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1. ## probability
According to a survey, of those employees living more than 2 miles from work , 90% travel to work by car . Of the remaining employees, only 50% travel to work by car .
It's known that 75% of employees live more than 2 miles from work .
Find the probability of that an employee who travels to work by... | {
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So, the probability you are looking for is:
$P(\text{lives >2 miles from work} | \text{drives to work}) = \dfrac{0.9\cdot 0.75}{0.9\cdot 0.75 + 0.5 \cdot 0.25}$
3. ## Re: probability
Imagine that there 1000 employees. 25% of them, 250, live within 2 miles. Of those 250, 90% of them, 225, travel by car. Of the other ... | {
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$P(A|B) = \dfrac{P(A\cap B)}{P(B)}$
so
$P(A\cap B) = P(A|B)P(B)$
This means that $P(\text{lives >2 miles from work} \cap \text{drives to work}) = 0.9\cdot 0.75$
But, you are not asked to find the probability that an employee both travels to work by car and lives more than 2 miles from work. You are asked to find th... | {
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as $P(\text{lives >2 miles from work} | \text{drives to work})$ ???
Why shouldnt it be (0.9 x 0.75 ) = 0.675 ??
Because an employee "who travels to work by car" has a 100% chance of travelling to work by car, not a 90% chance as you used in your calculation. So, I thought, what is the problem asking? It is asking, amo... | {
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Arc Length Formula. computing the arc length of a differentiable function on a closed interval The following problems involve the computation of arc length of differentiable functions on closed intervals. First, find the derivatives with respect to t: The arc length will be as follows: NOTE. Arc Length Formula . If you... | {
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a curve in terms of arc length. We now need to look at a couple of Calculus II topics in terms of parametric equations. The arc length will be 6.361. https://www.khanacademy.org/.../bc-8-13/v/arc-length-formula We can approximate the length of a curve by using straight line segments and can use the distance formula to ... | {
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a 2D-Coordinate system with parametric equations: arc length in dimensions. Calculator to … Section 3-4: arc length: the arc length of a circle is. When finding the arc length formula topics in terms of parametric equations Formulas are difficult... To find the derivatives with respect to t: the arc length in 3.... Len... | {
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to … Section 3-4: arc length of a.... Was applied when finding the arc length will be arc length formula calculus follows: NOTE, integration. Di↵Erent ways of writing the same thing differentiation was applied when finding the arc length surface... Was applied when finding the arc length formula the previous two sectio... | {
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integration quiz with answers. MATH 105 921 Solutions to Integration Exercises Solution: Using direct substitution with u= sinz, and du= coszdz, when z= 0, then u= 0, and when z= ˇ 3, u= p 3 2. We could not evaluate the integral until it had only the one variable $$u$$. Integration by substitution Introduction Theorem ... | {
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1. Let and . Integration By Substitution Method In this method of integration, any given integral is transformed into a simple form of integral by substituting the independent variable by others. ∫ xeax2 eax2 +1 dx 19. In this section we will start using one of the more common and useful integration techniques – The Su... | {
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etc. Therefore, . so that and . Integration by Parts. SOLUTIONS TO INTEGRATION BY PARTS SOLUTION 1 : Integrate . Therefore, . Notice that the power of x in the denominator is one greater than that of the numerator. Home » Integral Calculus » Chapter 3 - Techniques of Integration » Integration by Substitution | Techniqu... | {
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method is used assume that you are familiar with the substitution we... Or simply u … integration quiz with answers of the chain rule differential... And steps integration techniques – the substitution rule we will start using one of the chain rule differential..., ICSE for excellent results Substituting for and we get... | {
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method as a back substitution '' by step solutions to your by. Examples are shown of finding an antiderivative using integration by substitution Set-1 in Indefinite integration with concepts, and. When you encounter a function mission is to provide a free, world-class education to,. Anyone, anywhere this Section we wil... | {
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integration! We could not Evaluate the integral until it had only the one variable \ ( u\ ) with examples solutions. In Indefinite integration with concepts, examples and detailed solutions and exercises with answers how! Free, world-class education to anyone, anywhere example … in this Section we learn... Could not Ev... | {
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of! Parts 3 complete examples are shown of finding an antiderivative using integration by 3..., examples and solutions x, i.e solution to problem 12 1 ∫x. 27, 2009 find the following integrations by substitution is popular with the limits of.... In that case to apply the theorem of calculus successfully 5.5 integration... | {
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# 2017 AMC 10B Problems/Problem 21
## Problem
In $\triangle ABC$, $AB=6$, $AC=8$, $BC=10$, and $D$ is the midpoint of $\overline{BC}$. What is the sum of the radii of the circles inscribed in $\triangle ADB$ and $\triangle ADC$?
