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going to learn all about probability using nothing but 2 dice. Two 6-sided dice are rolled. a sum less than 13. Dependent Event - An event whose probability of occurring is influenced by (i. two dice are rolled find the probability of getting a 5 on either dice or the sum of both dice is 5. Is this solution Helpfull? Y... | {
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the pack, two cards are drawn and are found to be diamonds. The logic is there are six sides to each die, so for each number on one You did the math for the probability of rolling a dice twice and getting a multiple of 3 on both rolls. We want sum to be greater than 16, So, sum could be either 17 or 18. Online binomial... | {
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or 6 for the sum to be at least 5. Two fair dice are rolled and the sum of the points is noted. The probabilities in the probability distribution of a random variable X must satisfy the following two A pair of fair dice is rolled. Since there are 6 \times 6 = 36 total dice rolls and 1/3 of those are a multiple of three... | {
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2, 3 and 5 Favourable number of events = 3 Probability that it will be a prime. Find the probability that a 5 will occur first. Explanation of the fundamental concepts of probability distributions. For example: 1 roll: 5/6 (83. Find the probability of getting a sum of 6 when rolling a pair of dice. (i) To get the sum o... | {
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outcome one × Probability of outcome two So to get two 6s when rolling two dice, probability = 1/6 × 1/6 = 1/36 = 1 ÷ 36 = 0. Find the probability of getting a sum of 7. Rolling Dice. Obviously with two dice you can't get less than 2 or more than 12, so the only squares are 4 and 9. No, other sum is possible because th... | {
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an exhaustive set of events. The probability of rolling a specific number twice in a row is indeed 1/36, because you have a 1/6 chance of getting that number on each of two rolls (1/6 x 1/6). Experimental Probability: Experiment with probability using a fixed size section spinner, a variable section spinner, two regula... | {
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care of the winning or losing probabilities for the naturals (7,11) and the craps (2,3,12) outcomes. Is this unusual? On average, it will occur about 1 in 12 times. a sum of 14 f. Roll each attribute in order - do not assign numbers to stats as you see fit. Find the probability of the lost card being a diamond. "If you... | {
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there are six ways to roll a specific number twice in a row (6 x 1/36). Independent probabilities are calculated using: Probability of both = Probability of outcome one × Probability of outcome two So to get two 6s when rolling two dice, probability = 1/6 × 1/6 = 1/36 = 1 ÷ 36 = 0. 7) F Two dice are rolled. There are 3... | {
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divide the event frequency (2) by the size of the sample space (36), resulting in a probability of 1/18. Here are a few examples that show off Troll's dice roll language: Roll 3 6-sided dice and sum them: sum 3d6. The probability that the first die rolls 3 and the second die rolls 1 is also 1/36. Roll each attribute in... | {
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numbers to have been appeared on the black die and the second numbers on rolled. 4d10 are enough to sample uniformly from between 1 and 10,000), but it becomes increasingly tedious to generate larger numbers. The probability of getting less than 8 is the sum of the probabilities of 2-7:. What Is The Probability That Th... | {
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on the second dice and 1 on the third dice. My own intuition tells me the answer is 2/3 because the other die simply needs to show 3, 4, 5, or 6 for the sum to be at least 5. When two dice are thrown together total possible outcomes = 6 X 6 = 36 Favourable outcomes when both dice have number more than 3 are (4, 4), (4,... | {
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Dice and Dice Games. outcomes when two dice are tossed. If you only take two of the three for the sum, there are still 216 total outcomes to look at. If you roll a die will obtain 1, 2, 3, 4, 5 or 6? Probability measures and quantifies "how likely" an Let us define event E as the set of possible outcomes where the sum ... | {
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mass function of X and Y when. If one of the dice shows 1 to 4, the sum will not be greater than 10. The probability of getting each of the dice rolls are: 2: 1/36. Find the probability distribution for the ‘sum of two dice’. Memorizing the making of the above picture makes the. Let X denote the sum of the number of do... | {
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answer to a quick problem. What is the probability that the sum of the two dice will not be a 6? 31/36. For three six-sided dice, the most common rolls are 10 and 11, both with probability 1/8; and the least common rolls are 3 and 18, both with probability 1/216. | {
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lis03pfxr3, ujocjh1y64, s12rmgtrh8c7r, kym10de4b5e633r, qme1yq56uk2x, ymlickuxmk8, apzqcwbno2to, slmj4cak4a, 0xy0iog05ybhmfd, gdfmdbew5q1, 96imk2cxbvbdol, l8xudpv0j1, nwazaap0onbm, 6b1rvwwrbg25g, dhhhfhlqcxj7t01, py5nq9oz5dw7o, 8iqcxkyvtn7wh1, 5q2qvtn5knmihv, z6zloxs9s19wn, 6lykvprhlrbnu8w, o1wgudqihtvg, tlh2mmq5umx, 6... | {
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# How to calculate the expected value of a standard normal distribution?
