to justify a claim.
## Lesson 3
### Learning Focus
Examine characteristics of valid proofs.
### Lesson Summary
This lesson reviews the concept of a valid proof, examining examples of logical reasoning and problematic proofs. It discusses proofs about vertical angles and angles formed by parallel lines and a transversal, promoting thoughtful and strategic proof-writing.
## Lesson 4
### Learning Focus
Use theorems about angles formed by parallel lines and a transversal to prove properties of parallelograms.
### Lesson Summary
This lesson applies understanding of rigid transformations, triangle congruence criteria, and parallel lines to prove conjectures about parallelogram sides, angles, and diagonals.
## Lesson 5
### Learning Focus
Classify and justify types of parallelograms based on angle and diagonal characteristics.
### Lesson Summary
This lesson expands proof-writing skills by starting with given statements and creating diagrams to mark congruent parts. It demonstrates how sequencing proofs allows drawing upon theorems to prove more complicated theorems.
## Lesson 6
### Learning Focus
Examine properties of triangle medians, angle bisectors, and perpendicular bisectors. Construct the center of a circle passing through a triangle's vertices and the center of a circle touching all three sides. Construct a triangle's "balancing point".
### Lesson Summary
This lesson finds that a triangle's medians, angle bisectors, and perpendicular bisectors are concurrent, locating interesting points like the balancing point or circle centers. It constructs these points, demonstrating their significance in triangle geometry.
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openupresources.org
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en
| 0.927499
| 2023-03-25T20:52:14
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https://access.openupresources.org/curricula/our-hs-math-nc/nc/math-3/unit-1/family.html
| 0.998163
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To determine leap years, we can use a scheme based on the binary expansion of years. The goal is to find a method to assign leap years out of every 128 years.
First, consider the fraction 31/128, which equals 1/4 - 1/128. This implies that we should take every fourth year, except those divisible by 128.
Alternatively, 33/128 can be expressed as 1/4 + 1/128 or 1/2 - 1/4 + 1/128. The latter expression provides a scheme for assigning 33 leap years out of every 128. Include years that are even but not divisible by 4, and also include those divisible by 128.
In binary expansion, such years end in exactly one 0 (2 more than a multiple of 4) or at least seven 0s (a multiple of 128). For fractions of the form m/2^n, where 0 ≤ m < 2^n, we can write m/2^n as an alternating sum of powers of 1/2.
For example, 59/128 = 1/4 + 1/8 + 1/16 + 1/64 + 1/128 can be rewritten as 1/2 - 1/16 + 1/32 - 1/128. To assign 59 leap years out of every 128, include even years, exclude those divisible by 16, include those divisible by 32, and exclude those divisible by 128.
A more straightforward rule is that the binary expansion of a leap year must end in 1, 2, 3, 5, or 6 zeroes. This corresponds to the binary representation of 59/128 = .0111011_2, where there are 1s in the 2nd, 3rd, 4th, 6th, and 7th places after the decimal point.
In general, to determine if a year k is a leap year for a given proportion m/2^n, use the following scheme:
- Let p be the number of zeroes terminating the binary expansion of k.
- If the (p+1)st bit of m/2^n after the decimal point is 1, then k is a leap year; otherwise, it's a common year.
This method works because the years for which we examine the jth bit are exactly 1/2^j of all years.
For 31/128 = .0011111, a year is a leap year if its binary expansion ends in exactly 2, 3, 4, 5, or 6 zeroes. For 33/128 = .0100001, a year is a leap year if its binary expansion ends in exactly 1 or 6 zeroes.
One limitation of this scheme is that the set of leap years changes significantly as m/2^n passes through some small power of (1/2). The challenge remains to develop a similar scheme in decimal.
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blogspot.com
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en
| 0.938303
| 2017-12-14T20:45:19
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http://godplaysdice.blogspot.com/2008/03/leap-year-scheme-based-on-binary.html
| 0.97864
|
The Snake Lemma and other theorems in abstract algebra can be established in a unified setting of noetherian forms. This concept can be adapted to create a "noetherian information system," which is intended to be more agile in identifying applications.
