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

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0
votes
4answers
81 views

Finding the limit of a sequence $a_n = 1/(n+1) + 1/(n+2) +\cdots+1/(2n)$ where $n$ is natural number.

First, i should prove that this sequence converges, which is not that difficult since $$ a_{n+1} - a_n > 0 \\ \frac{1}{n+1} + \frac{1}{n+2} + \cdots+ \frac{1}{2n} + \frac{1}{2n+1} - \frac{1}{n+1} - ...
1
vote
3answers
49 views

Solve $y'=2 \sqrt{|y|}$, $y(0)=0$

Solve $y'=2 \sqrt{|y|}$, $y(0)=0$. $$\frac{dy}{2\sqrt{|y|}}=dx$$ The left side seems to be similiar to the derivative of $\sqrt{y}$ but the initial condition is $y(0)=0$ so it makes the denominator ...
1
vote
3answers
60 views

I would like to calculate the following limit: $\lim_ {n \to \infty} {\left( {n\cdot \sin{\frac{1}{n}}} \right)^{n^2}}$

I would like to calculate the following limit: $$\lim_ {n \to \infty} {\left( {n\cdot \sin{\frac{1}{n}}} \right)^{n^2}}$$ or prove that it does not exist. Now I know the result is $\frac{1}{\sqrt[6]{...
4
votes
1answer
49 views

Calc III Stoke's Theorem Calculate $\int_C v \cdot dr$

The sphere $x^2 + y^2 + z^2 = a^2$ intersects the plane $x + 2y + z = 0$ in a curve $C$. Calculate $\int_C \vec{v} \cdot d\vec{r}$, where $\vec{v} = 2yi -zj +2xk$ So I solved this question by taking ...
4
votes
2answers
56 views

Find $f(c)$ given the graph of $f'(x)$

I am given a graph of the derivative of some function and asked to find $f(c)$. I can't explain it well without the graph, so here is the image of the problem: First of all, should't the point ...
6
votes
2answers
97 views

I would like to calculate $\lim_ {n \to \infty} {\frac{n+\lfloor \sqrt{n} \rfloor^2}{n-\lfloor \sqrt{n} \rfloor}}$

I would like to calculate the following limit: $$\lim_ {n \to \infty} {\frac{n+\lfloor \sqrt{n} \rfloor^2}{n-\lfloor \sqrt{n} \rfloor}}$$ where $\lfloor x \rfloor$ is floor of $x$ and $x ∈ R$. Now I ...
2
votes
0answers
39 views

Change from Fourier Space to Real Space

I have a function in 3D fourier Space $$g(\textbf {k})=\frac{\hat{k}_i}{\hat{k_j}}f(\textbf {k}),$$ where $\hat{\alpha}$ is a fixed vector and $i$ and $j$ are the components of the relevant vector, ...
1
vote
3answers
33 views

Finding the integral variables

Is it okay to say, for example, $\displaystyle \int \dfrac{x^2}{x^3+1}dt$ if $t$ is a function of $x$? I was doing a trig substitution for a trig problem and I had intermediate steps where I had ...
1
vote
2answers
63 views

How to calculate limit of series

I have many limits for homework that I dont know how to solve them. I tried many things, but dont have any idea. Hope you can help me $$\lim_{n\to \infty} n*c^n $$ when $$\lvert c\rvert < 1$$ ...
3
votes
3answers
73 views

Calculating $\int \sin x e^x \Bbb d x$ by parts twice

Integrate $e^x \sin x $. I know I need to integrate by parts 2 times, but I'm stuck at the second integration. For the first I get $$-e^x \cos x - \int e^x\cdot (-\cos x) \,dx $$ Correct me if I'...
2
votes
4answers
107 views

Evaluate the Integral: $\int\frac{dx}{\sqrt{x^2+16}}$

I want to evaluate $\int\frac{dx}{\sqrt{x^2+16}}$. My answer is: $\ln \left| \frac{4+x}{4}+\frac{x}{4} \right|+C$ My work is attached:
4
votes
3answers
85 views

Find an unbound sequence in which $a_{n+1}$ - $a_n$ converges to zero

Find an unbound monotonic increasing sequence in which $a_{n+1}$ - $a_n$ converges to zero. I've been thinking about this one for a while. found it in an exam from seven years ago. I don't know if ...
3
votes
3answers
74 views

