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

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0
votes
2answers
15 views

Help with Infinite series + Integral Test + Improper Integrals

I am having some trouble with the infinite series $\displaystyle\sum_{n=2}^{\infty}\frac{1}{n\ln^2n}$ . I used the integral test and simplified it to $\int_{\ln 2}^b - \frac 1{\ln(n)}$ (implied ...
5
votes
2answers
20 views

Conceptual question on substitution in integration

In calculus we learn about the substitution method of integrals, but I haven't been able to prove that it works. I mainly don't see how manipulations of differentials is justified, i.e how $dy/dx = ...
0
votes
0answers
4 views

Set containment problem misunderstanding

I just wanted to know if I was misinterpreting this question, my interpretation given after the question below: Given An <= Bn and Cn <= Dn, and as An - Bn -> 0 and Cn - Dn -> 0 as ...
2
votes
1answer
12 views

Proof that the velocity vector is tangential to the path?

In calculus class my teacher asserted that the velocity vector is tangential to the path a point takes. I have tried to prove this but have gotten stuck. I computed $\dfrac{v_y}{v_x}$ to be ...
0
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3answers
59 views

How $\int_{-\infty}^{\infty}\frac{dx}{1+x^2}$ exists?

How $$\int_{-\infty}^{\infty}\frac{dx}{1+x^2}$$ exists? It is difficult question to me. i have tried to evaluate by using fact that $$\int_{-\infty}^{\infty} f(x) \ dx =\int_{-\infty}^{0} f(x)\, dx ...
2
votes
3answers
51 views

What is the integral of $\frac{\sqrt{x^2-49}}{x^3}$

I used trig substitution and got $\displaystyle \int \dfrac{7\tan \theta}{343\sec ^3\theta}d\theta$ Then simplified to sin and cos functions, using U substitution with a final answer of: ...
-1
votes
0answers
12 views

Is product of 2 positive concave function with different domain concave? [on hold]

Suppose f(x) and g(y) are two positive concave functions. h(x,y) = f(x)g(y) Is h(x,y) concave as well?
2
votes
5answers
77 views

Explain why the integral $\int_{-\infty}^\infty x \,dx$ does not exist

Why is it that $$\int_{-\infty}^\infty x \,dx$$ does not exist, but $$\lim_{N \to \infty} \int_{-N}^{N} x\,dx$$ does exist? I was thinking that it involves the fact that in the second case, the ...
0
votes
2answers
36 views

Finding all continuous functions so that $f^n(x)=x$ for some $n$.

I came up with this problem in class but I can't seem to solve it. I need to find all the functions $f$ with domain and codomain $\mathbb R$ such that there is an $n$ such that $f^n(x)=x$ for all $x$, ...
2
votes
1answer
43 views

Prove the sequence determined by $a_{n+1}={a_n\over \sin a_n}$ is convergent, and found its limit.

Let $\{a_n\}$ be a sequence defined by $0<a_1<{\pi \over 2}$, $a_{n+1}={a_n\over \sin a_n}$. $Attempt:$ $a_1>0$ and $\sin a_1>0$ and therefore the sequence begins positive and remains ...
1
vote
2answers
23 views

Rate of change of angle formed by $y=\sqrt[3]{x}$ and the origin.

the problem I am working on is this. Point $P(x,y)$ is on the curve $y=\sqrt[3]{x}$. The angle formed between the $x$-axis and the line that connects the origin and $P$ is $\theta$. As ...
1
vote
0answers
25 views

how to calculate this line integral $\int_{0}^{2\pi} (16\sin^2 3t +16\cos^2 4t)\sqrt{(144\cos^2 3t +256\sin^2 4t)}dt$

I am working on a line integral to calculate the amount of chocolate to cover a pretzel. the density of the pretzel is given by this formula $\lambda=3(x^2+y^2)$ and the parameter equation of a ...
0
votes
0answers
16 views

Proving the following multivariable limit using the definition. [on hold]

I am trying to prove that $ \lim_{(x, y) \to (0, 0)} x + 2xy + 2y + 6 = 6$ using the definition of a multivariable limit, but I am having no luck. Could someone please help me? Thanks!
1
vote
2answers
30 views

What is the indefinite integral of $x^2\sqrt{1+x^2}$

I get this but I don't know if it is correct. I used a reduction formula for $\tan^{2n}(x)\sec^{3}(x)$. Any help would be appreciated. My Final Answer: $$\frac{\sqrt{x^2+1} x}{8}+\frac{\sqrt{x^2+1} ...
1
vote
0answers
13 views

Find a vector equation and parametric equations for the line segment that joins $P$ to $Q$.