An information system over a network consists of devices and directed binary channels between them. A device can be a cellphone, and a cluster can be an approximate GPS location of the cellphone. Clusters can be part of others, and there are two ways to interpret this: classical interpretation, where one cluster implies another, and quantum interpretation, where one cluster has less attributes than another.
A transmission from one information system to another maps each cluster to a cluster in the target system, preserving the "part of" relation. An example of transmission is calculating distance range to a pinned location based on the approximate GPS location of the cellphone. A transmission category can be formed from an information system, where objects are devices and morphisms are composite transmissions.
Inputs and outputs are defined in terms of their properties. An input is a channel that maps clusters injectively, and any transmission with the same target as the input arises as a composite of the input transmission with another transmission. An output is a channel that maps clusters to reaches, and any transmission with the same source as the output arises as a composite of the output with another transmission.
An information system is prenoetherian if it satisfies three conditions: any transmission decomposes as an output followed by an isotransmission and an input, for any two inputs there is a third input whose reaches are the intersection of the reaches of the initial inputs, and for any two outputs there is a third output whose stashes are the intersection of the stashes of the initial outputs.
Mathematical examples of prenoetherian information systems include vector spaces and sets. In the case of vector spaces, devices are vector spaces, channels are linear maps, and clusters are subspaces. The transmission category is equivalent to the category of projective spaces. In the case of sets, devices are sets, channels are functions, and clusters are partitions of the set into equivalence classes.
A noetherian information system is a prenoetherian information system where clusters admit finite suprema and infima, and each transmission is a left adjoint in a Galois connection. A topological information system is a general information system where all information systems over individual devices are complete and all transmissions are continuous.
In a topological information system, every transmission can be reversed, and the reverse transmission recovers the largest possible cluster that can get transmitted into a given one. The law of Galois connection holds, and reverse transmission is monotone and preserves meets of clusters.
The concept of a noetherian information system has potential applications in machine learning, data science, and modeling the function of a living organism or cognitive function of a human being. The category of Hilbert spaces, which plays an important role in quantum mechanics, may have a noetherian form, and the physical universe may be modeled as a noetherian information system.
Some potential applications and questions for future research include:
1. Is there a useful real-life interpretation of a noetherian information system?
2. If yes, does it lead to the ability to usefully model real-life information systems as noetherian information systems?
3. In particular, are there any applications in machine learning or data science?
4. Or, is it perhaps possible to use noetherian information systems to usefully model function of a living organism, or maybe, cognitive function of a human being?
5. Does the category of Hilbert spaces, which is neither an abelian nor a semi-abelian category, but which plays an important role in quantum mechanics, have a noetherian form?
6. Can the physical universe be modelled as a noetherian information system?
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zurab.online
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en
| 0.892002
| 2024-09-07T14:30:59
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https://www.zurab.online/2022/02/noetherian-information-systems.html
| 0.89088
|
To convert 5/8 to a decimal, divide 5 by 8. The result of this division is 0.625. Therefore, 5/8 as a decimal is 0.625. This conversion can be done using long division or a calculator. The fraction 5/8 is equal to 0.625 in decimal form. To express 5/8 as a percentage, multiply 0.625 by 100, which equals 62.5%.
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earlyinvestor.org
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en
| 0.914379
| 2021-06-22T13:48:27
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http://earlyinvestor.org/mcuum2r8/jdfqrp.php?94b9bf=what-is-5%2F8-as-a-decimal
| 0.99757
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The base of a solid is the region bounded by the parabolas y = x^2 and y = 2x^2. To find the volume of the solid, we need to consider the cross-sections perpendicular to the x-axis, which are squares with one side lying along the base.
First, we find the intersection points of the two parabolas by setting x^2 = 2x^2, which gives us x = 0 and x = 1, but since the problem involves the region between the parabolas, we consider x = 1 as one of the limits. However, to accurately define the region, we recognize that the parabolas intersect at x = 0 and x = 1 is not a correct intersection for the given problem context, implying a need to identify the correct intersection points based on the parabolas' equations.
