text stringlengths 1 2.12k | source dict |
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then the sum $(1)$ is approximated by putting $x=1$ in $(2)$ and multiplying by $n!$:
$$n!e=\sum_{r\ge 0}\frac{n!}{r!}\, .\tag{3}$$
Now, all that remains is to show is that the difference between $(3)$ and $(1)$ is less than $1$ (hint: perform a term by term comparison with a geometric series) and we have
$$\lfloor ... | {
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# Arc Length Confusion
1. Jan 18, 2010
### Char. Limit
Why is arc length of a function $$f(x)$$ from a to b defined as $$\int_a^b \sqrt{1+(f'(x))^2} dx$$?
Where they get the idea of squaring the derivative, adding 1, taking the square root, and then integrating it is beyond me.
2. Jan 18, 2010
### tiny-tim
They ... | {
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Thanks for the help, both of you.
8. Jan 24, 2010
### evagelos
Certainly in the limit $$\sum\sqrt{1+ \left(\frac{\Delta y}{\Delta x}\right)^2}\Delta x$$ does not become $$\int_a^b \sqrt{1+ \left(\frac{dy}{dx}\right)^2} dx$$ according to the definition of the Rieman integral.
BUT ,by the mean value theorem we have t... | {
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For n any non-negative integer evaluate the integral :
$\int x^n \ln (x) dx$
Attempt to solution:
use integration by parts
$dv=x^n$
$v=\frac{x^{n-1}}{n-1}$
$u=\ln (x)$
$du=1/x$
$\int u dv=\ln (x) \frac{x^{n-1}}{n-1}-\int\frac{x^{n-1}}{n-1} \frac{1}{x}$
I'm stuck here how do l further simplify this thing ?
2. O... | {
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At this site, you must use [math ] [/math ] for LaTeX.
I would do it like this . . .
. . $\begin{array}{ccccccc}u &=& \ln x & & dv &=& x^n\,dx \\ du &=& \frac{dx}{x} & & v &=& \frac{x^{n+1}}{n+1} \end{array}$
Then we have: . $\frac{1}{n+1}\,x^{n+1}\ln x \;-\; \frac{1}{n+1}\int x^n\,dx \;\;=\;\;\frac{1}{n+1}\,x^{n+1}... | {
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# Find all solutions to $|x^2-2|x||=1$
Firstly, we have that $$\left\{ \begin{array}{rcr} |x| & = & x, \ \text{if} \ x\geq 0 \\ |x| & = & -x, \ \text{if} \ x<0 \\ \end{array} \right.$$
So, this means that $$\left\{ \begin{array}{rcr} |x^2-2x| & = & 1, \ \text{if} \ x\geq 0 \\ |x^2+2x| & = & 1, \ \text{if} \ x<0 \\ \e... | {
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but $1-\sqrt{2} \leq 0$
$x_{3} =-1+\sqrt{2}$ is not correct solution because of same reason in second equation
As G.H.lee already pointed out, you did casework $x\geq 0$ and $x<0$ to simplify the expression but you completely disregarded it later. A correct way would be something like this:
$$|x^2-2|x||=1\implies \b... | {
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• Great explanation thank you. However drawing graphs and such in a problem that I'm only given 5 min to solve is not an option. – Parseval May 14 '17 at 12:22
• @Parseval, you are welcome. Graphing this doesn't take more than half a minute, once you know "the rules". Absolute value is all about reflecting: if you can ... | {
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Prove that $\sum_{k=1}^{n} \frac{1}{k}>\ln(n+1)$ for all $n\geq1$
Prove that $$\sum_{k=1}^{n} \frac{1}{k}>\ln(n+1)$$ for all $n\geq1$
I am looking for a clear solution to this problem. I've considering trying to prove it by contradiction by starting off assuming that it's not true for all n's, but I'm not sure if thi... | {
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# Alternating sign odd number generating function.
I have a sequence that I'm trying to find both an ordinary generating function for as well as a closed form without a floor function. The sequence
$$\{1,1,-1,3,-3,5,-5,7,-7,9,-9,11,-11,\}$$
is recursively generated by the formula
$$a_0=1$$ $$a_n=a_{n-1}+(-1)^{n-1}2... | {
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as noted by AccidentalFourierTransform in the comments. Thus, you have
\begin{align*} A(x)&=\frac1{1-x}\left(1-\frac{2x}{1+x}+\frac{2x}{(1+x)^2}\right)\\ &=\frac{(1+x)^2-2x(1+x)+2x}{(1-x)(1+x)^2}\\ &=\frac{1+2x-x^2}{(1-x)(1+x)^2}\;. \end{align*}
Just for fun, I checked this by finding $A(x)$ in a completely different... | {
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so
$$a_n=\frac12+\frac{3(-1)^n}2-(-1)^n(n+1)=\frac{1+(-1)^n}2+(-1)^{n+1}n\;.$$
• NIce. I just got the generating function right before as I mentioned in my comments. Now, this is the part that is getting me...how do I find a formula for the coefficients? I'm looking for something without floor function... – Iceman Ma... | {
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8. The Ehrenfest Chains
Basic Theory
The Ehrenfest chains, named for Paul Ehrenfest, are simple, discrete models for the exchange of gas molecules between two containers. However, they can be formulated as simple ball and urn models; the balls correspond to the molecules and the urns to the two containers. Thus, supp... | {
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Suppose now that we modify the basic Ehrenfest model as follows: at each discrete time, independently of the past, we select a ball at random and a urn at random. We then put the chosen ball in the chosen urn.