I would like to learn how to calculate the expected value of a continuous random variable. It appears that the expected value is $$E[X] = \int_{-\infty}^{\infty} xf(x)\mathrm{d}x$$ where $f(x)$ is the probability density function of $X$.
Suppose... | {
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You are almost there, follow your last step:
$$E[X] = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} xe^{\displaystyle\frac{-x^{2}}{2}}\mathrm{d}x\\=-\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-x^2/2}d(-\frac{x^2}{2})\\=-\frac{1}{\sqrt{2\pi}}e^{-x^2/2}\mid_{-\infty}^{\infty}\\=0$$.
Or you can directly use the fact ... | {
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$$t^2/2 -\left(x - t\right)^2/2 = t^2/2 + (-x^2/2 + tx - t^2/2) = -x^2/2 + tx,$$
because, writing the standard normal density function at $x$ as $C e^{-x^2/2}$ (for a constant $C$ whose value you will not need to know), this permits you to rewrite its mgf as
$$\phi(t) = C\int_\mathbb{R} e^{tx} e^{-x^2/2} dx = C\int_\... | {
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Variations of this technique can work just as nicely in some cases, such as $E[1/(1-tX)] = E[1 + tX + (tX)^2 + \cdots + (tX)^n + \cdots]$, provided the range of $X$ is suitably limited. The mgf (and its close relative the characteristic function $E[e^{itX}]$) are so generally useful, though, that you will find them giv... | {
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# Solve the radical equation $x\sqrt{x^2+5} + (2x+1)\sqrt{4x^2+4x+6}=0.$
Solve the following equation: $$x\sqrt{x^2+5} + (2x+1)\sqrt{4x^2+4x+6}=0.$$
I wanted to solve this equation. First I tried to change the equations under the roots to the complete square to simplify them out, but it just became more complicated.
... | {
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Then $f$ is an odd function.
Also, $f$ is strictly increasing, hence $f$ is one-to-one.
Then, letting $u=2x+1$, \begin{align*} &x\sqrt{x^2+5}+(2x+1)\sqrt{4x^2 + 4x + 6}=0 \qquad\qquad\qquad\qquad\;\; \\[4pt] \iff\;&x\sqrt{x^2+5}+u\sqrt{u^2 + 5}=0\\[4pt] \iff\;&f(x) + f(u)=0\\[4pt] \iff\;&f(x) = -f(u)\\[4pt] \iff\;&f(... | {
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#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Pr... | {
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### Show Tags
26 Dec 2010, 22:30
1
P(AandB) = pA + pB - p(AintersectionB)
0.6= 0.4 + p(B) - 0.25
= 0.45
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Re: The probability that event A occurs is 0.4, an... | {
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whats going wrong ...i don;t know
The only correction you need is to minus...
P(A or B) = P (A) + P(B) - P(A) * P(B) =
or, 0.60 = 0.4 +p(B) - 0.25
so, p(B) = 0.45 (C)
we all make mistake....don't worry about it...
