For basic questions about limits, derivatives, integrals, and applications, mainly of one-variable functions.

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2answers
147 views

Zeroing the carrier measure of an exponential family

I'm trying to derive the general process of changing variables so that an exponential family has zero carrier measure. Distributions in the exponential family have cdf ...
2
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2answers
153 views

$s(x)$ is a arc length function, find $s'(x)$

Here is the problem is my textbook: Suppose $s(x)$ is the arc length function for the curve $y=\sin x$ taking $(0,1)$ as the starting point. Find $sā€™(x)$. According to arc length formula, I ...
1
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3answers
349 views

Application of Cauchy's convergence criterion

Use Cauchy's convergence criterion to prove convergence of $x_n$ $$x_n=1-\frac{1}{2}+\frac{1}{3}-\cdots+(-1)^{n+1}\frac{1}{n}$$ as far as I am concerned,supposing that $m>n$, it's apparently ...
3
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0answers
295 views

Fastest convergence Series which approximates function

The question is the following: Is there any proof that shows that the Taylor series of an analytical function is the series with the fastest convergence to that function? The motivation to this ...
0
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3answers
373 views

Show that this function is not increasing on any interval containing $0$:

$$f(x) = \begin{cases}x + 2x^2\sin\left(\frac1x\right),& x\ne 0\\0,& x = 0\;.\end{cases}$$ I am having a tough time answering this question in a rigorous mathematical way, here is what I have ...
5
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2answers
645 views

Deriving parameterization for hyperboloid

I know there is a parameterization of a hyperboloid $\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1$ in terms of $\cosh$ and $\sinh$, but I don't see how these equations are derived. I would ...
2
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2answers
148 views

Differential equation of $y = e^{rx}$

I am trying to find what values of r in $y = e^{rx}$ satsify $2y'' + y' - y = 0$ I thought I was being clever and knew how to do this so this is how I proceeded. $$y' = re^{rx}$$ $$y'' = r^2 ...
3
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3answers
240 views

$f(1)=-3$ and $ f'(x)\geq7$ how small is $f(5)$?

Here is a test question in my textbook. Suppose : $f(1)=-3$ and $f'(x)\geq7$ How small $f(5)$ can be possibly : a)$25$ b)$-21$ c)$28$ e)$31$ f) None of others. I just have this ...
8
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3answers
223 views

Integral of $\int \frac{\sin(x)}{3\cos^3(x)+\sin^2(x)\cdot\cos(x)}\,dx$

So, from here $$\int \frac{\sin(x)}{3\cos^3(x)+\sin^2(x)\cdot\cos(x)} dx$$ I divided by cos(x) and I got $$\int \frac{\tan(x)}{2\cos^2(x)+1} dx$$ But I'm stuck here. I tried to substitute $t=\cos(x)$ ...
0
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1answer
345 views

Derivative for Matrix function

I have a matrix kernel function which I am trying to find the derivative to. Function is K = c * exp[-1/2 * (P(X1 - X2))' * P(X1 -X2)] where uppercase are matrices and lower case are scalars (and ' ...
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2answers
63 views

Proof of a test for series

I would like to prove that given three sequences ${a_n}, {b_n}\text{ and }{c_n}$ and knowing that: They aren't necessarily of positive terms. $a_n \leq b_n \leq c_n, \forall n \geq 1$ $$\text{If ...
2
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2answers
123 views

Evaluating: $\int (x^{2}+x)\ln|x-1|dx$

Evaluating: $\int (x^{2}+x)\ln|x-1|dx$ My problem is due to the presence of the absolute value.
0
votes
2answers
85 views

The derivative of: $\sin\left(\int_{x^{3}}^{\sin(x^{2})}\sin t^{2}dt \right)$?

What is the derivative of $\sin\left(\int_{x^{3}}^{\sin(x^{2})}\sin t^{2}dt \right)$?
1
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2answers
107 views

determine whether this series converges for this value of z

Does $$f(z)=\displaystyle \sum_{n=0}^{\infty} \frac{2^n+n^2}{3^n+n^3}z^n$$ converge for $z=\frac{-3}{2}$?
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3answers
85 views

For which values of $d$ does $x - d\tanh(x) = 0$ have a positive solution?

For which $d$ values does the equation $$x - d\tanh(x) = 0$$ have a positive solution? I have tried rearranging this a number of different ways using the exponential form and and using hyperbolic ...
1
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1answer
164 views

Should students be taught that $\int dx/x = \log(|x|)$? [closed]

I think in precalculus students should be taught the following: Euler's identity for $e^{i \theta}$. The principal value of $log(x)$ for $x<0$. Then in Calculus they should be taught that $\int ...
2
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4answers
86 views

Derivative of $\|Xa\|_2 $ with respect to $X$

Can someone give me the answer to the following expression? $\frac{\partial}{\partial X}\|Xa\|_2 =?$ $X$ is a square matrix and $a$ is a vektor of the apropriate size. $\|\cdot\|_2$ is the euclidean ...
0
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1answer
32 views