The correct approach involves recognizing that the "square" has a vertical side that is the difference between the y-values of the two parabolas, (2x^2) - (x^2) = x^2. Thus, the area of the square is [x^2]^2 = x^4.
The volume element of the solid, dV, has a thickness of dx, so dV = x^4 * dx. The integration limits are from x = 0 to x = 1, where the parabolas intersect.
The volume V is given by the integral of x^4 with respect to x from 0 to 1:
V = ∫[0,1] x^4 dx
V = (1/5)x^5 | from 0 to 1
V = (1/5)(1^5) - (1/5)(0^5)
V = 1/5
However, considering the initial setup and the intention to calculate the volume based on the difference between the two parabolas and the correct formula for the area of the square cross-sections, let's re-evaluate the approach:
The side length of the square is the difference in y-values, which is 2x^2 - x^2 = x^2. Thus, the area of the square is (x^2)^2 = x^4.
Given this, the volume calculation should consider the correct area of the square cross-sections and integrate over the appropriate interval. The correct calculation involves recognizing the side of the square as 2(1 - x^2) when considering the vertical distance between the two curves, leading to the area of the square being [2(1 - x^2)]^2.
The volume element dV = [2(1 - x^2)]^2 * dx, and integrating from the correct limits based on the intersection points of the parabolas, which are x = 0 and x = 1, gives:
V = ∫[0,1] [2(1 - x^2)]^2 dx
V = ∫[0,1] 4(1 - 2x^2 + x^4) dx
V = 4 ∫[0,1] (1 - 2x^2 + x^4) dx
V = 4 [x - (2/3)x^3 + (1/5)x^5] from 0 to 1
V = 4 {[1 - (2/3)(1) + (1/5)(1)] - [0 - (2/3)(0) + (1/5)(0)]}
V = 4 [1 - 2/3 + 1/5]
V = 4 [(15 - 10 + 3) / 15]
V = 4 * (8 / 15)
V = 32 / 15
This calculation provides the volume based on the correct understanding of the square's side length and the integration over the defined interval.
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mathhelpforum.com
|
en
| 0.829013
| 2017-12-17T11:01:29
|
http://mathhelpforum.com/geometry/22033-base-solid-print.html
| 0.995954
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The holohedric Protopyramid gave rise to the pyramidal hemihedric Type I hexagonal bipyramid. The holohedric Deuteropyramid gave rise to the pyramidal hemihedric Type II hexagonal bipyramid. The holohedric Dihexagonal Bipyramid gave rise to the pyramidal hemihedric Type III hexagonal bipyramid. The holohedric Protoprism gave rise to the pyramidal hemihedric type I hexagonal prism. The holohedric Deuteroprism gave rise to the pyramidal hemihedric type II hexagonal prism. The holohedric Dihexagonal Prism gave rise to the pyramidal hemihedric hexagonal tritoprism. The holohedric Basic Pinacoid gave rise to the pyramidal hemihedric basic pinacoid.
From these pyramidal hemihedric Forms, we derive the Forms of the Hexagonal-pyramidal Class by suppressing their equatorial mirror plane. Starting with the pyramidal hemihedric type I hexagonal bipyramid, we remove the equatorial mirror plane, resulting in two independent monopyramids. The Naumann symbol of the generated Form expresses the fact that a pyramidal hemihedric Form has been subjected to hemimorphy, symbolized by "/ 2h".
Applying hemimorphy to the pyramidal hemihedric Type II hexagonal bipyramid yields two independent halves, i.e., two hemimorphous pyramidal hemihedric type II hexagonal monopyramids. Similarly, applying hemimorphy to the pyramidal hemihedric Type III hexagonal bipyramid generates two independent halves, namely two hemimorphous pyramidal hemihedric type III hexagonal monopyramids.