Find a vector equation and parametric equations for the line segment that joins $P$ to $Q$. Here $P(1,-1,7)$ and $Q(7,5,1)$. I have tried to find $r(t)$ by using the formula $r(t)=p+t(p-q)$ but ...
0
votes
0answers
16 views

Integral involving rapidly decreasing functions

Let $F,\varphi\in S(\mathbb{R}^n)$, the Schwartz space of rapidly decreasing functions. Is this enough to guarantee that the integral $$\int_{\mathbb{R}^n} F(x)\varphi(x)dx$$ is well-defined? Why or ...
1
vote
0answers
19 views

How to evaluate this (Fourier) integral? [duplicate]

Does somebody know how to evaluate $$\int_{\mathbb{R}^n}\frac{e^{i\langle\xi,x\rangle}}{\|\xi\|_2^2}d\xi$$ for some given $x\in\mathbb{R}^n$ and $n\in\{1,2,3\}$?
-3
votes
3answers
45 views

Easy Analysis question [on hold]

Prove $\{\sqrt{n+1}-\sqrt{n}\}$, $n ≥ 0$, is monotone, using just algebra
2
votes
3answers
38 views

What does it mean for a function to increase along a curve?

I think that if we were to say that, for instance, $y$ increases along the curve, (with no specific rate) then this means for the derivative to simply be positive. Or does it mean to choose the ...
1
vote
1answer
23 views

Improper Integral - Multiple Choice Problem - $I$

Let $f$ be a function defined $\forall~ x\geq 1$.Let $n$ denote a positive integer and let $I_n$ denote the integral $\int_1^nf(x)dx$ which is always assumed to exist. Which of the following ...
0
votes
0answers
20 views

If $S_n=\sum_{k=1} ^n{}a_k \to \infty$ as $n\to \infty$ then $\sum_{k=1}^{\infty}{a_k\over {1+a_k}}$ diverges. [duplicate]

Let $a_n>0$ and let $S_n=\sum_{k=1} ^n{}a_k \to \infty$ as $n\to \infty$. Prove $\sum_{k=1}^{\infty}{a_k\over {1+a_k}}$ diverges. I am confused by this sort of sequence\sum thing. How can I use ...
2
votes
0answers
69 views

Derivative under integral mixed with…

$$f(x,y)=\int_{e^{4y}}^{\ln^3(x)}{\frac{\sin(t)}{t}\,dt}$$ Whats the derivative $\frac{d f}{d t}$, if: $$x(t)=\cos(2+6t).4t^2$$ $$y(t)=\ln(2r+7e^{5t})$$ Really not much to say about this problem ...
4
votes
3answers
48 views

taylor of $\frac{1}{z}$ at $a=-2$

I want to find the taylor series representation of $f(z)=\frac{1}{z}$ at $a=-2$. The point of this exercise is not to find some pattern in the derivatives, infact we are not meant to find any ...
1
vote
4answers
79 views

Find $\lim_{n\to \infty}(\cos{x\over n})^{n^2}$

Find $$\lim_{n\to \infty}\left(\cos{x\over n}\right)^{n^2}$$ where $x\in \Bbb{R}.$ I tried using taylor series. A complete mess, and an area I am not very good at. I tried using $e$ which also gave me ...
1
vote
4answers
69 views

Computing the sixth derivative of $F(x) = \int_1^x\sin^3(1-t)\mathrm dt$

Compute the sixth derivative at $x_0 = 1$ of $$F(x) = \int_1^x\sin^3(1-t)\mathrm dt$$ It's from a multiple choice test. I was able to narrow down the choices to $0$ and $60$. I guessed $0$ and ...
1
vote
1answer
23 views

Is this theorem about integration with substitution wrong?

A theorem in my book states: If $g$ is differentiable, f is continuous, and F is an antiderivative of f, then : $\int f[g(x)]g'(x)dx=F[g(x)]+C$ The reason I am asking if this is correct, ...
1
vote
1answer
33 views

When does the integral converges?