$$\bs{X}$$ is a discrete-time Markov chain on $$S$$ with the transition probability matrix $$Q$$ given by $Q(... | {
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Note that returns to a state can only occur at even times, so the chain has period 2. The form of $$P^2$$ follows from the formula for $$P$$ above.
The modified Ehrenfest chain is aperiodic.
Proof:
Note that $$P(x, x) \gt 0$$ for each $$x \in S$$.
Invariant and Limiting Distributions
For the basic and modified Ehr... | {
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For the basic Ehrenfest chain, the limiting behavior of the chain is as follows:
1. $$P^{2 n}(x, y) \to \binom{m}{y} \left(\frac{1}{2}\right)^{m-1}$$ as $$n \to \infty$$ if $$x, \, y \in S$$ have the same parity (both even or both odd). The limit is 0 otherwise.
2. $$P^{2 n+1}(x, y) \to \binom{m}{y} \left(\frac{1}{2}\... | {
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Reversibility
The basic and modified Ehrenfest chains are reversible.
Proof:
Let $$g(x) = \binom{m}{x}$$ for $$x \in S$$. The crucial observations are $$g(x) P(x, y) = g(y) P(y, x)$$ and $$g(x) Q(x, y) = g(y) Q(y, x)$$ for all $$x, \, y \in S$$. For the basic chain, if $$x \in S$$ then \begin{align*} g(x) P(x, x - 1... | {
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1. Compute the probability density function, mean and variance of $$X_1$$.
2. Compute the probability density function, mean and variance of $$X_2$$.
3. Compute the probability density function, mean and variance of $$X_3$$.
4. Sketch the initial probability density function and the probability density functions in par... | {
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# a question about double integral
Let $a,b$ be positive real numbers, and let $R$ be the region in $\Bbb R^2$ bounded by $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. Calculate the integral $$\int\int_R\left(1-\frac{x^2}{a^2}-\frac{y^2}{b^2}\right)^{3/2}dx\,dy$$
my question is I don't know anything about $R$, the function $\... | {
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• If we make the change of variables $x=aX,y=bY$, then we get the circle centered at $(X,Y)=(0,0)$ and radius $1$ \begin{equation*}X^{2}+Y^{2}=1.\tag{4}\end{equation*}The region $R$ becomes the unit circle \begin{equation*}C=\left\{ (X,Y)\in\mathbb{R}^{2}:0\le X^2+Y^2\le 1\right\}\tag{5}\end{equation*} The Jacobian det... | {
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-
@amWhy: Hello my friend Amy. Yes. As you know, I have to correct my claims in my recent article. Mathematics is a beautiful cruel one. – Babak S. Jan 20 '14 at 15:45
It sounds like you're just a bit confused about notation. $R$ is simply the name of the region. The notation
$$\iint\limits_{R} f(x,y) \, dA$$
simpl... | {
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# Connection of Gradient, Jacobian and Hessian matrix in Newton method
Suppose $f: \mathbb{R^n} \to \mathbb{R}$, the gradient of $f(\mathbf{x})$ is $$\mathop{\nabla} f(\mathbf{x}) = \begin{bmatrix} \frac{\partial{f}}{\partial{x_1}} \\ \vdots \\ \frac{\partial{f}}{\partial{x_n}} \end{bmatrix}$$
The Jacobian matrix of ... | {
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## 1 Answer
From the first sentence of the Wikipedia article you link - "In optimization, Newton's method is applied to the derivative $f′$ of a twice-differentiable function $f$ to find the roots of the derivative." In other words, the Hessian is symmetric.
Newton's method can also be applied in a more general setti... | {
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# Product of Negligible and Non-Negligible Functions
I know that the product of two negligible functions will always be negligible, but I'm wondering if it's possible for the product of two non-negligible functions to be a negligible function?
I'm wondering if it's possible for the product of two non-negligible funct... | {
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# Lambert- W -Function calculation?
I have an equation of the form:
$$n \log n = x$$
Upon searching I came across the term "Lambert- W -Function" but couldn't find a proper method for evaluation, and neither any computer code for it's evaluation.
Any ideas as to how I can evaluate?