Applying hemimorphy to the pyramidal hemihedric type I hexagonal prism results in a hemimorphous pyramidal hemihedric type I hexagonal prism, with no change in shape but a lowering of symmetry. The same applies to the pyramidal hemihedric type II hexagonal prism and the pyramidal hemihedric hexagonal tritoprism.
From the pyramidal hemihedric basic pinacoid, we derive the hemimorphous pyramidal hemihedric pedion by applying hemimorphy. The basic pinacoid is dissolved into two independent halves, an upper and a lower one, i.e., an upper pedion and a lower pedion. A pedion is a Form consisting of just one horizontal face and can close a hemimorphous pyramidal hemihedric prism at its bottom or top.
We can also derive these Forms by subjecting the basic faces compatible with the Hexagonal Crystal System to the symmetry operations of the present Class. The seven basic faces are:
- face a : ~a : -a : c, which gives rise to the Type I hexagonal monopyramid
- face 2a : 2a : -a : c, which gives rise to a type II hexagonal monopyramid
- face [s/(s-1)]a : sa : -a : c, which gives rise to a type III hexagonal monopyramid
- face [s/(s-1)]a : sa : -a : ~c, which gives rise to a type III hexagonal prism
- face a : ~a : -a : ~c, which gives rise to a type I hexagonal prism
- face 2a : 2a : -a : ~c, which gives rise to a type II hexagonal prism
- face ~a : ~a : ~a : c, which gives rise to a pedion
These Forms can engage in combinations in real crystals. The facial approach involves subjecting each basic face to the symmetry operations of the Hexagonal-pyramidal Crystal Class, resulting in the generation of the corresponding Form. The position of each face and the generation of the corresponding Form are depicted stereographically in Figures 17-23.
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CC-MAIN-2024-38/segments/1725700651668.26/warc/CC-MAIN-20240915220324-20240916010324-00561.warc.gz
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metafysica.nl
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en
| 0.806401
| 2024-09-16T00:05:37
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http://metafysica.nl/hexagonal_4.html
| 0.634741
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# Worksheet: Kinetic Energy
This worksheet practices calculating the kinetic energy of objects with different masses and velocities.
**Q2:**
A 665 kg motorboat accelerates from rest to 340,480 J of kinetic energy. What is its velocity?
**Q3:**
A 40 g hummingbird flies 18 m in 12 s. What is its average kinetic energy?
**Q8:**
Which formula correctly shows the relationship between velocity (v), kinetic energy (KE), and mass (m)?
- A: KE = mv
- B: KE = mv^2
- C: KE = 0.5mv^2
- D: KE = 2mv^2
- E: KE = mv^3
**Q9:**
A 250 kg motorcycle moving at 32 m/s has four times the kinetic energy of a 640 kg car. What is the car's velocity?
**Q10:**
A graph shows an object's kinetic energy at different velocities.
- What is the kinetic energy at 2 m/s?
- What is the kinetic energy at 4 m/s?
- What is the object's mass?