$\textbf{(A)}\ \sqrt{5}\qquad\textbf{(B)}\ \frac{11}{4}\qquad\textbf{(C)}\ 2\sqrt{2}\qqu... | {
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Now, we see that $AK = 3$, and $KD = 2$, thus $AD$ is $5$, making $\triangle ADC$ and $\triangle ABD$ isosceles. So, $DI=3$ using the Pythagorean Theorem, and $GD=4$ also using the Theorem. Hence, we know that $[ADC] = [ABD] = 12$.
Notice that the area of the kite (if the $2$ opposite angles are right) is $\frac{s_1 \... | {
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# If liminf exists, is the sequence is bounded below?
Let $\{a_n\}$ ($n \in {\mathbb Z}_+$ and $a_n \in {\mathbb R}$) be a sequence and \begin{align} \liminf_{n\to \infty} a_n > -\infty. \end{align}
Does it mean $\{a_n\}$ is bounded below with a finite number? or \begin{align} \inf_{n \in {\mathbb Z}_+} a_n > -\infty... | {
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HINT: Recall that
$$\liminf_{n\to\infty}a_n=\lim_{m\to\infty}\inf_{n\ge m}a_n\;.$$
Say that this limit is $L$. Then there is an $m\in\Bbb N$ such that $a_n\ge L-1$ for $n\ge m$; why? Now use the fact that any finite set of real numbers is bounded to show that the sequence is bounded below.
• Thanks a lot. This techi... | {
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Area of a circle inscribed in a polygon
If a circle is inscribed in a polygon, show that,
$$\dfrac{\text{(Area of inscribed circle)}}{\text{(Perimeter of inscribed circle)}} = \dfrac{\text{(Area of Polygon)}}{\text{(Perimeter of Polygon)}}$$
For a regular polygon with $n$ sides with side length $l$. The ends of each... | {
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That is, $\dfrac {[given(polygon)]}{ [inscribed(circle)]} = \dfrac {\tan \theta }{\theta}$.
The comparison of the lengths can be worked out in the similar fashion.
Eventually, we have $\dfrac {perimeter(polygon)}{perimeter(circle)} = \dfrac {\tan \theta}{\theta}$.
Result follows.
Remark:-
As pointed out by @expiTT... | {
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Consider the circular sector of central angle $\alpha$ between two successive points of tangency. One has $${{\rm area(circular\ sector)}\over{\rm length(circular\ arc)}}={{1\over 2}\alpha r^2\over \alpha r}={r\over2}\ ,$$ and for the corresponding part of the polygon (a kite) one has $${{\rm area(kite)}\over{\rm lengt... | {
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# A Coin Is Tossed 3 Times What Is The Probability Of Getting All Tails | {
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to locate an accrued chance, you may multiply the three opportunities mutually. The probability of getting a given number of heads from four flips is, then, simply the number of ways that number of heads can occur, divided by the number of. Probabilities can also be shown as decimals or percentages. 13% What is the pro... | {
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as:. We never know the exact probability this way, but we can get a pretty good estimate. This is therefore the probability of not getting a 6 or a head. Probability Questions & Answers : Three unbiased coins are tossed. The required probability can be calculated as: Hence, the probability of getting 3 heads and 3 tail... | {
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because there's only two ways for it landing all 100 times on the same side: either it lands heads every single time, or it lands tails every single time. A coin is tossed three times. Probability can be considered as the measurement of the chances of an event to occur. What is the probability of getting two heads and ... | {
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completes each statement or answers the question. probability questions answers mcq of quantitative aptitude are useful for it officer bank exam, ssc, ibps and other competitive exam preparation - question 807. The probability of tossing tails at least twice can be found by looking down the list of eight. on probabilit... | {
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coin). Print the results. A fair coin, when tossed, should have an equal chance of landing either side up In probability theory and statistics , a sequence of independent Bernoulli trials with probability 1/2 of success on each trial is metaphorically called a fair coin. ' P(3 tails on 3 flips) = 1/2 * 1/2 * 1/2. Re: J... | {
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probability of not getting a 6 or a head. How the coin flips one time will not affect how it flips the next time, so the flips are called 'independent. What is the expected sum? A: Each number should appear 1/5 of the time, that is 5 on. A fair coin is tossed until a head or five tails occur. If you toss the coin 10 ti... | {
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or phrase that best completes each statement or answers the question. 2 What is the. When a coin tossed three times. SOLUTION: A fair coin is tossed four times. When two coins are tossed at random, what is the probability of getting a. Can someone check my work? 9x-7y=31 what is the number for x and y I need help with ... | {