For what $\alpha, \beta$ the integral $$\int_0^\frac{\pi}{2} \frac{(\frac{\pi}{2} - x)^\alpha}{(\cos x)^\beta} dx$$ converges? So first I've approved (using WolframAlpha) that $\frac{\pi}{2} - x ...
1
vote
2answers
57 views

How can I prove this integral?

I have to use the identity $b^4-a^4=(b-a)(b^3+b^2a+ba^2+a^3)$ to prove that: $\int_b^ax^3dx=\frac{b^4-a^4}{4}$. I know that you can just do $F(b)-F(a)$ and since the integral of $x^3$ is ...
0
votes
1answer
37 views

Getting ready for Calculus?

So I wanted to start a Masters program but they require that I have Calculus III. I want to take that course at the university, but I need to be ready for it. As I look at Khan Academy and do some ...
3
votes
3answers
160 views

Inequalities proven by real analysis or induction.

Let $t\in [-1,1]$. Prove that $(1+t)^p+(1-t)^p\ge2$ when $p\ge 1$ and that $(1+t)^p+(1-t)^p \le 2$ where $0 \le p\le 1$. I am not sure how I should solve it. I tried induction at first and it was ...
0
votes
4answers
65 views

Convergence of $\int_0 ^\infty \frac {dx}{\sqrt {1+x^3}}$

Convergence of $\int_0 ^\infty \dfrac {dx}{\sqrt {1+x^3}}$ Attempt: $\lim_{x \rightarrow \infty} \dfrac {x^{\frac{3}{2}}}{\sqrt {1+x^3}} =1$ Hence, $\dfrac {1}{x^{\frac{3}{2}}}$ and $\dfrac ...
0
votes
2answers
30 views

calculate $\int_{0}^{2\pi}\frac{1-\sin(t)}{2-\cos(t)}dt$

I need to calculate $\int_{\gamma} \frac{1-\sin(z)}{2-\cos (z)}dz$ where $\gamma$ is the upper hemisphere of the circle with center $\pi$ and radius $\pi$, with a positive direction. The original ...
0
votes
5answers
49 views

Evaluate the Limit $\lim_{x\to 0} {\left((e^x - (1+x)) \over x^n\right)}$

Evaluate the Limit: $$\lim_{x\to 0} {\left((e^x - (1+x)) \over x^n\right)}$$ I am trying to understand how to do this. I have to use series expansion and not L'Hospital. Any help would be great. ...
0
votes
0answers
33 views

$S_{1}\iff S_{2}$ in complex numbers

Let : $$a_0 , a_1 , a_2 \in \mathbb{C} \text{ and } :b_0 , b_1 , b_2 \in \mathbb{C}$$ : Show the following equivalence : $$\begin{cases} ( 1 + a_0 ) ( 1 + a_1 ) ( 1 + a_2 ) &=& ( 1 + b_0 ) ( ...
0
votes
0answers
20 views

Function with the opposite definition of Dirichlet function? [on hold]

I just happened onto the Dirichlet function today that states: $D(x)= \begin{cases} 0 & \text{if $x$ is irrational,}\\ 1 & \text{if $x$ is rational} \end{cases}$ which shows that points can ...
0
votes
1answer
26 views

If a power series converges uniformly on $\mathbb{R}$ then it must be to $0$?

Let $f(x) = \sum a_n x^n$. Let's assume that $f(x)$ has a radius $R=\infty$ and $f(x)$ converges uniformly. Now, obviously $f(0) = 0$. Meaning, $f(x)$ pointwise converging at $x=0$. Since we assumed ...
0
votes
1answer
16 views

Verifying transport equation solution

I have just started PDE's and I have the transport equation $u_t + au_x = 0$ which has the general solution $u(x,t) = f(x - at)$ In a book I'm reading it says this can be verified by substitution ...
0
votes
1answer
10 views

What is the connection between slant/oblique asymptote to the polynomial part of the function and polynomial division?