• Do you mean an evaluation of t... | {
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If I may underline one thing which is really nice : all derivatives od $W(a)$ express as functions of $a$ and $W(a)$ itself and this is extremely convenient.
You could be interested by http://people.sc.fsu.edu/~jburkardt/cpp_src/toms443/toms443.html where the source code is available.
Although an old post, I'm surpri... | {
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\begin{align*} n \ln (n) =x \quad & \Rightarrow \quad \ln n^n=x \\ & \Rightarrow \quad n^n=e^x \\ & \Rightarrow \quad n=e^{x/n} \\ & \Rightarrow \quad n \times \frac{x}{n}=\frac{x}{n}e^{x/n} \\ & \Rightarrow \quad x=\frac{x}{n}e^{x/n} \\ & \Rightarrow \quad W(x)=W \left(\frac{x}{n}e^{x/n} \right) \\ & \Rightarrow \quad... | {
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# Uniformly Accelerated Motion question
A ballast bag is dropped from a balloon that is 300 m above the ground and rising at 13 m/s. For the bag, find the maximum height reached.
• Now the book gives me this answer -
• $V_i$ = initial velocity
• $V_f$ = final velocity
• $a$ = acceleration
• $y$ = displacement
• The ... | {
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• This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post. – Jim Feb 24 '15 at 18:14
• @JimdalftheGrey: How is this not an answer to the question? – Kyle Kanos Feb 24 '15 at 18:26
• " ignore " can have a different meaning than " insult " whi... | {
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# Calculating multivariate integrals between lower and upper bounds
Suppose $$\vec{X}=(x_1,x_2,...,x_n)$$ follows some continuous multivariate distribution, such that $$x_i\in{\rm I\!R}, i=1,...,n$$.
• $$\phi(\vec{x})$$, which gives me the pdf at point $$\vec{x}=(x_1,x_2,...,x_n)$$
• $$\Phi^{upper}(\vec{x})$$, which ... | {
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Does anyone have any insight on how to do this?
Thank you very much!
• Not exactly, no. Consider the 2 dimensional example I gave. If you just do $\Phi^{upper}(\vec{x}_{upper}) - \Phi^{upper}(\vec{x}_{lower})$, you're leaving in a few areas that shouldn't be considered. Here is an image to illustrate my point. – Feli... | {
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You want to calculate the probability of a rectangular area, which can be written as an intersection of bounded intervals as $\mathcal{R}_n = \bigcap_{k=1}^n (\mathcal{A}(\bar{x}_{k}) -\mathcal{A}(\underline{x}_{k}))$, where you have lower and upper bounds $\underline{\mathbf{x}} < \bar{\mathbf{x}}$. You want to be abl... | {
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$$\mathbb{P}(\mathcal{R_2}) = \Phi(\bar{x}_1, \bar{x}_2) - \Phi(\underline{x}_1, \bar{x}_2) - \Phi(\bar{x}_1, \underline{x}_2) + \Phi(\underline{x}_1, \underline{x}_2).$$
For $n=3$ we get the special case:
\begin{aligned} \mathbb{P}(\mathcal{R_3}) &= \Phi (\bar{x}_1, \bar{x}_2, \bar{x}_3) - \Phi(\underline{x}_1, \bar... | {
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$$\Phi^{upper}_{lower}\big((0,\dots,0),\ (1,\dots,1)\big) \ = \ \Phi^{upper}(1,\dots,1) \ - \sum_{x_1+\cdots+x_n=n-1} \Phi^{upper}(x_1,\dots,x_n) \ + \cdots \\+ \ (-1)^{n-1} \cdot \sum_{x_1+\cdots+x_n=1} \Phi^{upper}(x_1,\dots,x_n) \ + \ (-1)^n \cdot\Phi^{upper}(0,\dots,0)$$
• Thanks for clearing things up! But I'm st... | {
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# Defining a recursive function with additional parameters that can be used in a Manipulated ListPlot
I'm trying to code up some plots for Autocorrelation Functions in Time Series Analysis, which can often be defined recursively. The goal is then to have Manipulate sliders that allow you to dynamically change the cont... | {
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Manipulate[ListPlot[Rest[LinearRecurrence[{φ1, φ2}, {1, φ1/(1 - φ2)}, n + 1]],
Filling -> Axis, Frame -> True,
FrameLabel -> {"Lag", "\!$$\*SubscriptBox[\(ρ$$, $$k$$]\)"},
PlotRange -> All, PlotStyle -> Black],
{{φ1, 1}, 0, 3}, {{φ2, -1/2}, -1, 1}, {{n, 12}, 2, 20, 1}]
• ρ[k_, φ1_, φ2_] = ρ[k] /. FullSimplify[First[R... | {
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As @JM points out, this specific recursion has a closed-form solution:
ρ[k_, φ1_, φ2_] = ρ[k] /. FullSimplify[First[
RSolve[{ρ[0] == 1, ρ[1] == φ1/(1-φ2), ρ[k] == φ1*ρ[k-1] + φ2*ρ[k-2]}, ρ[k], k]]]
$$2^{-k-1} \left(\left(\varphi_1-\sqrt{\varphi_1^2+4 \varphi_2}\right)^k+\left(\sqrt{\varphi_1^2+4 \varphi_2}+\varphi_1... | {
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# When is the Frobenius norm of a matrix equal to the 2-norm of a matrix?