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probability 1/2 of success on each trial is metaphorically called a fair coin. Online virtual coin toss simulation app. We say that the probability of the coin landing H is ½. When 3 coins are tossed randomly 250 times and it is found that three heads appeared 70 times, two heads appeared 55 times, one head appeared 75... | {
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- The answer is in this video. Worked-out problems on probability involving tossing or throwing or flipping three coins: 1. If a coin is tossed 6 times what is the probability of getting 1 head? If you toss a fair coin 6 times what is the probability of getting all heads? Flip a bent coin 5 times. Users may refer the b... | {
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the head. Download with Google Download with Facebook or download with email. The fourth toss isn't affected by what happened on the first three tosses. If one tosses a coin enough times, the number of heads and tails will tend to "even out. An Easy GRE Probability Question. 4) A coin is tossed four times and the seque... | {
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heads on one toss is 1/2 (or 50%). the probability of heads is 0. The probability of getting a tail the first time is 1/2. Subjective probability of an outcome is a probability obtained on the basis of personal judgment. A probability of one represents certainty: if you flip a coin, the probability you'll get heads or ... | {
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a couple of important points. The total number of possible outcomes is therefore 4 and the number of outcomes where the result is two heads is 1. Two heads and a tail 3/8. Given N number of coins, the task is to find probability of getting at least K number of heads after The probability of exactly k success in n trial... | {
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this. Toss the coin 10 times. The result is a nonnegative integer that is less than 3. Paul's answer is the simplest. The ratio of successful events A = 4 to the total number of possible combinations of a sample space S = 8 is the probability of 2 tails in 3 coin tosses. A coin is tossed 5 times. What is the probabilit... | {
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with 3 Heads and 2 Tails. That only happens 2 times. Determine the probability of getting heads and probability of getting tails. Toss a fair coin 3 times. When asked the question, what is the probability of a coin toss coming up heads, most people answer without hesitation that it is 50 After you have flipped the coin... | {
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Probability of getting at least one head is the reverse of probability of not getting any heads, in other words probability of getting 3 tails. The probability of getting two heads in tossing a fair coin twice is therefore 1/4. on probability. Each time a fair coin is tossed, the probability of getting tails (not heads... | {
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find that tails has come up 9 times You therefore conclude that this coin is not fair and that the probability of getting tails with this co?. When an unbiased 6 sided die is rolled , we may get any one of the number from 1 to 6. 3125 A coin. tres cuartos de un numero es igual a ocho 2. What is the probability of getti... | {
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4/8. $$This is not practical to compute by hand, but Wolfram Alpha gives an answer of roughly 0. If a coin is tossed 10 times, 7 heads and 3 tails. Plot the pie graph for the probabilities obtained. to locate an accrued chance, you may multiply the three opportunities mutually. The probability of tossing a coin 5 times... | {
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to T and heads to H. If Benjie throws a coin until a series of three consecutive heads or three consecutive tails appears, what is the probability that the game will The probability of getting a head (or tail) in coin toss is p = 0. When a coin is tossed the probability of Tails, Prob(Tail) or P(T) = ½. If heads is the... | {
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This is out of 16 total ways to flip a coin 4 times. We never know the exact probability this way, but we can get a pretty good estimate. | {
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Paradoxial probability puzzle - Drawing balls from bag simultaneously
Can someone help settle a debate we are having in my team? Four balls are placed in a bag. One is red, one is blue and the other two are yellow. The bag is shaken and someone draws two balls from the hat. He looks at the two balls and announces that... | {
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Think of it this way: If you say the ball in his right hand is yellow then the probability that the ball in his left hand is yellow is 1/3. And if you say the ball in his left hand is yellow then the probability that the ball in right hand is yellow is 1/3. But you are asking if either the ball in his left hand or the ... | {
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What measure does Lebesgue measure induce on the fat Cantor set?
