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

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1
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1answer
57 views

$Pr(X+Y \geq \frac{\pi}{2})$

I want to find $Pr(X+Y \geq \frac{\pi}{2})$ for joint pdf $f_{X,Y}(x,y) = x \cos y, 0 \lt x \lt \frac{\pi}{2}, 0 \lt y \lt x, 0$ otherwise. I believe I have found conditional pdf of $Y$ given $X=x$ ...
1
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2answers
57 views

Example or disprove for an $f:\mathbb{R}\rightarrow \mathbb{R}$

Does there exists an $f:\mathbb{R}\rightarrow \mathbb{R}$ differentiable everywhere with $f'$ discontinuous at some point?
0
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4answers
88 views

Integral of $16/(1-\cos8x)$

Can someone please help me with this question: $$ \int \ \frac{16}{1-\cos8x} \ \ dx \ \ . $$ I tried substitution by letting $u=1-\cos8x$, it got messy after the substitution. I used the ...
1
vote
1answer
549 views

Simpson's Rule like Method for Arc Length Approximation

Simpson's Rule is of course a very important method for Numerical Integration. The basic idea behind it is to the intervals as a quadratic curve, and to calculate the area under that curve (yes, I ...
0
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1answer
47 views

what will be the answer of this limit.

$\displaystyle \lim_{n\to\infty} \cos^{2n} x =?$ The given answer is $0$. I tried solving it by using the fact that $0 \le \cos^2 x \le 1$, after this I am unable to solve further. Any help is ...
1
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1answer
30 views

Triple Integral Troubles

I'm having trouble calculating this integral. I can do the first one just fine, but it's in simplifying and calculating the third integral where I get stuck. ...
1
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1answer
38 views

Finding conditionally expected $y$ given a specific $x$ from a joint distribution function!

I want to determine expected $y$, given $x=2$ given joint pdf shown below $$\frac{1}{2y} * e^{-\frac{y^2 + \frac{x}{2}}{y}}$$ for $x,y \gt 0$ and $0$ otherwise I believe this means I want ...
1
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0answers
48 views

Justifying that the derivative doesn't exist

Is the right way of justifying that the derivative of $\sqrt[3]{x}$ at $x=0$ calculting, by th definition, $$\lim_{x\to0} \frac{1}{\sqrt[3]{x^2}}$$ via left and right havd limits? I mean, it is ...
2
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1answer
72 views

Finding the conditional pdf of $Y$ given $X=x$ from a joint pdf. Answer confirmation!

I have a continuous joint pdf, and I am working out the conditional pdf of Y given X=x. Is my method correct? I am given: $f_{X,Y}(x,y) = x\cos y , 0 \lt x \lt \frac{\pi}{2}, 0 \lt y \lt x$ ...
-1
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1answer
49 views

Find the Taylor Series generated by $\frac1x$ at $x = a$

Can someone help me find the Taylor series for the following equation: $f(x) = \frac1x$ at $a = 10$
5
votes
1answer
261 views

Integral $\int_0^\infty \log(1+x^2)\frac{\cosh \pi x +\pi x\sinh \pi x}{\cosh^2 \pi x}\frac{dx}{x^2}=4-\pi$?

Hello am looking for a solution to proving this. $$ I:=\int_0^\infty \log(1+x^2)\frac{\cosh \pi x +\pi x\sinh \pi x}{\cosh^2 \pi x}\frac{dx}{x^2}=4-\pi. $$ This one is related to Marvelous ...
0
votes
1answer
946 views

Limit of a function with absolute values

So I've got this limit: $$\lim_{x\to 3^-} \frac{x^2-9}{|x-3|}$$ My (wrong) answer was zero. I figured that since the numerator approaches zero then regardless of what the denominator was, the whole ...
1
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0answers
78 views

Wedge volume of sphere problem

The Clare College bridge at Cambridge is decorated with 14 stone spheres, but one of it missed a wedge. I took a photo to estimate the volume of the missing part of the sphere. I am not confident ...
1
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1answer
31 views