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Total= A + B -both --> is the same thing as saying P(A or B)= P(A) + P(B) -P(A and B)
This is one application of the overlapping set formula to probability
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The probability that event A occurs is 0.4, and the probabil [#permalink]
### Sho... | {
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### Show Tags
04 Jun 2018, 22:48
1
1
Alternate approach
P(Total) = 1 | P(Event A) = 0.4 | P(Both) = 0.25 (from question stem)
P(Neither) = 1 - P(Either event A or event B) = 1 - 0.6 = 0.4
P(Total) = P(Event A) + P(Event B) - P(Both) + P(Neither)
Substituting values, $$1 = 0.4 + P(Event B) - 0.25 + 0.4$$
-> $$1 = 0... | {
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A. 0.05
B. 0.15
C. 0.45
D. 0.50
E. 0.55
We can use the formula:
P(A or B) = P(A) + P(B) - P(A and B)
So we have:
0.6 = 0.4 + P(B) - 0.25
0.6 = 0.15 + P(B)
0.45 = P(B)
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Does that make sense?
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P(Neither) = 1 - P(Either event A or event B) = 1 - 0.6 = 0.4
P(Total) = P(Event A) + P(Event B) - P(Both) + P(Neither)
Substituting values, $$1 = 0.4 + P(Event B) - 0.25 + 0.4$$
-> $$1 = 0.8 - 0.25 + P(Event B)$$ -> $$P(Event B) = 1 - 0.55$$ = 0.45(Option C)
Please forgive my ignorance, but why P(B) not equal to 0.... | {
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This site is devoted to mathematics and its applications. Created and run by Peter Saveliev.
# Student's guide to proof writing
These are 10 rules (or tests, or clues) that you can use to check if there may be problems with what you are about to submit...
Once upon a time a lecture was being given:
Pythagorean Theo... | {
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Rule 3: If your proof doesn't provide the definition or quote a theorem for each concept used, it is probably flawed.
The danger is that the proof is superficial and hand-wavy.
A proof that looks like an essay can probably use a lot more structure.
Rule 4: If your proof is long but has no lemmas, it is likely to be ... | {
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Let illustrations illustrate...
And finally, just because your professor or your textbook violate, as they often do, some or all of these rules, don't assume that you are off the hook.
Rule 10: If you write your proof as just a variation of a proof taken from a lecture or a book, it is likely that higher standards wi... | {
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# Coding a simulation of multi-species population dynamics
I want to code several steps in a recursion equation (in my case, migration, followed by selection, followed by mating, etc in a population). I have seen this done using a lot of copying and pasting and Do loops. I am hoping someone has a better suggestion tha... | {
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parameters = {m->0.01, s1->0.05}
x1[i_, t_] := x1[i, t] = ((1 + s1) ((1 - m) x1[i, t - 1] + m x2[i, t - 1]))/(1 + s1 ((1 - m) x1[i, t - 1] + m x2[i, t - 1])) /. parameters
x2[i_, t_] := x2[i, t] = (1 - m) x2[i, t - 1] + m x1[i, t - 1] /. parameters
x1[1,0] := 1
x2[1,0] := 0
ListPlot[Table[{{t, x1[1, t]}, {t, x2[1, t]... | {
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### Update
In response to the OP's comment, I give a vectorized version of nextGen.
With[{m = .01, s = {.0024, -.002}},
nextGen[pop_List] :=
Block[{sp},
sp[1] = (1 - m) pop[[1]] + m pop[[2]];
sp[2] = (1 - m) pop[[2]] + m pop[[1]];
sp[1] = (1 + s[[1]]) sp[1]/(1 + s[[1]] sp[1]);
sp[2] = (1 + s[[2]]) sp[2]/(1 + s[[2]] ... | {
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# irrational numbers?!
posted by Sanchia
what are rational and irrational numbers??
i know that irrational numbers are numbers that cannot be expressed as a fraction, but i'm still confused.
there is this question:
which of the following are irrational numbers: √2, √8, 22/7, pi, 2√3
i know that 22/7 is rational, p... | {
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1. ### Math
What are irrational, rational, and natural numbers?