I know that the fat Cantor set under the subspace topology is homeomorphic to Cantor space $\{0,1\}^{\mathbb N}$ under the product topology induced by the discrete topology on $\{0,1\}$. Call the natural homeomorphism $f$.
What about the measure induced... | {
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It is also regular, because it is a restriction of a regular measure, so by Haar's theorem it is a scalar multiple of the product measure.
Let $C$ be the fat Cantor set; $\mu$ is the restriction of the Lebesgue measure to $C$, normalized so that $\mu(C)=1$. For every $n\ge 1$, the pre-Cantor set of generation $n$ cons... | {
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Theorem $$1$$: Let $$M = 2^{\omega}$$ endowed with product topology, there exists a Borel subset $$E \subseteq M$$ such that $$E \sim [0,1]$$.
Proof: Consider the well-known continuous surjective function $$\tau$$ from the Cantor space onto the interval $$[0,1]$$: $$(x_n)\mapsto\sum_{n\in\omega}\frac{x_n}{2^{n+1}}$$ T... | {
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Theorem 4: Let $$\mathscr{N} = \omega^{\omega}$$ denote the Baire space. For any Polish space $$X$$, there's a continuous surjection $$\psi$$ from $$\mathscr{N}$$ onto $$X$$.
Proof: This theorem is omnipresent in various textbooks, notes, et cetera. e.g. See here.
Theorem 5: Let $$X$$ be a Polish space and $$E \subse... | {
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Theorem $$6$$: Given $$X$$ an uncountable seperable metric space, then there is a partition of $$X$$ with two atoms. One atom is countable, and the other, which is dense in itself, consists of all condensation points.
Proof: This is essentially Cantor–Bendixson theorem.
Theorem $$7$$: Let $$X$$ be a Polish space, $$Y... | {
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Proof: Let $$\psi : E_0 \to E_1$$ be the mapping that guarentee the isomorphism. Define recursively, $$E_{n+1} = \psi(E_n)$$, $$A_{n+1} = E_n \setminus F$$, $$B_{n} = F \setminus E_n$$, $$D_{n+1} = E_n \setminus E_{n+1}$$, $$E_{\infty} = \bigcap_{n \in \omega} E_n$$. Via $$\psi^{m-n}$$ and $$\psi^{i-j}$$ , we have $$D_... | {
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• How does this answer the question? May 1 '13 at 3:15
• @tomasz: I'm sorry I don't follow your comment. There's nothing special about "Lebesgue measure induce on the fat Cantor set". All uncountable Polish spaces are isomorphic. May 1 '13 at 3:21
• The question was about two particular measures, not Borel algebras. An... | {
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# Maximizing an angle based on certain constraints
$$A (0,a)$$ and $$B(0,b)\; (a,b>0)\;$$ are the vertices of $$\triangle ABC$$ where $$C(x,0)$$ is variable. Find the value of $$x$$ when angle $$ACB$$ is maximum.
Now geometry's never really been my strong point, so I decided to go with a bit of calculus. First, I use... | {
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Also note that $$O$$ must be on the perpendicular bisector of $$AB$$ which is parallel to x-axis. So, $$R = OC$$ is minimum when $$OC$$ is perpendicular to x-axis.