Limit of an integral expression

Does anyone have any idea on how to solve this problem ? I don't have the faintest clue on how to even start it . I would sure appreciate a solution , thank you for your patience !
0
votes
1answer
58 views

Solving IVP $y'=t|y|^\alpha, \ y(0)=1$

Intro: This is a follow up to my post Application of Picard-Lindelöf to determine uniqueness of a solution to an IVP, where I am trying to verify that the below IVP has a unique solution in some ...
1
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4answers
62 views

Use logarithmic differentiation to find the derivative of $y = (1 +\frac1x)^{2x}$

Can someone guide me through solving this problem? $$\dfrac{\mathrm d}{\mathrm dx}\left(1 +\dfrac1x\right)^{2x}$$
2
votes
4answers
377 views

A particle moves along a path described by $y = 4 − x^2$ . At what point along the curve are $x$ and $y$ changing at the same rate?

Why is the answer $\left(-\frac 1 2 , \frac{15}{4} \right)$? I have no idea how to approach this problem. Can someone guide me/explain it to me step by step? I have a final on this in less than 2 ...
2
votes
1answer
21 views

Given $g(1) = 6$, $g'(1) = -1$, find $d/dx(2 g(x)/(x^2 + 1))$ when $x = 1$

Why is the answer $-7$? I plugged $1$ into the equation and I ended up with $12/2$ and got $6$. Can someone explain to me what I did wrong?
1
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1answer
196 views

Precise definition of oscillation behavior of functions like $\sin(\frac1{x})$

I tried today precisely defining the oscillation behavior present in functions like $\sin(\frac1{x})$ i.e: To do this, I started with the domain of function as in limits, and let $f(x)-L$ have ...
6
votes
2answers
249 views

Solving this integral?

Well, this might be a really simple one. But still... What will be the soln. to --- \begin{aligned} \int\frac{1}{1+x^n} dx \end{aligned} Is substituting \begin{aligned} 1+x^n \end{aligned} by tan z ...
0
votes
2answers
29 views

Continuity of a function when domain and rangle are not real numbers

Let $f:\mathbb{Q}\rightarrow \mathbb{Q}, f(x)=x$ and choose $\epsilon=\delta$ then $\left| f(x)-f({ x }_{ 0 }) \right|<\epsilon$ and $\left| x- x _ 0\right|<\delta$ for $x_0,\epsilon,\delta\E ...
0
votes
1answer
19 views

A limit calculation problem.

I see in a book the following : $$\lim_{\rho \rightarrow0}\frac{e^{\frac{f(i)}{\rho}}}{\sum_{i=1}^{n}e^{\frac{f(i)}{\rho}}}$$ is 1 for $i=\arg \max_{i}f(i)$ and $0$ otherwise. I don't see it ...
10
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0answers
250 views

The closed form of $\sum_{n=0}^{\infty} \arcsin\bigl(\frac{1}{e^n}\bigr)$

In my study on some type of integrals I met the series below that I don't how to approach it. Of course, one of the obvious questions is: does it have a closed form? Before answering that, I need to ...
4
votes
4answers
69 views

Simplifying a u-substitution for $\int \frac{x} { \sqrt {4-3 x^4 } } \, dx$

this is a calculus one problem I cannot figure out. I may be making a simple assumption in my substitutions, please help. (I hope I typed this correctly, this is my first time using the MathJaX ...
3
votes
2answers
211 views

Application of Picard-Lindelöf to determine uniqueness of a solution to an IVP

I am still struggling quite a lot with the Picard-Lindelöf-Theorem (also known as the Cauchy-Lipschitz-Theorem). Problem: Consider the following IVP with $\alpha \neq 1$ $$\begin{cases} y'&= ...
1
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1answer
33 views

Taking a limit for a modified difference quotient without L'Hopital

This limit is immediate using L'Hopital rule: $$ \lim_{h\rightarrow 0} \frac{f\big(x + h(\alpha - x)\big) - f(x)}{h}= \lim_{h\rightarrow 0} f'\big(x + h(\alpha - x)\big) (\alpha - x)= f'(x) ...
2
votes
3answers
76 views