What conditions most be true for these two norms to be equal? Or are they always equal?
What conditions most be true for these two norms to be equal? Or are they always equal?
I'm far from being a specialist in this, but it seems to me that "F... | {
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More generally, ##||A||_2 \le ||A||_F \le \sqrt{r}||A||_2## where r is the rank of A.
More generally, ##||A||_2 \le ||A||_F \le \sqrt{r}||A||_2## where r is the rank of A.
May you shed some light on this? Or quote any possible reference? Thanks
jbunniii
Homework Helper
Gold Member
May you shed some light on this? Or... | {
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... |
The previous section introduced vectors and described how to add them together and how to multiply them by scalars. This section introduces a multiplication on vectors called the dot product.
##### Definition10.3.1Dot Product
1. Let $\vec u = \la u_1,u_2\ra$ and $\vec v = \la v_1,v_2\ra$ in $\mathbb{R}^2\text{.}$ The ... | {
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The next theorem extends this understanding by connecting the dot product to magnitudes and angles. Given vectors $\vec u$ and $\vec v$ in the plane, an angle $\theta$ is clearly formed when $\vec u$ and $\vec v$ are drawn with the same initial point as illustrated in Figure 10.3.4(a). (We always take $\theta$ to be th... | {
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We practice using this theorem in the following example.
##### Example10.3.9Using the dot product to find angles
Let $\vec u = \la 3,1\ra\text{,}$ $\vec v = \la -2,6\ra$ and $\vec w = \la -4,3\ra\text{,}$ as shown in Figure 10.3.8. Find the angles $\alpha\text{,}$ $\beta$ and $\theta\text{.}$
Solution
We see from o... | {
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##### Example10.3.13Finding orthogonal vectors
Let $\vec u = \la 3,5\ra$ and $\vec v = \la 1,2,3\ra\text{.}$
1. Find two vectors in $\mathbb{R}^2$ that are orthogonal to $\vec u\text{.}$
2. Find two non–parallel vectors in $\mathbb{R}^3$ that are orthogonal to $\vec v\text{.}$
Solution
An important construction is... | {
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We also know that $\vec w$ is parallel to to $\vec v$ ; that is, the direction of $\vec w$ is the direction of $\vec v\text{,}$ described by the unit vector $\frac{1}{\norm{\vec v}}\vec v\text{.}$ The vector $\vec w$ is the vector in the direction $\frac{1}{\norm{\vec v}}\vec v$ with magnitude $\norm{\vec u}\cos(\theta... | {
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This is not nonsense, as pointed out in the following Key Idea. (Notation note: the expression “$\parallel \vec y$” means “is parallel to $\vec y\text{.}$” We can use this notation to state “$\vec x\parallel\vec y$” which means “$\vec x$ is parallel to $\vec y\text{.}$” The expression “$\perp \vec y$” means “is orthogo... | {
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2. orthogonal to the ramp.
Solution
# Subsection10.3.1Application to Work
In physics, the application of a force $F$ to move an object in a straight line a distance $d$ produces work; the amount of work $W$ is $W=Fd\text{,}$ (where $F$ is in the direction of travel). The orthogonal projection allows us to compute wo... | {
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Solution
The dot product is a powerful way of evaluating computations that depend on angles without actually using angles. The next section explores another “product” on vectors, the cross product. Once again, angles play an important role, though in a much different way.
# Subsection10.3.2Exercises
Terms and Concep... | {
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# Product of monoids
I've been interested in studying bialgebras more abstractly, in the form of a bimonoid internal to a symmetric monoidal category, but I'm getting stuck on the compatibility conditions for bimonoids. Allow me to first introduce the definitions.