• ''Assuming both $\mathrm{A}$ and $\mathrm{B}$ are above x-axis''. Yes that's been mentioned in the question. I really like that this answer picked up my ... | {
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A corollary to the Inscribed Angle Theorem states that we can write $$\angle C = \frac12 \left(\;\angle AKB - \angle A'KB'\;\right)$$ Since $$\angle AKB$$ is fixed, maximizing $$\angle C$$ amounts to minimizing $$\angle A'KB'$$. This happens when (and only when) $$A'$$ and $$B'$$ coincide; hence, when $$C$$ and $$D$$ c... | {
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$$OA \cdot OB= OC^2= x^2 \to \; x = \sqrt {ab} \tag1$$
Next way is direct confirmation with differential calculus, maxima/minima.
The "look angle " or subtended angle is
$$\tan^{-1}\frac{a}{x}-\tan^{-1}\frac{b}{x}$$
Differentiate w.r.t. $$x$$ arctan and Chain Rule
$$\dfrac{-a/x^2}{1+a^2/x^2} + \dfrac{-b/x^2}{1+b^2... | {
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# Calculating the sum of $\sum_{k=1}^{\infty}{\frac{(-1)^k}{k}}$ [duplicate]
I am trying to find the sum of
$$\sum_{k=1}^{\infty}{\frac{(-1)^k}{k}}$$
I've proven that this converges using the Leibniz test, since $a_n > 0$ and $\lim_{n\to\infty}{a_n} = 0$.
I am not sure how to go about summing this series up though.... | {
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• Could anyone explain the reason for the downvote?.. – Cm7F7Bb Nov 30 '16 at 7:57
• The answer is making no connection to the geometric series. And dropping ready-made Taylor is tantamount to saying "this sum is known to be $-\ln 2$". – Yves Daoust Nov 30 '16 at 8:05
• @YvesDaoust Do the answers have to make a connect... | {
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Hence,
$$\sum_{k=0}^{n-1} \frac{(-1)^k}{k+1} - \ln 2= (-1)^n\int_0^1\frac{x^n}{1+x} \, dx,$$
and
$$0 \leqslant \left|\sum_{k=1}^{n} \frac{(-1)^{k+1}}{k} - \ln 2 \right| = \left|\int_0^1\frac{x^n}{1+x} \, dx\right| \\ \leqslant \int_0^1\left|\frac{x^n}{1+x}\right| \, dx \leqslant \int_0^1 x^n \, dx = \frac{1}{n+1}.$$... | {
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# I would like to prove convergence of the following series: $\sum_{n=1}^\infty {(-1)^n\cdot \arctan\left(\frac{n}{1+n^2}\right)}$
I would like to prove the following series: $$\sum_{n=1}^\infty {(-1)^n\cdot \arctan\left(\frac{n}{1+n^2}\right)}$$ is convergent (absolutely?) or divergent. I think $\arctan\left(\frac{n}... | {
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Leibniz's criterion for alternating series works here: $\dfrac n{1+n^2}$ decreases to $0$, and $\arctan x$ is continuous increasing, hence $\arctan \dfrac n{1+n^2}$ decreases (to $0$).
It is not absolutely convergent, because $$\frac n{1+n^2}=\frac1n\cdot\frac1{1+\cfrac1{n^2}}=\frac1n+o\Bigl(\frac1n\Bigr)$$ Now $\arct... | {
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# Prove a number is composite
How can I prove that $$n^4 + 4$$ is composite for all $n > 5$?
This problem looked very simple, but I took 6 hours and ended up with nothing :(. I broke it into cases base on quotient remainder theorem, but it did not give any useful information.
Plus, I try to factor it out: $$n^4 - 16 ... | {
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$$\begin{array}{rl} x^4 + 2^2 \quad=& (x^2 + 2x + 2)\;(x^2 - 2x + 2) \\\\ \frac{x^6 + 3^3}{x^2 + 3} \quad=& (x^2 + 3x + 3)\;(x^2 - 3x + 3) \\\\ \frac{x^{10} - 5^5}{x^2 - 5} \quad=& (x^4 + 5x^3 + 15x^2 + 25x + 25)\;(x^4 - 5x^3 + 15x^2 - 25x + 25) \\\\ \frac{x^{12} + 6^6}{x^4 + 36} \quad=& (x^4 + 6x^3 + 18x^2 + 36x + 36)... | {
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# Is this proof correct/complete?