I would like to calculate $\lim_{x \to \frac{\pi}{6}} \frac{2 \sin^2{x}+\sin{x}-1}{2 \sin^2{x}-3 \sin{x}+1}$

I want to calculate the following limit: $$\lim_{x \to \frac{\pi}{6}} \frac{2 \sin^2{x}+\sin{x}-1}{2 \sin^2{x}-3 \sin{x}+1}$$ or prove that it does not exist. Now I know the result is $-3$, but I am ...
4
votes
2answers
140 views

Limit of a periodic function

I stumbled upon this question in my course, and I am out of ideas. Let $f$ be a periodic function $$f(x)=f(x+l), \qquad l>0$$ Prove that if it is not constant, then $\lim_{x\to 0}f\left(\frac1x\...
0
votes
5answers
90 views

A simple integration problem.

I must find the following: $$\int (x^2 + 1)^{50}\,dx$$ Would rewriting it as $50\ln(x^2+1)$ and then integrating be wrong? Using regular the substitution $u= x^2+1$, I get the answer to be $$\...
0
votes
3answers
63 views

List all degree one, two, three polynomials.

All degree one polynomials over $\mathbb{C}$ are of the form $ax+b$, where $a,b$ are complex numbers. How two write down all degree two (and three) polynomials over $\mathbb{C}$? I only need to write ...
28
votes
4answers
1k views

An infinitely powered expression [duplicate]

Here's an expression I am struggling to evaluate: $$\LARGE {\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\:\cdot^{\:\cdot^{\:\cdot}}}}}} $$ The value turns out be $2$, but I don't understand how do we get it. Can ...
6
votes
1answer
169 views

How to Prove the Chain Rule for Limits Using a $\varepsilon-\delta$ Argument?

I came across the chain rule for limits the other day and it interested me quite a bit and surprisingly I couldn't find the proof on the internet anywhere. From what I understand the chain rule for ...
3
votes
1answer
85 views

$\lim_{x\to 0}\frac{2+\cos x}{x^3\sin x}-\frac{3}{x^4}=\frac{1}{60}$ without using L Hospital rule or series expansion

Prove that $\lim_{x\to 0}\frac{2+\cos x}{x^3\sin x}-\frac{3}{x^4}=\frac{1}{60}$ without using L Hospital rule or series expansion. I tried $\lim_{x\to 0}\frac{2+\cos x}{x^3\sin x}-\frac{3}{x^4}=\...
2
votes
1answer
88 views

Is there any error in my solution?

Let $f(x)$ is continuous function for all real values of $x$ and satisfies $$\int^x_0 f(t)dt =\int^1_{x} t^2f(t)dt +\frac{x^{16}}{8}+\frac{x^6}{3}+a$$ Then the value of a is equal to: (a) $\frac{-...
7
votes
2answers
147 views

Integral $\int_0^1\arctan(x)\arctan\left(x\sqrt3\right)\ln(x)dx$

I need to evaluate this integral: $$\int_0^1\arctan(x)\arctan\left(x\sqrt3\right)\ln(x)dx$$ Apparently, Maple and Mathematica cannot do anything with it, but I saw similar integrals to be evaluated in ...
0
votes
1answer
45 views

Need to prove $\int_{0}^{1}\frac{x^{2n+1}}{\sqrt{1-x^{2}}}\;dx = \int_{0}^{\pi/2}\sin^{2n+1}\theta\;d\theta$ [closed]

Please help me. I'm having trouble trying to prove $\int_{0}^{1}\frac{x^{2n+1}}{\sqrt{1-x^{2}}}\;dx = \int_{0}^{\pi/2}\sin^{2n+1}\theta\;d\theta$
4
votes
2answers
55 views

Convergence of a series $\sum {\left( {1 - \frac{{\sin {a_n}}}{{{a_n}}}} \right)} $

Given ${a_n}$ a converging series $\sum {{a_n}}$, and also that for every $n$, $a_n\ne0$. Does the series $\sum {\left( {1 - \frac{{\sin {a_n}}}{{{a_n}}}} \right)} $ converge? For every example I've ...
1
vote
1answer
57 views

What's the $\otimes$-operator in the proof of Reynolds' transport theorem at Wikipedia?