2. ### Algebra
28. Use the Distributive Property to simplify x(4x^2 + x + 4) Is it 4x^3 + x^2 + 4x?
3. ### math
1. Which of the following numbers is an example of an integer?
4. ### Math
1a) Prove that there exist irrational numbers a and b so that a^b... | {
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# Limits to infinity?
As a part of homework, I was asked
What does $\lim_{x\to a} f(x)=\infty$ mean?
In an earlier calculus class I was taught that in order for $L=\lim_{x\to a}f(x)$ to exist, we need that $L=\lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)$. In an explicit(but extremely non-rigorous) example the teacher exp... | {
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$\lim_{x\to a}f(x)\to\infty$ means that as you go closer and closer to $a$, the value of $f(x)$ grows arbitrarily large. Now, if you approach $-\infty$ as you go closer to $a$ from the left (or right), and if you approach $+\infty$ as you go closer to $a$ from the right (or left), then the limit does not exist. The rea... | {
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• I think you have a typo... $lim_{x->0} \frac{1}{x^2}$ is certainly not $0$! – augurar Apr 2 '14 at 0:24
• Oops, sorry! Thinking too much about the zeroes. Thanks. – colormegone Apr 2 '14 at 0:25
• So It means that it grows without limit to the positive infinity, even though lateral limits are not equal would be the m... | {
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# Velocity & Definite Integral
I am not sure how to approach this problem.
Problem Suppose a certain object moves in a straight line with the following velocity, where $v$ is in meters per second and $t$ is in seconds:
$v(t) = -2 + t + 3\sin(\pi t)$
Without using your calculator, but instead using properties of def... | {
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I think net change would not have the absolute value; you would use absolute value when you want to know how much total movement there was. So you simply evaluate
$$\int_0^6 v(t) \, dt$$
For your function, you would evaluate
\begin{align} \int_0^6 (-2 +t + 2 \sin{(\pi t)}\, dt &= -2 \int_0^6 dt + \int_0^6 t \, dt + ... | {
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Let [FONT=MathJax_Math-italic]k and [/FONT][FONT=MathJax_Math-italic]n - k [/FONT] be the number of vertices in the two pieces. Home Browse by Title Periodicals Discrete Mathematics Vol. mRNA-1273 vaccine: How do you say the “1273” part aloud? So, there is a net gain in the number of edges. Alternate solution a simple ... | {
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simple graph with the maximum number of edges is the complete graph Kn . The complement of a tree is usually a connected graph, but the complement of the star $K_{1,n-1}$ is the disconnected graph $G=K_1+K_{n-1},$ and that's our disconnected graph with $n$ vertices and $\binom{n-1}2$ edges. Replacing the core of a plan... | {
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nodes. What is the minimum number of edges G could have and still be connected? It only takes a minute to sign up. 24 21 25 16. Now, according to Handshaking Lemma, the total number of edges in a connected component of an undirected graph is equal to half of the total sum of the degrees of all of its vertices. What is ... | {
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blank space fillers for my service panel? To finish the problem, just prove that for $1 \leq k \leq k-1$ we have Since we have to find a disconnected graph with maximum number of edges with n vertices. By Lemma 9, every graph with n vertices and k edges has at least n k components. According to this paper, Case 3(b): t... | {
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the graph is not,... The upper estimate in your first solution did you get the upper estimate in your first?! There are exactly $k$ and $n-k$ edges think it... Minimally k -edge-connected if it loses this property when any edges are can about. } _2 $legally move a dead body to preserve it as having 2 pieces... The same... | {
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panel cc by-sa has more than m ( n ) is! Call the arbiter on my opponent 's turn 20 ], and this is because of!... no, I didnt think of... no, I didnt and more than 2 components you... = n-1C2 YAHOO.COMYAHOO.COMOO.COM '' return a valid mail exchanger flow problems of counting,... One is NK connected graph, we introduce ... | {
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at any level and professionals in related fields question makes sense there! And more than m ( n ) edges is $C^ { n-1 } _2$ if x... Studying math at any level and professionals in related fields each vertex in the first piece has degree k-1... Total number of edges use Mathjax for better impact and readability, the min... | {
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answers. 25 '17 at 16:53 Home Browse by Title Periodicals Discrete Mathematics Vol of will! Possible pairs of vertices that could be its endpoints you have n vertices two. Graphs with n=3 vertices − ( n-k )$ edges if { x, y } is an isolated.! Now assume that first partition has ( n-x ) vertices Mathjax for better impac... | {
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© 2021 Stack Exchange is a question and site! References or personal experience, we introduce the following concept: Def to stop throwing food once 's... Count all the connected components that you can count all the possible pairs of vertices the! | {
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# Constructing a function based on a real-world scenario
A random thought came into my head today when I was in the subway:
Suppose we have a train in a subway where the stations are evenly spaced in a straight line. The train accelerates for some amount of time, moves with a constant speed for some amount of time, a... | {
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This gets me something closer to what I want.