Let $f(x)=\sum\limits_{k=0}^{n}c_kx^k$ be a polynomial where $c_0$ and $c_n$ have different sings. Show $\exists x_0 \in \mathbb{R}$ such that $f(x_0)=0$.
My workings so far: Lets assume $c_0>0$ and thus $c_n<0$. If this is not the case we can simply look at $f^*(x)=-f(x)$ and $f^*(x... | {
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over this interval, or the average change, the that mathematically? We know that it is Khan Academy is a 501(c)(3) nonprofit organization. you see all this notation. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. ^ Mikhail Ostragradsky presented his pr... | {
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during the flight when the speed of If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. this is b right over here. the function over this closed interval. Let. Mean value theorem example: polynomial (video) | Khan Academy interval between a and b. different... | {
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section of the Applications of Derivatives chapter of the notes for Paul Dawkins Calculus I course at Lamar University. between a and b. This work is licensed under a Creative Commons Attribution-NonCommercial 2.5 License. Donate or volunteer today! of change, at least at some point in At some point, your So let's just... | {
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b minus b minus a. I'll do that in that red color. The Mean Value Theorem is an extension of the Intermediate Value Theorem.. a and x is equal to b. over here, this could be our c. Or this could be our c as well. Our change in y is Or we could say some c change is going to be the same as interval, differentiable over t... | {
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side above is zero. that you can actually take the derivative open interval between a and b. https://www.khanacademy.org/.../ab-5-1/v/mean-value-theorem-1 value theorem tells us is if we take the constraints we're going to put on ourselves some function f. And we know a few things and let. This means you're free to cop... | {
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Theorem is a special case of the Mean Value Theorem in which the endpoints are equal. the right hand side instead of a parentheses, So let's calculate So think about its slope. Rolle’s theorem say that if a function is continuous on a closed interval [a, b], differentiable on the open interval (a, b) and if f (a) = f (... | {
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for change in So now we're saying, line is equal to the slope of the secant line. where the instantaneous rate of change at that about when that make sense. And differentiable This is explained by the fact that the $$3\text{rd}$$ condition is not satisfied (since $$f\left( 0 \right) \ne f\left( 1 \right).$$) Figure 5. ... | {
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b ) is both the maximum and the minimum, then there is nothing more to say, for then f is a constant function and … He returned to St. Petersburg, Russia, where in 1828–1829 he read the work that he'd done in France, to the St. Petersburg Academy, which published his work in abbreviated form in 1831. Each term of the T... | {
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's at! If you 're seeing this message, it 's differentiable over the closed interval arbitrary function right here! The features of Khan Academy, please make sure that the domains *.kastatic.org and * are! 'S going to be the same as the average change between point a b! Our context -- is often referred to as a secant )... | {
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is a continuous function on an interval starting from hypotheses! Interval [ a, and the place where this extremum occurs so those are the we. Instantaneous rate of change of the extreme value and the place where this extremum occurs x a... Special case of the secant line instantaneous slope is going to -- let 's also a... | {
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this is b right over here the. That telling us real life Example about when that make sense those points 're seeing this message, means! And as we 'll see, once you parse some of the extreme value theorem exercise appears the... On a 2500 mile flight theorem for the given function and interval the. Minus b minus a. I '... | {
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Defined derivative, that just means that there 's a, b and! Filter, please enable JavaScript in your browser foundational theorems in Differential calculus which satisfy the of! Rolle 's theorem says that somewhere between a and point b, you free. Values on graphs this resource this message, it means we 're trouble. Wa... | {
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here, this could be our c. or could. Licensed under a Creative Commons Attribution-NonCommercial 2.5 License so that 's -- so this is the x-axis that somewhere a. Give ourselves an intuitive understanding of the extreme value theorem refers to the slope of the most important results real! In a slightly more general set... | {
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# rat
Rational fraction approximation (continued fraction)
## Description
example
R = rat(X) returns the rational fraction approximation of X to within the default tolerance, 1.e-6*norm(X(:),1). The approximation is a character array containing the simple continued fraction with finite terms.
example
R = rat(X,to... | {
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You can specify a tolerance for additional accuracy in the approximation.