In the proof of Reynolds' transport theorem at Wikipedia, they use the identities $$\nabla\cdot(v\otimes w)=v(\nabla\cdot w)+\nabla v\cdot w$$ and $$(a\otimes b)\cdot n=(b\cdot n)a\;,$$ where $n$ is ...
4
votes
0answers
64 views

Find $\int_{0}^{\frac{\pi}{2}} \dfrac{\sin^n(x)}{\sin^n(x)+\cos^n(x)} dx$. [duplicate]

Find $\displaystyle \int_{0}^{\frac{\pi}{2}} \dfrac{\sin^n(x)}{\sin^n(x)+\cos^n(x)} dx$. I was told to switch the limits of integration then add them. How can I do that here?
0
votes
1answer
57 views

write in form of $x+iy$

Can somebody help me, can somebody give a hint to solve this question $\dfrac{\log z}{z^{3}-(1+i)(z)} $ at $z=1+i$ actually, I want to but it in the form of $x+iy.$
1
vote
3answers
47 views

Prove that $\sin(2\sin^{-1}(\alpha)) = 2\alpha \sqrt{-\alpha^2+1}$.

Prove that $\sin(2\sin^{-1}(\alpha)) = 2\alpha \sqrt{-\alpha^2+1}$. I was doing a trigonometric substitution problem in Calculus and came across this and wanted to know the proof of it.
1
vote
2answers
129 views

Will integral be $\frac{\pi}{2}$?

Show that $\int \frac{1-cosx}{x^2}\ dx=\frac{\pi}{2}$. I used Taylor's series for cosx to find integral but I don't see intergal becoming equal to $\frac{\pi}{2}$ without any limits of integration. ...
0
votes
2answers
50 views

Limit of a sequence: proof $\lim(a_n/b_n) \to L/K$

$$ \lim_{n\to \infty} (a_n) = L$$ $$ \lim_{n\to \infty} (b_n) = K$$ I need to proof that: $$ \lim_{x\to \infty} \frac{a_n}{b_n} = \frac{L}{K}$$ I have to get to something like that: $$ \left| \...
2
votes
4answers
35 views

Problem with solving a specific limit

So, here is the limit $\lim_{n\to\infty}\left(\frac{3n^{2}+4n-5}{3n^{2}-7n+9}\right)^{n}$ I'm not really sure how I should approach this limit. Would really appreciate if someone could say the in ...
2
votes
4answers
123 views

Find $ \int \frac{1}{2\sin(x)-3\cos(x)}dx$.

Find $\displaystyle \int \dfrac{1}{2\sin(x)-3\cos(x)}dx$. My book said to solve this by saying $u = \tan \left(\dfrac{x}{2} \right)$ since $\cos(x) = \dfrac{1-u^2}{1+u^2}$ and $\sin(x) = \dfrac{2u}{...
1
vote
0answers
89 views

How to calculate the indefinite integral of $\frac{\sin(x)}{\sqrt x}$?

So my question simply is: what is $\int \frac{\sin(x)}{\sqrt x}\,d x $?
3
votes
3answers
91 views

Calculus contradiction?

Why is $$ \int \dfrac{1}{2x+2}dx \neq \dfrac{1}{2}\int \dfrac{1}{x+1} dx?$$ According to WolframAlpha: http://www.wolframalpha.com/input/?i=integral+of+1%2F%282x%2B2%29+%3D+1%2F2%28integral+of++1%2F%...
4
votes
3answers
203 views

Differentiable function satisfying $f(x+a) = bf(x)$ for all $x$

This is an exercise from Apostol Calculus, (Exercise 10 on page 269). What can you conclude about a function which has derivative everywhere and satisfies an equation of the form $$ f(x+a) = ...
5
votes
3answers
130 views

$\lim_{x\to 0}\frac{\sin 3x+A\sin 2x+B\sin x}{x^5}$ without series expansion or L Hospital rule

If $$f(x)=\frac{\sin 3x+A\sin 2x+B\sin x}{x^5},$$ $x\neq 0$, is continuous at $x=0$, then find $A,B$ and $f(0)$. Do not use series expansion or L Hospital's rule. As $f(x)$ is continuous at $x=0$,...
2
votes
4answers
109 views

Computing the Integral $\int \frac{\sqrt{x}}{x^2+x} dx$

Find $\displaystyle \int \dfrac{\sqrt{x}}{x^2+x} dx$. What would be the best way to integrate this? I saw the answer to this and it looked simple so that might mean the steps would be too?
2
votes
0answers
431 views