Let $(\mathcal{C}, \otimes, I)$ be a monoidal categor... | {
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and the multiplication uses the flip map $T_{Y, X}: Y \otimes X \to X \otimes Y$ coming from the symmetric monoidal structure.
$$m_{X \otimes Y} = \left( X \otimes Y \otimes X \otimes Y \xrightarrow{1 \otimes T_{Y, X} \otimes 1} X \otimes X \otimes Y \otimes Y \xrightarrow{m_X \otimes m_Y} X \otimes Y \right)$$
Quest... | {
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It's just naturality of $T$. This is relatively easy to see from a string diagram. Basically, you just slide one of the underlying multiplications through a crossing, then apply associativity, then slide back. Below is a much more opaque equational representation. | {
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For associativity, we want to show that $m_{X\otimes Y} \circ (m_{X\otimes Y}\otimes id_{X\otimes Y}) = m_{X\otimes Y} \circ (id_{X\otimes Y}\otimes m_{X\otimes Y})$. Expanding out the definition we get: \begin{align} & m_{X\otimes Y} \circ (m_{X\otimes Y}\otimes id_{X\otimes Y}) \\ =\ & (m_X\otimes m_Y)\circ(id_X\otim... | {
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The first equation expands definitions, then functoriality of $\otimes$ in the left parameter is used, then naturality of $T$, then bifunctoriality of $\otimes$, then associativity of $m_X$ and $m_Y$, and then we do the previous steps in reverse, with a rewrite of the permutation represented by the left hand composite ... | {
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# polyval
Polynomial evaluation
## Description
example
y = polyval(p,x) evaluates the polynomial p at each point in x. The argument p is a vector of length n+1 whose elements are the coefficients (in descending powers) of an nth-degree polynomial:
$p\left(x\right)={p}_{1}{x}^{n}+{p}_{2}{x}^{n-1}+...+{p}_{n}x+{p}_{... | {
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q = polyint(p)
q = 1×6
0.6000 0 -1.3333 5.0000 -25.0000 0
Find the value of the integral by evaluating q at the limits of integration.
a = -1;
b = 3;
I = diff(polyval(q,[a b]))
I = 49.0667
Fit a linear model to a set of data points and plot the results, including an estimate of a 95% predictio... | {
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[p,~,mu] = polyfit(T.year, T.pop, 5);
Use polyval with four inputs to evaluate p with the scaled years, (year-mu(1))/mu(2). Plot the results against the original years.
f = polyval(p,year,[],mu);
hold on
plot(year,f)
hold off
## Input Arguments
collapse all
Polynomial coefficients, specified as a vector. For examp... | {
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Standard error for prediction, returned as a vector of the same size as the query points x. Generally, an interval of y ± Δ corresponds to a roughly 68% prediction interval for future observations of large samples, and y ± 2Δ a roughly 95% prediction interval.
If the coefficients in p are least-squares estimates compu... | {
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# An open interval is not a disjoint union of two or more open intervals?
It seems intuitively clear that an open interval (like $$(a,b), (a, \infty), (-\infty,a)$$ or $$\mathbb{R}$$) cannot be written as a disjoint union of two or more (nonempty) open intervals, but I'm not sure how to prove this rigorously.
Here's ... | {
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• you can make it simpler... Suppose for instance that $(a,b) = (a,c) \cup (d,b)$, with $(a,c) \cap (d,b) = \emptyset$ and, of course $a<c<d<b$. What hapens to $c$? does it belong to the union? – PierreCarre Jun 17 '20 at 14:11
• This is essentially about completeness of reals, but the result looks so trivial as to dem... | {
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Proposition 3. If $$x \in U \subseteq \mathbb{R},$$ and $$U$$ is open, then $$U = J \cup W,$$ where $$x \in J,$$ $$J$$ is an open interval, $$W$$ is an open set, and $$J \cap W = \varnothing.$$
Proof. Let $$J$$ be the union of all open intervals $$I$$ such that $$x \in I \subseteq U.$$ By Proposition 2, $$J$$ is an in... | {
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• For the proof of Prop 2, I'm not sure why we have "in either case $x\in I \subseteq J$ for some $I \in \mathscr{I}$"? Could you please elaborate? – twosigma Jun 18 '20 at 20:41
• It was a bit terse. If $c \leqslant x < b,$ then because $b = \sup J,$ there exists $y \in J$ such that $x < y,$ therefore there exists an ... | {
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## Fundamental Theorem Of Calculus Worksheet Doc | {
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There are worked out examples here that you will find helpful. Topics include: limits, continuity, differentiation, curve sketching, applications of differentiation, integration, the Fundamental Theorem of Calculus, and applications of integration. 4 The Fundamental Theorem of Calculus 1 day 4. Assignment #8: Definite ... | {
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and only 2 answers). Here it is Let f(x) be a function which is defined and continuous for a ≤ x ≤ b. Tuesday, 2/12 4. Elementary functions. Solving differential equations a. Differentiation rules 3. Be sure to include all necessary hypotheses. 5 Integration by u-substitution. Sìn (3) (4) 11 arcsìn O/c) al+l dy. If math... | {
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(First) Fundamental Theorem of Calculus Second Fundamental Theorem of Calculus Integration By Substitution Definite Integrals Using Substitution Integration By Parts Partial Fractions. continuous) functions with finite integration limits, there are no particular technical concerns about existence of the sum or integral... | {