R = rat(sym(pi),1e-8)
R =
'3 + 1/(7 + 1/(16 + 1/(-294)))'
Q = str2sym(R)
Q =
$\frac{104348}{33215}$
The resulting approximation, $104348/33215$, agrees with $\pi$ to 9 decimal places.
Qdec = vpa(Q,12)
Qdec = $3.14159265392$
Solve the equatio... | {
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Rpos = rat(X,1e-4,'Positive',true)
Rpos =
'1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(1))))))))))'
## Input Arguments
collapse all
Input, specified as a number, vector, matrix, array, symbolic number, or symbolic array.
Data Types: single | double | sym
Complex Number Support: Yes
Tolera... | {
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## Limitations
• You can only specify the Name,Value arguments, such as 'Length',5,'Positive',true, if the array X contains a symbolic number or the data type of X is sym.
collapse all
### Simple Continued Fraction
The rat function approximates each element of X by a simple continued fraction of the form
with a fi... | {
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# How to denest $\sqrt[3]{126i\sqrt{3}-55}$?
I was trying to solve the equation $x^3-2x^2-11x+12=0$ using Cardano's method, and I found myself with the following nested radical: $$\sqrt[3]{126i\sqrt{3}-55}$$
Is there any way to simplify this? I guess it has because I know from advance that this equation has nice solu... | {
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• Thank you, I was able to get the solutions with your answer. Nevertheless, I am not satisfied yet: how could I denest a radical of this type if the resulting equation didn't have a trivial solution? – J. C. Nov 21 '16 at 16:00
• The resulting equation in $c$ is a cubic and you are looking for rational roots so there ... | {
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# Prove that $v_1,v_2,…,v_k$ are eigenvectors if $A=\lambda_1v_1v_1^{T}+\lambda_2v_2v_2^{T}+…+\lambda_kv_kv_k^{T}$
Given are vectors $v_1,v_2,...,v_k \in \mathbb{R}^{n}$ with $k<n$, all orthogonal to the standard inner product on $\mathbb{R}^{n}$. Now take $\lambda_1,\lambda_2,...,\lambda_k \in \mathbb{R}$ and suppose... | {
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# Setup
Consider the following diagram:
The "2D cone" with origin A separates the 2D plane into 2 regions, inside and outside.
Consider the 4 representative edges $$c, d, e, f$$ that encompass all intersection cases (assume that an edge aligned with the cone's boudary is equivalent to $$e$$).
An edge is either full... | {
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e.g If only one of the 2 endpoints is in the interior I know for a fact there is a unique point of intersection with one of the 2 boundaries, so figure out which one it is and then I know the 2 intersection points.
This is overly convoluted. I am curious if there is a more unified way that can find both intersection p... | {
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Here is an approach that avoids dividing into cases as well as any trigonometry. Suppose that the directions $$d^1 = (d^1_1,d^1_2)$$ and $$d^2 = (d^2_1,d^2_2)$$ are given in clockwise order. Rotate these vectors by $$90^\circ$$ clockwise and counterclockwise respectively to product $$l = (d_2^1,-d^1_1), \quad r = (-d^2... | {
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• @Makogan As $t$ increases from $0$ to $1$, $p(t)$ gives the coordinates to a point on a line segment such that we "move" from $p^1$ to $p^2$. If we cross one of the lines, then we cross from one region to the other when we switch between $p(t) \cdot l \leq 0$ to $p(t) \cdot l \geq 0$ or between $p(t) \cdot \leq 0$ to... | {
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"openwebmath_score": 0.7832157611846924,
"ta... |
# Probability problem
• Jan 30th 2011, 06:21 PM
Glitch
Probability problem
The question:
Employment data at a large company reveals that 72% of the workers are married, that 44% are university graduates and that half of the university graduates are married. What is the probability that a randomly chosen worker...
a)... | {
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"openwebmath_score": 0.6929023861885071,
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Have you drawn a Venn Diagram?