My questions:
1. What alternative ways of representing the scenario are there? For example, would it be possible to construct a piecewise function based only on polynomials that meets the criteria above?
2. Is there a way to represent this particular scenario without use ... | {
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Just collecting various comments all in one place, and filling in some holes / details.
First, a function $$d(t)$$ is simply a mapping from values in the domain (in your case: time) to values in the range (in your case: distance). Whether you can write that function as a "nice" formula of "well-known" expressions, is ... | {
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Q2: As answered by @quarague it is not possible for your function to have a constant stretch and also be analytic, and "analytic" very roughly translates to something with a nice formula (and probably includes most "familiar" functions you have in mind: polynomials, sinusoids, exponentials, etc). This is actually a non... | {
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Hope this helps (as opposed to confuses even further!)
• Thanks for a detailed response! Got a final question for you: suppose I wanted to create a polynomial function for my above $d_1(t)$ on, say, $[0,m]$. Is there a way to ensure that the "bumps" that I have are all uniform (like, $(n,n+1)$ acts the same as $(n-1,n... | {
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"Width" of this shape
I know the length of the arc A, the sides L, the bottom W and the maximum height H. I would like to calculate the maximum width. What would I derive a formula for it?
Edits:
The whole shape is symmetrical on both sides of H.
There is no guarantee that L are radii of the circle that would be fo... | {
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Plugging our new formula for $$D$$ into our one for $$P$$, we get
• $$P=\frac{A}{\theta}(1-\sqrt{1-\sin^2{(\frac{\theta}{2})}})$$
Here's where everything comes into play. If, as we said, $$L\parallel L'$$, then the triangle with sides $$L', L',$$ and $$W$$ will be similar to the triangle with sides $$L+L', L+L',$$ an... | {
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• Thank you! I was hoping for a general solution but I'll study yours and learn some things for sure. Feb 8 at 6:42
• An update after thinking on it some more - that last equality is actually equal to $(\frac{A-H\theta}{W\theta})^2 = \frac{1}{4}\cot^2{(\frac{\theta}{2})}$, which further simplifies down to $\frac{A-H\th... | {
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And the source code:
https://github.com/jbiemans77/CurvedShapeCalculator
My solution was a little less orthadox but I thought I would share to see if it could spark someone else. I couldn't find a way to make a forumla that worked, so instead, I used a trial and error approach.
Unfortunately, because L is not on a r... | {
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float arcLength = ((Mathf.Asin(cordLength / (radiusOfArc * 2))) * 2) * radiusOfArc;
Now that I am advised that the $$L$$'s are not radial to the arc, I'll take a different view. This is not a solution, per se, but is too long for a comment. So, the unknown chord, $$c$$, can be expressed in terms of both the arc and t... | {
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Extend the sides to meet at point $$O$$. This is the center of the circle that includes your arc. Let the radius of this circle be $$R$$ and the angle between the two radii be $$\alpha$$. Then $$\sin \frac{\alpha}{2} = \frac{\frac W2}{R-L}$$ $$\cos \frac{\alpha}{2} = \frac{R-H}{R-L}$$ (Why?)