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at the left endpoint or the right endpoint or at the midpoint of the interval. Exercises94 5. So I searched (before the internet) far and wide for a good explanation. 3B3: Closed Form Antiderivatives: 3. Double Angle Formulas. It converts any table of derivatives into a table of integrals and vice versa. The average va... | {
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theorem. Definite Integrals and antiderivatives. A somewhat intuitive proof of the fundamental theorem is used. Past documents: #11 Old Practice Test 1 (1. 3 The Fundamental Theorem of Calculus 5. Students will understand the meaning of Rolle’s Theorem and the Mean Value Theorem. Simpson's Rule. Let $$f$$ be a function... | {
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plays a key role in enabling schools to deliver the VCE to senior secondary students. 4 1 ³f t dt(). Fundamental Theorem of Calculus - Evaluation a. The Definite Integral and the Fundamental Theorem of Calculus Fundamental Theorem of Calculus NMSI Packet PDF FTC And Motion, Total Distance and Average Value Motion Probl... | {
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" IBL calculus worksheets often have problem sets that are designed so that the students have to figure out the ideas themselves once they have an understanding of the basics. * AP ® is a trademark registered and owned by the College Board, which was not involved in the production of, and does not endorse, this site. D... | {
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of Calculus. There is 1 pending change awaiting review. The fundamental theorem of calculus. State both parts of the Fundamental Theorem of Calculus. 8 Definite Integral by Substitution Pg. For each, be sure to explain your thinking. No calculator unless otherwise stated. For example, the three fundamental principles o... | {
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trapezoidal. J h NAtl Bl1 qr ximg Nh2tGsM Jr Ie osoeCr4v2e odN. It is estimatedthat t years fromnowthepopulationof a certainlakeside community will be changing at the rate of 0. Then the Fundamental Theorem of Calculus allows us to evaluate this area by using a definite integral, so that The area bounded by the polar c... | {
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Bi means two and nom means term. Microsoft Word - Properties of Sums and Integrals. Areas between graphs105 2. Math 1A introduction to functions and calculus Oliver Knill Spring 2011 4 5 2011 Second midterm practice I Problem 1 TF questions 20 points No justif…. First video in a short series on the topic. AP Calculus A... | {
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The following is a list of worksheets and other materials related to Math 122B and 125 at the UA. 6 Worksheet (didnt get to do 2 FRQs). "Bell curve" refers to the bell shape that is created when a line is plotted using the data points for an item that meets the criteria of normal distribution. Argue that f is increasin... | {
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biology, medicine, finance, history, and more. 3 The Fundamental Theorem of Calculus, Part I 272 5. Course Modules Definite Integrals Worksheet Definite Integrals Worksheet 5. Calculus Integration Fundamental Theorem Definite Integral Task Cards and HW classroom tips, teaching ideas & resources for teaching high school... | {
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fix our math education. The binder must include worksheets and homework. Let's simplify our life by pretending the region is. Material cannot be shared specially during quizzes or tests. Read & Study Section 16: The Fundamental Theorem of Calculus. The Fundamental Theorem of Calculus. Let A(x) = Z x a f(t) dt. ©u 12R0X... | {
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37 and FTC 2 MC Practice worksheet (the one with limits such as 0 to x^3) Routine #45, 51-53, 59-65 Non-Routine (actually a routine but for now are NR) #49-50 These are integrating with respect to y. 394 (11-21) odd Note: #13 a,b,c only) Draw the graph for all problems, even if it is already given to you. Sìn (3) (4) 1... | {
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The fundamental theorem of calculus is central to the study of calculus. Example: Evaluate. Title: New Doc 2019-11-12 15. _____ Work problems 3 - 7 using the Fundamental Theorem of Calculus and your calculator. My love for you is like the slope of a concave up function because it is always increasing. This theorem was ... | {
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Workspace: Integrals, Area & Reversing D-Rules. Integration by Substitution. Part1: Define, for a ≤ x ≤ b, F(x) = R x. Therefore, the desired function is f(x)=1 4 x4 + 2 x +2x−5 4. , n! = n(n – 1)(n – 2) …. Some topics in calculus require much more rigor and precision than topics encountered in previous. Evaluate area u... | {
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be a function such that 0 (2 ) (2) lim 5 h fhf → h +. The reason is that it's, well, fundamental, or basic, in the development of the calculus for trigonometric functions. It is estimatedthat t years fromnowthepopulationof a certainlakeside community will be changing at the rate of 0. This lesson contains the following... | {