Partial integration of $e^x\ln(1+e^x)$

I am trying to solve $$\int_0^1e^x\ln(1+e^x)dx.$$ I tried to do a partial integration $\displaystyle\left.e^x \ln(1+e^x)\right|_0^1- \int_0^1\frac{e^{2x}}{1+e^x}dx$ but this leaves me quite a bit ...
0
votes
4answers
131 views

Prove that $f(x) = 4 - 15x$ is continuous

Need help showing this. Show that the function $f\colon \mathbb{R} \to \mathbb{R}:$ defined by $f(x) = 4 - 15x$ is continuous at every point $c\in\mathbb{R}$. Thanks.
0
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1answer
51 views

what will be the definite integration of the following equation [duplicate]

What will be the definite integration from 0 to inf of the cosine function $$\int_0^\infty\cos(r)\,\mathrm dr$$
2
votes
2answers
96 views

Dirac's delta integration

What about the following integral? $$\int_0^a x^3 \delta(x-1) dx$$ If $a$ is more or less than 1 it's all clear, but what if $a=1$. Is the integral is equal to $1/2$ ? Edit: this is my motivation, ...
1
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2answers
39 views

Finding solution of first order non linear equation

I have $$m\frac{dv}{dt}=mg-kv^2$$ and I want to find v(t). I tried to separate the derivative over both sides but I am getting no where. At the moment I have $$v+\frac{v^2}{gt}=\frac{kt}{m}$$ Can ...
0
votes
1answer
31 views

$|a_{n}| \leq C e^{-|n|} \implies \sum_{n\in \mathbb Z} a_{n} e^{in(x+iy)} $ converges absolutely for $|y|<1$?

Suppose $\{a_{n}\} \subset \mathbb C$ with $|a_{n}| \leq C e^{-|n|}, n\in \mathbb Z$ and fix $C >0.$ My Question is: How to show the series, $$\sum_{n\in \mathbb Z} a_{n} e^{in (x+iy)}; (x, ...
1
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1answer
39 views

question about changing variables in integration

Let $R = \{ (x,y,z) : x,y,z \geq 0, \; \; x+y+z \leq 1 \} $. WAnt to find $Vol(R) $. Well, I know that the volume is given by $$ Vol(R) = \int_0^1 \int_0^{1-x} \int_0^{1-x-y} dzdydx = \frac{1}{6} $$ ...
4
votes
1answer
78 views

Find general solution of first order non-linear in a transcendental function

I have the function $$\frac{dV}{dT}=1-V^2$$ Just looking to see if my working is okay. $$dV=1-V^2dT$$ $$\frac{1}{1-V^2}dV=dT$$ Integrate $$\int{}\frac{1}{1-V^2}dV=\int{}dT$$ Let $V=\tanh(x)$ ...
1
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2answers
62 views

Notation for Derivatives with Respect to Time

My math teacher writes $\frac{dx}{dt}$ as $x'$, and makes us do the same for homework. (in a particle motion context, where $t$ is time) I am aware that the standard notation is usually $\dot{x}$, ...
11
votes
4answers
310 views

Elegant proof of $\int_{-\infty}^{\infty} \frac{dx}{x^4 + a x^2 + b ^2} =\frac {\pi} {b \sqrt{2b+a}}$?

Let $a, b > 0$ satisfy $a^2-4b^2 \geq 0$. Then: $$\int_{-\infty}^{\infty} \frac{dx}{x^4 + a x^2 + b ^2} =\frac {\pi} {b \sqrt{2b+a}}$$ One way to calculate this is by computing the residues at the ...
2
votes
2answers
56 views

$\int_0^{\pi\over2}{\cos x\sin x\over(x+1)}dx$ = ${1\over2}({1\over2}+{1\over\pi+2}$- $\int_0^\pi{\cos x\over(x+2)^2}dx)$

$$A=\int_0^\pi{\cos x\over(x+2)^2}dx$$ Then prove that, $\displaystyle\int_0^{\Large\pi\over2}{\cos x\sin x\over(x+1)}dx={1\over2}\left({1\over2}+{1\over\pi+2}-A\right)$
23
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4answers
994 views