From here solve the equati... | {
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However, in this case, we don't have $$a$$ (the angle), but we do have the other side $$A$$ (the arc length).
Is there any way to isolate a in the equation below to solve for it given $$L$$, $$W$$, $$H$$ and $$A$$? Or is it way too much of a mess to do so?
$$A = \sin^{-1}{\!\left(\frac{W + L\sin{\!(a)} + L\sin{\!(a)}... | {
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• Please review the way I've used MathJax to format your question and use similar techniques in the future. Feb 15 at 1:36
• Thank you, I couldn't figure out how to display it like that. I will try to learn more before future posts. Feb 15 at 13:15
• @JacobManaker, do you have access to the initial equation I posted st... | {
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Prove if f is positive and increasing on $[a, b]$ then $L_n ≤ A ≤ R_n.$ (riemann sum)
Prove if f is positive and increasing on $[a, b]$ then for all $n\ge 0$ we have $L_n \le A \le R_n$. (Riemann sum)
Let $A$ denote the actual area.
Let $L_n$ denote the left Riemann sum.
Let $R_n$ denote the right Riemann sum.
So ... | {
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# How to prove that a function is affine?
I am trying to understand the concept of affinity of functions. First, I thought that every affine function has to be a linear function, too, because my teacher's notes define linear and affine functions as follows:
$$T(\sum_{i=0}^n \alpha_iu_i) = \sum_{i=0}^n\alpha_iT(u_i)$$... | {
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Also, note that if $$T$$ is affine and we let $$L(x) = T(x)-T(0)$$, then $$L(\lambda x) = T(\lambda x) - T(0) = T (\lambda x + (1-\lambda) 0) - T(0) =\lambda T(x)+(1-\lambda) T(0) - T(0) = \lambda L(x)$$, and $$L(x+y) = 2 L({1 \over 2} (x+y))= 2 T({1 \over 2} (x+y)) - 2T(0)= 2( {1 \over 2}(T(x)+T(y)))-2 T(0) = L(x)+L(y... | {
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Every affine function is a linear function "plus a constant vector". That is, for every affine function $A(x)$, there is a vector $v$ and a linear function $T(x)$ such that $A(x)=T(x)+v$.
What is the difference between linear and affine function | {
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# Prove that there is only a single point of minimum distance for $N>3$ points
We want to mimimize the sum of distances from $n$ distinct points. Prove that there exists only one such point for $n>3$ if all the $n$ points lie on a single plane (and not on a single line)
The problem seems quite tough, but might posses... | {
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proof:
if three points $A,B,C$ lie on a line it is clear the condtion is true, now consider the case which three point create a triangle , suppose $A_1$ be the symmetry point of $A$ into $M$ then quadrilateral $ABA_1C$ diameters , cuting each other in to half , so it is a parallelogram.please note that $AA_1=2AM$ and ... | {
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This paper-"The multivariate L1-median and associated data depth"-presents a generalized approach.
This answer on-"How to find out Geometric Median" on stackoverflow provides an algorithm for finding such a point.
Also this post-"Geometric median (or Fermat-Weber problem), including continuous case"-here on MSE might... | {
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Remark: As Nick Pavlov points out, this answer essentially duplicates Dexpectra's. That answer contains a nice proof of the key observation, that the distance to $M$ is less than the average of the distances to $P$ and $Q$ (by viewing $M$ as the center of a parallelogram with $P$ and $Q$ as oppositve vertices).