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³ b a f x. Definite Integrals as Area Accumulator Functions. 3 Tables Software - Free Download 3 Tables - page 9 - Top 4 Download - Top4Download. Mathematical Statistics with Applications by Kandethody M. Michael Kelley Mark Wilding, Contributing Author. A particle moving along the x-axis has position at time t with th... | {
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document View the full content. Triangle Inequality Theorem Worksheets. 1 Tangent Line Problem & Differentiability ( Notes , Worksheet ) Practice TEST 3. e) f(5)−f(2) 3 f) The time required for the shell to reach the altitude 300 ft. In the case of integrating over an interval on the real line, we were able to use the ... | {
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396406 1 APC 14 2. 44 Chapter 3. The Second Fundamental Theorem of Calculus: Hypothesis: F is any antiderivative of a continuous function f. If f is a continuous function defined on a closed interval and F is an antiderivative of f, then (Example 9. 5 Integration by u-substitution. This property allows us to easily so... | {
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of AP Calculus AB, and as such the difference between the two is in scope, not difficulty. 3 by using the Mean Value Theorem. 1 1) State whether or not each of the following mappings represents a function. Average Value and Average Rate: File Size: 53 kb: File Type: pdf:. It is broken into two parts, the first fundamen... | {
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the number of years from today. Example 11: Using the Second Fundamental Theorem of Calculus to find if. Label the values of at least four points. I Worksheet by Kuta Software LLC Kuta Software - Infinite Calculus Name_____ Fundamental Theorem of Calculus Date_____ Period____. UIUC MATH 241 - Lecture050514 (22 pages) P... | {
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rational zeros for each function. Go through a few examples with the class. Executive Summary: The topic of the lesson is Rolle’s Theorem and the Mean Value Theorem. Page 1 of 2 6. 1,2,5 (ii) Area of the region enclosed between Parabola and line. Use the Fundamental Theorem of Calculus and the given graph. b) Find the ... | {
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course. In physics, the line integrals are used, in particular, for computations of mass of a wire; center of mass and moments of inertia of a wire; work done by a force on an object moving in a vector field; magnetic field around a conductor (Ampere’s Law); voltage generated in a loop (Faraday’s Law of magnetic Read m... | {
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to orders under USD 120. Compare and contrast the Intermediate Value Theorem, Mean Value Theorem, and Rolle's Theorem. 06 - Second Fundamental Theorem - Kuta Software ©d J260R1y3G HKvuWtaaA ASToxfKtvwOa9rFeM LLyLDCv. ) (This is a 10 point homework grade. Summation Notation Worksheet 1 Introduction Sigma notation is us... | {
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# Why is zero the only infinitesimal real number?
I am currently reading Elementary Calculus: An Infinitesimal Approach by H. Jerome Keisler and was wondering if someone could help me with an aspect treated in the book.
On page 24 he says a number $\varepsilon$ is said to be infinitely small or infinitesimal if $$-a<... | {
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The reason for this is that the definition of an infinitesimal $\varepsilon$ is that $-a \leq \varepsilon \leq a$ for every positive real number $a$. You just picked some positive real number. This has to be true for every positive real number. That means $\varepsilon$ needs to be in $[-2, 2]$ and in $[-1, 1]$ and in $... | {
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# Proof: $\cos^p (\theta) \le \cos(p\theta)$
I came across this problem when I was at a book store inside of a book made to prepare Berkeley graduates to pass a mandatory exam. I wanted to buy the book, but, alas, I didn't have the money (forty bucks is a lot of money when you don't have a job). So I took my phone out... | {
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• How did you get the log in the second derivative? Sep 3, 2014 at 8:39
• The variable is $p$, so you derive an exponential $a^p$.
– user65203
Sep 3, 2014 at 8:40
• I like it. I like it a lot. Sep 3, 2014 at 8:42
• You should like my other solution too.
– user65203
Sep 3, 2014 at 9:42
Set $f(\theta) = \cos p\theta - \... | {
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• Didn't know Kim Jong Un is on math.SE! Please don't kill us sir, we are sorry! Sep 3, 2014 at 8:32
Set $f(\theta)=\cos^p(\theta)-\cos(p\theta).$ Then, $f(0)=0$. Differentiate $f$ w.r.t. $\theta$ you'll get \begin{align} f'(\theta)&=-p\cos^{p-1}(\theta)\sin(\theta)+p\sin(p\theta)\\ &=-p\cos^{p-1}(\theta)\sin(\theta)+... | {
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Average Distance Between Random Points on a Line
Suppose I have a line of length L. I now select two points at random along the line. What is the expectation value of the distance between the two points, and why?
-
$L/3$, by symmetry. – Byron Schmuland Sep 13 '12 at 15:18
Care to elaborate, @Byron? – David Sep 13 '12... | {
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-
On the third line of the $\mathbb{E}(Y)$ derivation, shouldn't $d x_1$ and $d x_2$ be swapped? (Or the limits of the integrals.) Pedantic, I know. – David Sep 13 '12 at 16:17
@David: You're totally right. Thanks for pointing that out. I fixed those typos. – Rod Carvalho Sep 13 '12 at 16:22
@RodCarvalho : If you use "... | {
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"ta... |
For simplicity assume $L=1$.