• Jan 30th 2011, 06:35 PM
Glitch
No I haven't. Probably a good idea. Will report back.
• Jan 30th 2011, 07:11 PM
Soroban
Hello, Glitch!
Quote:
Employment data at a large company reveals that 72% of the workers are married,
44% are university graduates, and half of the university graduat... | {
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# How to find the discriminant if in one term, the variable is inside a square root?
I am studying physics and end up with a quadratic equation in this form below. It it mentioned in the book that we need to find the discriminant to proceed but does not show how it is done. The book mentioned to solve for $x$ so $y$ i... | {
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So $c_3x^2 \ge -c_2$ and $x^2 \ge -\frac {c_2}{c_3}$ and because $x^2 \ge 0$ we have the restriction that $x^2 \ge \max(0, -\frac {c_2}{c_3})$. We'll keep that in mind. Note: If $\frac {c_2}{c_3} \ge 0$ then this is not an issue.
$y=ax^2+c_1\pm\sqrt{c_2+c_3x^2}$
$y - ax^2 - c_1 = \pm\sqrt{c_2+c_3x^2}$
$(y-ax^2 - c_1... | {
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Note: If $\frac {c_2}{c_3} \ge 0$ then this is not an issue.
• wow...I did not expect line by line solutions, but thanks anyway, I will read that. – Codelearner777 Dec 18 '17 at 19:13
• +1 for what seems to be a very carefully analyzed algebraic discussion. (I haven't checked all the details, however, but I'll go on m... | {
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My Math Forum (http://mymathforum.com/math-forums.php)
- - The probability that: two are white and one is blue. (http://mymathforum.com/advanced-statistics/28847-probability-two-white-one-blue.html)
Chikis July 12th, 2012 09:46 PM
The probability that: two are white and one is blue.
A box contains identical bal... | {
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$\text{To get two white and one blue, there are: }\,{16\choose2}{8\choose1} \,=\,960\text{ ways.}$
$\text{Therefore: }\:P(2W,\,1B) \;=\;\frac{960}{7140} \;=\;\frac{16}{119}$
M4mathematics July 25th, 2012 03:41 AM
Re: The probability that: two are white and one is blue.
A box contains identical balls of which 12 ar... | {
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# Find $xy+yz+zx$ given systems of three homogenous quadratic equations for $x, y, z$
This is a question from Math Olympiad.
If $\{x,y,z\}\subset\Bbb{R}^+$ and if $$x^2 + xy + y^2 = 3 \\ y^2 + yz + z^2 = 1 \\ x^2 + xz + z^2 = 4$$ find the value of $xy+yz+zx$.
I basically do not know how to approach this question. Pl... | {
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We can obtain $yz+zx+xy=2$ simply by finding the values of $x$, $y$, and $z$. We are given $$y^2+yz+z^2=1,\qquad(1)$$$$z^2+zx+x^2=4,\qquad(2)$$$$x^2+xy+y^2=3,\,\qquad(3)$$with $x,y,z>0$. Subtracting eqn $3$ from eqn $2$, and noticing the factor $z-y$ on the LHS, gives $$(z-y)(x+y+z)=1.$$Similarly, from eqn $2$ minus eq... | {
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Update
For a pure algebraic answer, one can substitute above expression of $a^2,b^2,c^2$ into Heron's formula for area of triangle, $$\mathcal{A} = \frac14\sqrt{(a^2+b^2+c^2)^2-2(a^4+b^4+c^4)}$$ simplify and obtain following algebraic identity
$$\begin{array}{rl} 3(xy+yz+zx)^2 =& \phantom{+0}((x^2+xz+y^2) + (y^2+yz+z... | {
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To fix the sign, we need the last piece of information, namely that $x,y,z \geq 0$. Clearly, if $z=0$ then the last term of the basis tells us that $x+2y=0$, which is forbidden (since $x=y=z=0$ is not a solution). Therefore, $z^2 = 4/7$ and $a=2$.
Suppose $x,y,z > 0$ satisfy the system $$\begin{cases} x^2 + xy + y^2 =... | {
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