How large need $n$ be taken to ensure that $T_n(x)$ gives a value of $\ln(1.3)$ which has an error of less than $0.0002$?

$$ f(x) = \ln(1+x)$$ The previous part of this question required me to write down the remainder term for the taylor polynomial of order n. My remainder term worked out to be: $$R_n(x) = (-1)^n ...
6
votes
2answers
850 views

Show that $f'$ is not continuous at 0 for the following function:

$$ f(x) = \begin{cases} x + 2x^2\sin(1/x) & \text{ for }x \neq 0 \\ 0 & \text{ for } x = 0\end{cases} $$ This is another exam practice question I am working on. I simply took the ...
4
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2answers
80 views

How are the limits of this integral transformed?

Using $$\ln(x) = \int_1^x \frac{1}{t} dt$$ Show that for $x > 0$, $\ln\left(\frac{1}{x}\right) = -\ln(x)$ I am following a provided answer and didn't quite understand the following ...
8
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2answers
356 views

Is there a differential limit?

I'm wondering if there's such a concept as a "differential limit". Let me give an example because my nomenclature is my own and unofficial, but hopefully indicative of the concept. For some function ...
0
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1answer
43 views

probability inequality resolution finite field

I'm trying to find out whether my communication protocol should have redundant information padded, in order to help the receiver correct the error (error correction code, ECC) without needing a ...
3
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4answers
1k views

Hydrostatic pressure on a square

Vertically inserted into the water I have a rectangle 6 feet wide and 4 feet high that is submerged under the water with 2 feet of water above it. Using a riemann sum how do I find the pressure? I ...
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3answers
505 views

Hydrostatic pressure on a triangle

I am attempting to follow that horrendous site, Paul's calculus notes, but there are so many omissions or possible mistakes, I am not sure which. I am following ...
5
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3answers
150 views

What other substitutions could I use to evaluate this integral?

Consider the integral $$ \int x^2\sqrt{2 + x} \, dx$$ I need to find the value of this integral, yet all its (seemingly) possible substitutions don't allow me to cancel appropriate terms. Here are ...
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3answers
4k views

Centroid of a region

$$y = x^3, x + y = 2, y = 0$$ I am suppose to find the centroid bounded by those curves. I have no idea how to do this, it isn't really explained well in my book and the places I have looked online ...
7
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1answer
906 views

Multi variable integral : $\int_0^1 \int_\sqrt{y}^1 \sqrt{x^3+1} \, dx \, dy$

$$\int_0^1 \int_\sqrt{y}^1 \sqrt{x^3+1} \, dx \, dy$$ Here is my problem in my workbook. If I solve this problem by definition, that find integral for $x$, after that solve for $y$. so ...
3
votes
2answers
106 views

The value of $\int_0^{2\pi} g(re^{i(\theta + \phi)}) \, d\phi$ is independent of $\theta$?

I want to prove this geometrically. For function $g : \mathbf{C} \rightarrow \mathbf{R}$ and g is some continuous function. The value of $\int_0^{2\pi} g(re^{i(\theta + \phi)}) \, d\phi$ is ...
1
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1answer
192 views

Evaluating $ \int_0^{\infty } \exp\left(-g x-\frac{x^2}{2}-\frac{x^2 z}{1-z}\right) x^k \sin(hx) \, dx $

I'm attempting to evaluate the following integral, so far, with little success. Any help would be appreciated: $$ \ \int_0^{\infty } \exp\left(-g x-\frac{x^2}{2}-\frac{x^2 z}{1-z}\right) x^k \sin(hx) ...
2
votes
7answers
136 views

Limit finding of an indeterminate form

here is the limit I'm trying to find out: $$\lim_{x\rightarrow 0} \frac{x^3}{\tan^3(2x)}$$ Since it is an indeterminate form, I simply applied l'Hopital's Rule and I ended up with: ...
1
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0answers
114 views

Integrating by parts .

I have a equation something like this: $\int_U |\Delta u|^2dx \le C\int_U |D^2 u|^2dx$ . where $u \in C_c^{\infty}$ implies that u belongs to a compactly supported smooth function . I would like ...
3
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1answer
61 views

Area of a revolution of $x=\frac{1}{3}\left(y^2+2\right)^\frac{3}{2}$

I think my biggest problem here is I can not find a good way to find the square root in this problem $$x=\frac{1}{3}\left(y^2+2\right)^\frac{3}{2} \ \ \ \ 1 \le x \le 2$$ $$\int_1^2 2 \pi \cdot ...
3
votes
1answer
772 views

Area of a surface of revolution of $y = \sqrt{4x+1}$

$y = \sqrt{4x+1}$ for $1 \leq x \leq 5$ I really have no idea what to do with this problem, I attempted something earlier which I will not type up because it took me two pages. $$y = \sqrt{4x+1}$$ ...
6
votes
4answers
1k views