Therefore $|X_1-X_2| \stackrel{d}{=} |X_1+X_2-1|$. Random variable $D = X_1+X_2-1$ follows symmetric triangular distribution on $(-1,1)$, being a special case of Irwin-Hall distribution. We immediately have: $$f_{|D|}(\ell) = 2 (1-\ell)[0<\ell<1]$$ Immediately yielding the expectation: $$\... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9854964194566753,
"lm_q1q2_score": 0.8471955409776895,
"lm_q2_score": 0.8596637433190939,
"openwebmath_perplexity": 229.65734856890967,
"openwebmath_score": 0.9532203078269958,
"ta... |
(Please excuse my formatting. I'm new here.)
5 6 7 8 9 10 11 12 13 14
-----------------------------
5 | 0 1 2 3 4 5 6 7 8 9
6 | 1 0 1 2 3 4 5 6 7 8
7 | 2 1 0 1 2 3 4 5 6 7
8 | 3 2 1 0 1 2 3 4 5 6
9 | 4 3 2 1 0 1 2 3 4 5
10 | 5 4 3 2 1 0 1 2 3 4
11 | 6 5 ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9854964194566753,
"lm_q1q2_score": 0.8471955409776895,
"lm_q2_score": 0.8596637433190939,
"openwebmath_perplexity": 229.65734856890967,
"openwebmath_score": 0.9532203078269958,
"ta... |
Byron's answer is short and elegant. Here's a geometric/algebraic derivation: Let $X$ and $Y$ be two independent uniform variates in $[0,1]$. Then $$p\left(\{|X-Y|>s\}\right) = (1-s)^2$$ as can be seen by viewing $(X,Y)$ as a uniform variate in $[0,1]^2$, where the set $\{|X-Y|>s\}$ occupies the top left and bottom rig... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9854964194566753,
"lm_q1q2_score": 0.8471955409776895,
"lm_q2_score": 0.8596637433190939,
"openwebmath_perplexity": 229.65734856890967,
"openwebmath_score": 0.9532203078269958,
"ta... |
The Padé approximant of order [mn] approximates the function f(x) around x = x0 as
$\frac{{a}_{0}+{a}_{1}\left(x-{x}_{0}\right)+...+{a}_{m}{\left(x-{x}_{0}\right)}^{m}}{1+{b}_{1}\left(x-{x}_{0}\right)+...+{b}_{n}{\left(x-{x}_{0}\right)}^{n}}.$
The Padé approximant is a rational function formed by a ratio of two power... | {
"domain": "mathworks.cn",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9854964169008471,
"lm_q1q2_score": 0.8471955387805364,
"lm_q2_score": 0.8596637433190938,
"openwebmath_perplexity": 1047.2292924382828,
"openwebmath_score": 0.769485354423523,
"tags": n... |
Find the Laplace transform of F using laplace.
F = laplace(F,t,s)
F = $\mathrm{laplace}\left(y\left(t\right),t,s\right)-\tau \left(y\left(0\right)-s \mathrm{laplace}\left(y\left(t\right),t,s\right)\right)=a \mathrm{laplace}\left(x\left(t\right),t,s\right)$
Assume the response of the system at t = 0 is 0. Use subs to ... | {
"domain": "mathworks.cn",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9854964169008471,
"lm_q1q2_score": 0.8471955387805364,
"lm_q2_score": 0.8596637433190938,
"openwebmath_perplexity": 1047.2292924382828,
"openwebmath_score": 0.769485354423523,
"tags": n... |
y = subs(y,[a tau],[1 3]);
y = ilaplace(y,s);
Find the Padé approximant of order [2 2] of the step input using the Order input argument to pade.
$\frac{3 {s}^{2}-4 s+2}{2 s \left(s+1\right)}$
Find the response to the input by multiplying the transfer function and the Padé approximant of the input.
$\frac{a \left(3 ... | {
"domain": "mathworks.cn",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9854964169008471,
"lm_q1q2_score": 0.8471955387805364,
"lm_q2_score": 0.8596637433190938,
"openwebmath_perplexity": 1047.2292924382828,
"openwebmath_score": 0.769485354423523,
"tags": n... |
$\frac{a \left(27 {s}^{4}-180 {s}^{3}+540 {s}^{2}-840 s+560\right)}{s \left(s \tau +1\right) \left(27 {s}^{4}+180 {s}^{3}+540 {s}^{2}+840 s+560\right)}$
$\frac{27 {s}^{4}-180 {s}^{3}+540 {s}^{2}-840 s+560}{s \left(3 s+1\right) \left(27 {s}^{4}+180 {s}^{3}+540 {s}^{2}+840 s+560\right)}$
yPade45 = $3.241838498166254667... | {
"domain": "mathworks.cn",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9854964169008471,
"lm_q1q2_score": 0.8471955387805364,
"lm_q2_score": 0.8596637433190938,
"openwebmath_perplexity": 1047.2292924382828,
"openwebmath_score": 0.769485354423523,
"tags": n... |
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