Any sort of real world application for volumes of solids (revolution)$?$

Just wondering if you've ever encountered an actual situation that is related to the concept of integration of cross sectional area to find $3d$ volume.
0
votes
1answer
34 views

Domain of piecewise-defined functions composition

I'm wondering what is the right way to perform function composition on those two functions: $$f\left(x\right) = \left\{ \begin{array}{lr} 1/x & : x \ne 0\\ 0 & : x = 0 ...
2
votes
1answer
51 views

On the sup convolution of upper semi continuous function

Let $f$ is a bounded upper semi continuous function on $[-1,1]$ and for any $\epsilon>0$, we define $$f^{\epsilon}(t):=\max\limits_{s\in[-1,1]}\left\{f(s)+\epsilon-\frac{1}{\epsilon}|t-s|^{2}\...
1
vote
4answers
121 views

Integrate $\int \frac{1}{x^4+4}dx$

Integrate $\displaystyle \int \dfrac{1}{x^4+4}dx$. I could try breaking this up into two quadratic trinomials, but that seems like it would be a lot of work. If that is the best way here how do I ...
6
votes
2answers
99 views

Is there standard terminology to describe the not-quite-a-limit behavior of ${\tan( \log x) \over x}$ as x approaches infinity?

Suppose I want to describe the long term behavior of ${\tan(\log x) \over x}$ as x increases towards positive real infinity. Now, $$\lim_{x \rightarrow \infty}{\tan(\log x) \over x}$$ obviously ...
3
votes
3answers
52 views

Limit of $\frac{1-2\cos(x)+\cos^2(2x)}{x^2}$

I tried to find the value of this limit without L'Hopital , but no luck $$\lim_{x\to 0}\frac{1-2\cos(x)+\cos^2(2x)}{x^2}$$
1
vote
2answers
34 views

applications to calculus questions

A farmer has 80m length of fencing. He wants to use it to form 3 sides of a rectangular enclosure against an existing fence, which provides the 4th side. find the maximum area that he can enclose and ...
11
votes
1answer
117 views

Exponential integral: something is wrong.

Consider the function $$ E(z)=\int_{-\infty}^z\frac{e^t}{t}dt.\quad (1) $$ Substituting $t\mapsto -u$ one obtains $$ E(z)=-\int_{-z}^{\infty}\frac{e^{-u}}{u}du\equiv Ei(z).\quad (2) $$ It is ...
11
votes
1answer
140 views

Prove that $f_n(x)=\frac{x}{n}$, $n=1,2,\ldots$ does not converge uniformly on $\mathbb{R}$

Prove that $f_n(x)=\frac{x}{n}$, $n=1,2,\ldots$ does not converge uniformly on $\mathbb{R}$. It's clear this function converges pointwise to $0$ function. We have to show that there is $\epsilon&...
2
votes
1answer
229 views

Double Integral with a Delta Function

Consider the integral $$\int_0^b\int_0^a\delta(x-y)f(x,y)dxdy$$ where $b>a$. I know that we need to integrate over the larger range first (i.e do the $y$ integral) and then do the remaining ...
4
votes
2answers
61 views

Calculating $\lim_{n \to \infty}\frac{n^{3}}{(3+\frac{1}{n})^{n}}$

I need help to calculate this limit: $$\lim_{n \to \infty}\frac{n^{3}}{(3+\frac{1}{n})^{n}}$$
1
vote
2answers
56 views

Delta epsilon argument in general

When I want to prove something in mathematics fe an expression goes to zero, I can either use basic rules of 'limits' or I can use the epsilon-delta method. I have a feeling that it's more consistent ...
1
vote
2answers
58 views

Find the value of $\lim_{x \to 2} \frac {xf^2(x)-9}{(x-2)}$

Suppose that $$\lim_{x\to 2}\frac{x f(x)-3}{x-2}=5.$$ Then, what is the value of $$\lim_{x\to 2}\frac{x f^2(x)-9}{x-2}?$$
5
votes
3answers
34 views

Show that $\lim\limits_{r\to\infty} \textrm{\{} \|x-ru\|-\|y-ru\| \textrm{}\} = \left<y-x,u\right>$

Let $x,y,u \in \mathbb{R}^2, r\in\mathbb{R}$ and $\|\cdot\|$ be the norm. Show that $$\lim\limits_{r\to\infty} \textrm{\{} \|x-ru\|-\|y-ru\| \textrm{}\} = \left<y-x,u\right>$$ I have to tried ...