Arclength of the curve $y= \ln( \sec x)$ $ 0 \le x \le \pi/4$

Arclength of the curve $y= \ln( \sec x)$ $ 0 \le x \le \pi/4$ I know that I have to find its derivative which is easy, it is $\tan x$ Then I put it into the arclength formula $$\int \sqrt {1 - ...
3
votes
3answers
548 views

Limits without l'Hôpital

Find: $$\lim_{x\to 0}\ \frac{\dfrac{\sin x}{x} - \cos x}{2x \left(\dfrac{e^{2x} - 1}{2x} - 1 \right)}$$ I have factorized it in this manner in an attempt to use the formulae. I have tried to use that ...
2
votes
1answer
71 views

multiplication of a trigonometric series

Let $f(x)$ be the value of a trigonometric series, which converges uniformly on $\left[ -\pi, \pi\right]$. If I multiply $f(x)$ with $e^{iax}$ where $a\in\mathbb{N}$ will the result then be a ...
0
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1answer
104 views

Uniform convergent trigonometric series

If $f:[-\pi, \pi] \rightarrow\mathbb{C}$ is the value of an uniform convergent trigonometric series, can I then deduce that the $2\pi$-periodic normalized extension is an uniform convergent ...
1
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5answers
102 views

Proving that $A=\left\{\dfrac{n}{2n+1}:n \in \mathbb{N}\right\}$ is bounded by $\dfrac{1}{2}$

Let $A=\left\{\dfrac{n}{2n+1}:n \in \mathbb{N}\right\}$. I want to prove that $supA=\dfrac{1}{2}$ so I need to show that $$\forall\epsilon\gt0 \exists a\in A:a\gt\dfrac{1}{2}-\epsilon$$ So suppose by ...
6
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4answers
284 views

Proving that $A=\{(-2)^n : n \in \mathbb{N} \}$ is unbounded

I am trying to prove that $A=\{(-2)^n : n \in \mathbb{N} \}$ is unbounded. What I did was first to show that for every $n \in \mathbb{N}$ if $n$ is even then $(-2)^n = 2^n$ and if $n$ is odd then ...
1
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2answers
66 views

Finding Polynomial Limit

I am asked to find the Limit for: $$\lim_{x\rightarrow -āˆž}(x^4+x^5) $$ The first thing I am tempted to do is divide the numerator and denominator of this fraction by the highest power of x, in ...
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1answer
574 views

Solving a word problem using derivatives

I don't understand the question at all. It is very confusing. Can anyone help? The sum of two positive numbers is 5. Find the numbers such that: a. Their product is a maximum. b. The sum of their ...
0
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1answer
570 views

Solving a word problem using derivatives to find the minimum

I need help solving this equation. Can anyone help? Find the least amount of material needed to make a square-based open box that has volume of 4000 cubic meters.
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1answer
653 views

Solving a word problem using derivatives to find the maximum

I am having a lot of trouble with this word problem. Can anyone help? There are 900 units of fencing available to enclose a rectangular plot of ground with a fence down the middle and parallel to two ...
0
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1answer
92 views

Finding the minimum in a word problem by using derivatives

I need help with this problem. I have no idea where to start and how to get the answer. Find three numbers such that the first is the sum of the second and third, the second is the square of the ...
0
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1answer
49 views

Using derivatives to find the max

I had some trouble figuring out how to solve this problem. Can anyone help? A trough with a rectangular cross section is to be made from a long sheet of metal 24 meters wide by turning up strips ...
2
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5answers
257 views

Show that the largest rectangle with a perimeter of $20$ meters is a square.

Show that the largest rectangle with a perimeter of $20$ meters is a square.
2
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0answers
71 views

polar form of a double integral

given the following region $R=\lbrace m,n \geq0$, $1 \geq m+n \geq 2\rbrace$ where $(m,n) \in \mathbb{R}^2$.write in polar coordinates $(r, \theta)$ the following double integral $\int\int_R m \,dA$
1
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1answer
812 views

How to find the dimensions of a rectangle if its area is to be a maximum?

A rectangle has two of its vertices as the $x$-axis. The other two vertices are on the parabola whose equation is $y=18-x^2$. What are the dimensions of the rectangle if its area is to be a maximum?
6
votes
5answers
1k views

Michael Spivak in “Calculus” asserts that $\sqrt2$ cannot be proven to exist, and that such a proof is impossible. What does he mean by “exist”?

Michael Spivak in "Calculus" asserts that $\sqrt2$ cannot be proven to exist, and that such a proof is impossible. What does he mean by "exist"? How are you to prove that any number "exists"? Why ...
5
votes
4answers
158 views

Evaluate $\lim_{x \to \infty} \frac{1}{x} \int_x^{4x} \cos\left(\frac{1}{t}\right) \mbox {d}t$

Evaluate $$\lim_{x \to \infty} \frac{1}{x} \int_x^{4x} \cos\left(\frac{1}{t}\right) \mbox {d}t$$ I was given the suggestion to define two functions as $g(x) = x$ and $f(x) = ...