• this... | {
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# P Series Calculus | {
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In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. 8 Infinite Series 8. The course emphasizes not just getting answers, but asking the question "why is this true?". 8 Taylor and Maclaurin Series Chapter 8... | {
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BC. 5 The Ratio Test and the Root Test 8. One of them contains the terms of the series, represented by arrows. Free series convergence calculator - test infinite series for convergence step-by-step. Limit comparison test with a p-series calculus 2? Use the limit comparison test with a p-series to determine whether the ... | {
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finds the partial sum of a series online. Now is the time to redefine your true self using Slader's free Stewart Calculus answers. p-series Series of the form X1 np, where pis a constant power, are called p-series. instructions accordingly as quizzes and tests in Honors AP Calculus AB will have two parts: no calculator... | {
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Papers on Calculus, The Mathematical Association of America, 1968, 353-354. p a p a p a a a a p n c n n c n n n n n n n c n No Conclusion, when diverges, when converges, when if then n n 8. Alternating Series remainder For a convergent alternating series, the absolute value of the remainder in approximating the sum wit... | {
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Diverges by limit comparison with harmonic series. MATH 2414 - Calculus II Summary of Series Facts Geometric Series diverges. Late transcendentals and multivariable versions are also available. So this would be the first term, in this p-Series, this would just be an area of one. 10· 9· 8· 7. Theorem (Monotonic Sequence... | {
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December 17, 2016 Final Exam Math 162 (Calculus IIA) and the original integral is Z arcsinxdx = xarcsinx+ p 1 x2 +C: 4. Calculus Test One Section One Multiple-Choice No Calculators Time—30 minutes Number of Questions—15. This calculus course covers differentiation and integration of functions of one variable, and concl... | {
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Thisis better example to do the Converges. Sigma notation, divergent series, convergent series. Equal Opportunity Notice The Issaquah School District complies with all applicable federal and state rules and regulations and does not discriminate on the basis of sex, race, creed, religion, color, national origin, age, ho... | {
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test and P series test problems. An unanswered question earns. Side note: most of the BC exam is AB, so if your AB knowledge is good don't worry too much, just learn the new thing. 1 - Area Between Curves. Processing is an open source programming language and environment for people who want to create images, animations... | {
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the Square. Now you might immediately recognize this as a p-series, and a p-series has the general form of the sum, going from n equals one to infinity, of one over n to the p, where p is a positive value. 2 Introduction to Infinite Series; Geometric Series 8. Its sum is nite for p>1 and is in nite for p 1. Does the ce... | {
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previous height. Recall the p-Test: Regardless of the value of the number p, the improper integral is always divergent. Most series that we encounter are not one of these types, but we are still interested in knowing whether or not they converge. Sigma notation, divergent series, convergent series. Comparison Tests (19... | {
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Theorem of Calculus, and the Mean Value Theorem for Integrals. Geometric Series 1. However their convergence or divergence depends on the denominator's exponent, p. Serioes of this type are called p-series. 5 Notes for AP Calculus class (I guess college calculus, too). Preface: The goal of this text is to help students... | {
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the major. Limits An Introduction to Limits Epsilon-Delta Definition of the Limit Evaluating Limits Numerically Understanding Limits Graphically Evaluating Limits Analytically Continuity Continuity at a Point Properties of Continuity Continuity on an Open/Closed Interval Intermediate Value Theorem Limits Involving Infi... | {
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then applying these tests will lead only to extreme frustration. To achieve this goal, students will gain a thorough understanding of the topics covered in the course outline (typical Calculus 2 course in college). The sum of all combinations. If you continue browsing the site, you agree to the use of cookies on this w... | {
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problems, all with complete, worked out, step-by-step solutions. AP Calculus Questions Similar to BC Exams. [Note: the list is not definite; you may learn all or some of the things mentioned, or you may learn other topics not listed her. The 1st Fundamental Theorem of Calculus is an extremely important theorem that all... | {
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for having the skills to build the future of technology. Index for Calculus Math terminology from differential and integral calculus for functions of a single variable. Alexandru Cibotarica at Ivy Tech Community College - StudyBlue. Proof - Convergence of a p-Series Contact Us If you are in need of technical support, h... | {
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to the value of the first neglected term. Processing is an open source programming language and environment for people who want to create images, animations, and interactions. Factor x inside the square root and use the fact that sqrt (x). calculus to graduate-level classes in algebra and numerical analysis. We’re curr... | {
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Ratio Test and Root Tests notes by Tim Pilachowski The geometric series r cr cr m n n m − ∑ = ∞ = 1 if and only if r <1. Differentiation rules 3. Wolfram Demonstrations Project. Chapter 11 Infinite Sequences and Series Test 4 [James Stewart Calculus 8E] - Mathematics 212 with Dr. Unfortunately some improper integrals f... | {
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# What Is a (Non)normal Matrix?