Integral $\int_0^\infty \log(1+x^2)\frac{\cosh{\frac{\pi x}{2}}}{\sinh^2{\frac{\pi x}{2}}}\mathrm dx=2-\frac{4}{\pi}$

Hi I am trying to show$$ I:=\int_0^\infty \log(1+x^2)\frac{\cosh{\frac{\pi x}{2}}}{\sinh^2{\frac{\pi x}{2}}}\mathrm dx=2-\frac{4}{\pi}. $$ Thank you. What a desirable thing to want to prove! It is a ...
1
vote
0answers
228 views

2008 AP Calculus AB multiple choice

The function f is twice differentiable with f(2)=1, f'(2)=4, and f"(2)=3. What is the value of the approximation of f (1.9) using the line tangent to the graph of f at x = 2? (A) 0.4 (B) 0.6 (C) ...
0
votes
1answer
36 views

Given the derivative is $g'(x) = x^3(x-2)^2(x+8)^9$, where does $g$ have a local maximum?

Consider a function $g(x)$ with derivative $g'(x) = x^3(x-2)^2(x+8)^9$. For what values of $x$ does $g(x)$ have a local maximum? I know the answer is -8, but how do you solve this?
2
votes
2answers
106 views

If $f$ has only removable discontinuities, show that $f$ can be adjusted to a continuous function

I was working on this problem from Spivak's text, and I thought I'd post my answer, in case someone can improve on it. In particular, I wonder if there is a proof that can generalize to topological ...
2
votes
3answers
494 views

Most important things to be proficient in before Calculus 1?

What are the main things one should be proficient in before taking Calculus 1? Please be specific.
0
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2answers
77 views

Evaluate the definite integral of 3x(1-x^2)^5 dx from x =0 to x = 2

I know the answer is -182, but how do you work out this problem?? How in the world do you take the anti derivative of a function like that?
1
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1answer
95 views

How to solve this limit without using L'Hôspital?

This limit wildly appeared in a test and, well, I could not solve it without using L'Hôspital rule - which were not allowed in the test. Can anyone help me? $$\lim_{x \to ...
1
vote
2answers
29 views

Evaluate The Indefinite Integral With U-Sub

I can't get beyond the u-sub in this problem, here's what i've tried: $\int{x^2\sqrt{2+x}}dx$ Let $u=\sqrt{2+x}$ Let $z = 2 + x$ $du = (z)'(\sqrt{z})'dx$ <- Chain Rule $du = ...
1
vote
3answers
233 views

Evaluate $\lim_{x \to 0} \frac{2 - \cos(3x) - \cos(4x)}{x}$

How do you evaluate $$\lim_{x \to 0} \frac{2 - \cos(3x) - \cos(4x)}{x}?$$ I have looked at this problem for a while and cannot think of a way of algebraically determining this limit although I know ...
2
votes
2answers
47 views

Solution of a limit

Find all $(a, b)$ such that $\displaystyle\lim_{x\to 0}\left(\frac{ax-1+e^{bx}}{x^2}\right)=1.$ I figured that $a=-b$, but can't solve the values.
0
votes
1answer
136 views

AP Calculus AB multiple choice number 16

I know we're supposed to use the chaine rule then implicit differentiation. However, can someone show me a step by step explaination?
1
vote
1answer
83 views

Limit of a Cosh function

Evaluate $$\lim_{t\to\infty} (\cosh x)^{1/x}.$$ I tried to use L'Hopital's but I think I made a mess of the differentiation, and the differentiation doesn't seem like it'll help much.
0
votes
1answer
69 views

point wise and uniform convergence of function series

i need some help to understand point wise and uniformly convergence and solve the following: Let f be a series of functions defined by $f_n(x) := \dfrac{1}{n}e^{-n²x²}$. Show that $f'_n(x)$ ...