What is the connection between slant/oblique asymptote calculation to the polynomial part of the function and polynomial division? To find the slant asymptote $y=mx+n$ we can can calculate it in two ...
4
votes
2answers
41 views

$\lim_{n \rightarrow \infty} \frac {1^{a+1}+2^{a+1}+\cdots+n^{a+1}}{n.(1^{a }+2^{a }+\cdots+n^{a })} $

The value of $$\lim_{n=\infty} \dfrac {1^{a+1}+2^{a+1}+\cdots+n^{a+1}}{n.(1^{a }+2^{a }+\cdots+n^{a })} $$ Attempt: $S = \lim_{n \rightarrow \infty} \sum_{n=0} ^\infty \dfrac {k^{a+1}} {n.( 1^{a ...
1
vote
1answer
45 views

Computing a strange integral

Prove that $(-1)^n \int_{-1}^1 (x^2 - 1)^ndx = \frac{2^{2n+1}(n!)^2}{(2n+1)!}$ This one has me stumped. I've tried the obvious (using binomial theorem and then integrating termwise, or computing the ...
1
vote
1answer
35 views

Help with troublesome limit [on hold]

I need help in computing the following limit: $$\lim_{x\to 0} \frac{a}{x}~\exp \left(-\frac{a^2(\log(bx))^2}{2}\right)$$
2
votes
2answers
51 views

How to solve $\int{\frac{1}{\sqrt{3-2x-x^2}}\,dx}$?

$$\int{\frac{1}{\sqrt{3-2x-x^2}}\,dx}$$ I tried to do it by substitution with no sucess. Anyone can solve it?
-4
votes
2answers
35 views

How to evaluate $\lim_{x\rightarrow +\infty } \frac{\ln(a+be^x)}{\sqrt{a+bx^2}} =? $

Suppose $b>0$ $\lim_{x\rightarrow +\infty } \frac{\ln(a+be^x)}{\sqrt{a+bx^2}} =? $ I know that this is an indetermination of the form $\infty / \infty$ . I tried to use L'Hôpital, but I didn't ...
2
votes
3answers
41 views

How to evaluate $ \lim_{x\rightarrow +\infty } \sqrt[x]{a^x+b^x} = ? $

If a>0 and b>0, $ \lim_{x\rightarrow +\infty } \sqrt[x]{a^x+b^x} = ? $ What I was trying to do: Suppose a>b. Then, for sufficiently large values of x, $ a^x >> b^x $; so $\sqrt[x]{a^x+b^x} ...
1
vote
3answers
97 views

Anyone can integrate $e^{-\frac{x^2}{3}}$ by hands?

I just used wolfram integral calculator and the result is weird, there is something called error function. $$ \int_{-\infty}^\infty e^{-\frac{x^2}{3}}\,\mathrm dx $$ Hint says that change of variable ...
2
votes
2answers
30 views

Which of the following statements are true $(NBHM - 2015)$?

Let $X =\{f \in C[-5, 5] : f(-5)= f(5) = 0 \} $ There exist $ f \in X$ such that $ f \equiv 2$ on $[-1, 0]$ and $ f \equiv 3$ on $[1 , 2] \cup [3 , 4]$ For every $ f \in X$, there exist ...
1
vote
1answer
36 views

Can a divergent alternating series by rearrangement of terms be made to converge to a value?

Riemann discovered that a conditionally convergent series, through rearrangement of it's terms, can be made to converge to any value. But, if $S$ is a divergent alternating series, through ...
4
votes
2answers
25 views

Investigate the convergence of $\int_1^\infty \frac{\cos x \ln x}{x\sqrt{x^2-1}}$

Investigate the convergence of $$\int_1^\infty \frac{\cos x \ln x}{x\sqrt{x^2-1}}$$ so first of all let's split the integral to: $$I_1 = \int_1^2 \frac{\cos x \ln x}{x\sqrt{x^2-1}}, I_2 = ...
-2
votes
0answers
24 views

Find area of surface $x^2+y^2+z^2=4$ which lies directly above cardioid $p=1-\cos t$ [on hold]

It doesn't appear like a hard problem since the overall formula would be: $$S=\int_0^\pi\int_0^{1-\cos\theta}{\frac 2{\sqrt[2]{4-\rho^2}}}\, \rho\, d\rho d\theta,$$ but then I get stuck on solving the ...
0
votes
0answers
25 views

Need help solving this integral

$\int_1^\infty du$$\int_{-2}^2 dv(u-v)e^{-u}$ Do I just evaluate the integrals separately and then multiply the answers together?