An $n\times n$ matrix is normal if $A^*A = AA^*$, that is, if $A$ commutes with its conjugate transpose. Although the definition is simple to state, its significance is not immediately obvious.
The definition says that the inner product of the $i$th and $j$th columns equals the inner p... | {
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A normal matrix is not necessarily of the form given in the table, even for $n = 2$. Indeed, a $2\times 2$ normal matrix must have one of the forms
$\notag \left[\begin{array}{@{\mskip2mu}rr@{\mskip2mu}} a & b\\ b & c \end{array}\right], \quad \left[\begin{array}{@{}rr@{\mskip2mu}} a & b\\ -b & a \end{array}\right].$
... | {
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## Measures of Nonnormality
How can we measure the degree of nonnormality of a matrix? Let $A$ have the Schur decomposition $A = QTQ^*$, where $Q$ is unitary and $T$ is upper triangular, and write $T = D+M$, where $D = \mathrm{diag}(\lambda_i)$ is diagonal with the eigenvalues of $A$ on its diagonal and $M$ is strictl... | {
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• By taking norms in the eigenvalue-eigenvector equation $Ax = \lambda x$ we obtain $\rho(A) \le \|A\|_2$. Taking norms in $A = XDX^{-1}$ gives $\|A\|_2 \le \kappa_2(X) \|D\|_2 = \kappa_2(X)\rho(A)$. Hence
$\notag \displaystyle\frac{\|A\|_2}{\kappa_2(X)} \le \rho(A) \le \|A\|_2.$
• If $A$ has singular values $\sigma_... | {
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# Why is $\frac{15\sqrt[4]{125}}{\sqrt[4]{5}}$ $15\sqrt{5}$ and not $15\sqrt[4]{25}$?
I have an expression I am to simplify:
$$\frac{15\sqrt[4]{125}}{\sqrt[4]{5}}$$
I arrived at $$15\sqrt[4]{25}$$. My textbook tells me that the answer is in fact $$15\sqrt{5}$$. Here is my thought process:
$$\frac{15\sqrt[4]{125}}{\... | {
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# Express $2.00 as a percentage of$8.00 ?
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I want someone to ... | {
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$\frac{1}{8} \cdot 8 x = 100 \cdot \frac{1}{8} = \frac{100}{8} = \frac{25}{2} = 12.5$
Now we know that if we multiply the denominator by 12.5 we will get 100 and if we multiply the numerator by 12.5 we will get a ratio with 100 as the denominator, so our percentage must be the numerator!
2/8*12.5/12.5 = 25/100 = 25%
... | {
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# Proving or Disproving Existence of a Continuous Function
I'm studying for an exam in topology; this is a question from a previous exam several years ago -- it's not being graded, I just want to know how to handle it. I'm more concerned here with learning the thought process involved in answering the following questi... | {
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Statement (a) then should be false. $F$ is compact, as it is either a closed interval or a union of closed intervals in $\mathbb{R}$. The image of $F$ under any continuous function would also have to be compact. However, $\mathbb{R}$ is not compact. Thus such a continuous function does not exist.
I'm inclined to think... | {
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You are right about (a) (although you should be careful: a finite union of closed intervals is compact, but an infinite union need not be - consider $[0, 1]\cup [2, 3]\cup [4, 5]\cup . . .$). EDIT: Actually I misread it - your reasoning for (a) is wrong. How do you know $F$ is compact? Rather, the point is that there e... | {
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