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CC-MAIN-2020-05/segments/1579250598217.23/warc/CC-MAIN-20200120081337-20200120105337-00311.warc.gz
|
nagwa.com
|
en
| 0.882634
| 2020-01-20T09:10:18
|
https://www.nagwa.com/en/worksheets/947146475307/
| 0.830821
|
Get Ready for Statistics: Corequisites - Inequalities
To find the percent \(P\) of people that will vote for a certain candidate, a sample of \(100\) people is polled. The given numbers are:
\[\overline{x}=42, z=2.58, \sigma=5,\text{ and } n=100\]
The actual percent of the population that will vote for the candidate is given by the expression \[\overline{x}\pm z\cdot\dfrac{\sigma}{\sqrt{n}}\]
Substituting the given numbers into the expression yields:
\[42 \pm 2.58 \cdot \dfrac{5}{\sqrt{100}}\]
Simplifying, we get:
\[42 \pm 2.58 \cdot \dfrac{5}{10}\]
\[42 \pm 2.58 \cdot 0.5\]
\[42 \pm 1.29\]
Converting the results to an inequality gives:
\[42 - 1.29 \leq P \leq 42 + 1.29\]
\[40.71 \leq P \leq 43.29\]
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CC-MAIN-2024-38/segments/1725700652130.6/warc/CC-MAIN-20240920022257-20240920052257-00483.warc.gz
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mathtv.com
|
en
| 0.716703
| 2024-09-20T04:32:57
|
https://www.mathtv.com/topic/2105
| 0.999862
|
**Surface Tension by Capillary Rise Method**
A capillary tube of uniform bore is dipped vertically in a beaker containing water. Due to surface tension, water rises to a height h in the capillary tube. The surface tension T of the water acts inwards, and the reaction of the tube R acts outwards. R is equal to T in magnitude but opposite in direction. This reaction R can be resolved into two rectangular components:
1. Horizontal component *R sin* θ acting radially outwards
2. Vertical component *R cos* θ acting upwards
The horizontal component cancels out, while the vertical component balances the weight of the water column in the tube. The total upward force is given by F = 2πr R cos θ or F = 2πr T cos θ.
As the water column is in equilibrium, the upward force is equal to the weight of the water column acting downwards: F = W. The volume of water in the tube is assumed to be made up of a cylindrical water column of height h and water in the meniscus above the plane CD. The total volume of water in the tube is πr²(h + r/3).
The weight of water in the tube is W = πr²(h + r/3) ρg, where ρ is the density of water. Substituting the equations, we get T = (h + r/3)r ρg / 2cos θ. Since r is very small, r/3 can be neglected compared to h, giving T = hr ρg / 2cos θ. For water, θ is small, so cos θ = 1, resulting in T = hr ρg / 2.
**Experimental Determination of Surface Tension of Water by Capillary Rise Method**
A clean capillary tube of uniform bore is fixed vertically with its lower end dipping into water. A needle is fixed with the capillary tube, and a traveling microscope is focused on the meniscus of the water in the capillary tube. The reading R₁ corresponding to the lower meniscus is noted, and the microscope is lowered and focused on the tip of the needle, giving the reading R₂. The difference between R₁ and R₂ gives the capillary rise h.
The radius of the capillary tube is determined using the traveling microscope. If ρ is the density of water, the surface tension of water is given by T = hrρg / 2, where g is the acceleration due to gravity.
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CC-MAIN-2023-14/segments/1679296948932.75/warc/CC-MAIN-20230329023546-20230329053546-00194.warc.gz
|
brainkart.com
|
en
| 0.842721
| 2023-03-29T04:26:45
|
https://www.brainkart.com/article/Experimental-determination-of-surface-tension-of-water-by-capillary-rise-method_3066/
| 0.409312
|
## Key Concepts in Coding Theory
To find a code that satisfies C = C⊥, C has to be a [2k, k]-code satisfying specific conditions. Every codeword in C has even parity, and all codewords differ in an even number of zeros.
### Properties of Codewords
- Every codeword in C has even parity, implying u · u = 0 for all u in C.
- All codewords differ in an even number of zeros.
- Considering bases with codewords containing exactly two ones is sufficient.
### Counting Bases
The number of ways to choose k independent vectors in V(n, q) is given by (qn - 1)(qn - q) · · · (qn - qk-1). The number of ways to choose a basis for a particular k-dimensional subspace is (qk - 1)(qk - q) · · · (qk - qk-1).
### Gaussian Polynomial
The number of different k-dimensional subspaces of V(n, q) is given by the Gaussian polynomial G(n, k), calculated as:
\[ \frac{(q^n - 1)(q^n - q) \cdot \cdot \cdot (q^n - q^{k-1})}{(q^k - 1)(q^k - q) \cdot \cdot \cdot (q^k - q^{k-1})} \]
### Linearity of Codes
Given a binary code C of length n, if C is a linear code, then C is also a linear code, where C = {x1 · · · xnxn+1 | x1 · · · xn ∈ C, xn+1 = 0 if the weight of x1 · · · xn is even, and xn+1 = 1 if the weight of x1 · · · xn is odd}.
### Proof of Linearity
1. For all u, v in C, u + v is in C.
2. For all u in C and a in GF(2), au is in C.
### Slepian Array and Cosets
A [n, k]-code has qn-k cosets, each consisting of qk words. For a [5, 3]-code, there are 4 cosets, each having 8 words.
### Generator Matrix
The generator matrix of a code can be constructed from its basis. For a code C with a basis of n-1 vectors, adding a parity bit creates a new code C with a generator matrix in standard form.
### Parity Check Matrix
The parity check matrix H of a code C is used to define the code: x is in C if and only if xH^T = 0. The weight of the non-zero word of minimum weight in C gives the distance of C.
### Example: Hamming Code
The parity check matrix of a Hamming code Ham(r, 2) is a r × (2r - 1) matrix. Every row has exactly 2r-1 ones and 2r-1 - 1 zeroes, and summing any rows creates a word with a weight of 2r-1.
### Minimal Non-zero Solution
For a given system of equations derived from the parity check matrix, finding the minimal non-zero solution gives the word with the minimum weight in the code, which determines the code's distance.
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CC-MAIN-2019-04/segments/1547583850393.61/warc/CC-MAIN-20190122120040-20190122142040-00128.warc.gz
|
pharmapdf.com
|
en
| 0.868646
| 2019-01-22T13:23:14
|
http://pharmapdf.com/s/school.lisk.in1.html
| 0.986699
|
# Simple Guidance for You in 4D Toto
## Overview of the Topic
The objective is to assess how players get started with 4D Toto. Players can purchase both Toto and 4D together, but the results are announced on different days. The combination of both games allows players to participate in the draw, known as 4D Toto. The cost of a Toto lottery ticket is lower than 4D, but buying 4D draws has its advantages, increasing the odds of winning.
## Aspects Related to Game Rules
In 4D, players pick four digits to form a number between 0000 and 9999. The minimum bet is $1, and draws take place on Saturday, Sunday, and Wednesday. There are 23 winning numbers for each draw, and the prize money depends on the number of winners in each category. Players can place a big or small bet, with big bets winning a prize if the 4D result appears in any prize category, and small bets winning only if the result appears in the top three categories.
## Features Associated with System Entry
System Entry allows players to select a specific sequence of four digits, but the bet cost increases with the number of combinations selected. This means players can purchase multiple combinations, but at a higher ticket cost.
## Basic Feature of 4D Roll
Players can buy a 4D Roll with a minimum bet of $10. They select the first three digits and one "rolling digit" (0-9), which can match the winning number in any position. The numbering order is crucial in matching the winning number.
## Assessment of the QuickPick Feature
QuickPick generates random numbers for players who struggle to choose. This feature can enhance the 4D winning prize, as it guarantees a specific number. Players can use QuickPick to generate numbers, such as car plate numbers, birthday dates, or anniversary dates.
## Understanding of the Game Rules
The easiest version of 4D Toto is the Ordinary Bet, where players pick at least six numbers between 1 and 49, with a minimum bet of $1. Players win a prize if their numbers match at least three winning numbers. The jackpot is the highest prize, requiring all six numbers to match. QuickPick can generate random numbers without additional cost.
## Total Amount a Player Can Win
The total amount a player can win depends on the split of the final prize among winners. If there are no winners, the next ticket is selected, and the process continues until winners are finalized.
## Conclusion
This guidance explains the key aspects of 4D Toto, including the requirements for winning and the situations where a player may not win. A detailed analysis of the topic has been provided to make the guidance helpful for readers.
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CC-MAIN-2023-14/segments/1679296945315.31/warc/CC-MAIN-20230325033306-20230325063306-00485.warc.gz
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weclub88.cc
|
en
| 0.960388
| 2023-03-25T05:43:52
|
https://www.weclub88.cc/news-simple-guidance-for-you-in-4d-toto
| 0.499396
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