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

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4
votes
2answers
71 views

Find: $\lim\limits_{x\to 0}{x^{\alpha}\int_{x}^{1}{f(t)\over t^{\alpha +1}}dt}$.

Let $f$ be continuous on $[0,1]$, and let $\alpha>0$. Find: $\lim\limits_{x\to 0}{x^{\alpha}\int_{x}^{1}{f(t)\over t^{\alpha +1}}dt}$. I tried integration by parts, but I am not sure if $f$ is ...
0
votes
0answers
14 views

Proving multi-variable differentiability using the limit definition

I'm doing advanced calculus and I find it challenging to solve multi-variable limits while proving differentiability, more specifically 2 variable limits. could you show me how do I solve this limit?: ...
1
vote
6answers
150 views

What are the limits of these sine functions as x approaches infinity?

I have two different limit sine functions: $$\lim_{x\to \infty} {\sin x\over x}$$ $$\lim_{x\to \infty}x^2\sin\left({3\over x^2}\right)$$ My thoughts are that as $x$ becomes infinitely large, then ...
1
vote
1answer
52 views

Convergence of $\int _{-\infty}^{+\infty}\sin(cx)dx$

At this forum there is an abundance of questions regarding the convergence of integrals and sums of infinite series. The mathematicians who answer these questions emphasize that only under strict ...
0
votes
1answer
363 views

How to find the limits of integration to get the area for a loop of a lemniscate?

I know how to integrate the squared radius to get the equation that'll give me the area, like such for a lemniscate with $r^2=8\sin(2\theta)$ : $$1/2\int 8sin(2\theta) = 4 \int \sin(2\theta) = 4 * ...
2
votes
2answers
214 views

Integration: Method of partial fractions - Any standard method of finding constants in hard to solve expressions?

I've been computing many indefinite integrals using the method of partial decomposition. The integrals are usual on the form $$\int \frac {x^2-29x+5} {(x-4)^2(x^2+3)} dx$$ which is equal to $$\int ...
2
votes
0answers
26 views

Continuity of $f^{(n-1)}$ in Taylor's Theorem with Mean-value remainder

I refer to Rudin's proof of Taylor's Theorem with the Mean-value form of the remainder. I'm not sure if I'm understanding the proof correctly. Why must $f^{(n-1)}$ be continuous on $[a,b]$? I ...
1
vote
2answers
68 views

On proving the total differential.

I am following an open-course on multi variable calculus provided by MIT taught by Denis Auroux. The question I am about to ask is from this lecture. In the lecture Denis Arnoux gives a sketch proof ...
1
vote
0answers
33 views

Using Multipule Scale Analysis to solve a non-linear differential equation

I would like to know if there are other methods to solve equations such as this one below. I don't really understand the theory behind the multiple scale analysis and why it works I understand some of ...
0
votes
2answers
26 views

Domain of derivative on open interval is open

Let $f : (a, b) \to \mathbb{R}$. Suppose that the derivative $f'$ exists at every point of a set $E \subseteq (a,b)$. Is it true that the domain $E$ of $f'$ is open? And if it is not true, is it true ...
0
votes
2answers
37 views

Optimization Problem: Fence with adjacent sides rather than opposing sides

I'm unsure if I got the following right on a test I just took: A farmer wants to build a rectangular fence using both wood and metal and wants adjacent sides to be of the same material. Metal costs ...
1
vote
2answers
186 views

Volume of a rotated region?

How can I find the volume of the solid generated when the region enclosed by $y=0, x=0, x=1$ and $(1+e^{-2x})^{0.5}$ is rotated through $360^\circ$ about the x axis?
9
votes
2answers
141 views
+100

Proof of strictly increasing nature of $y(x)=x^{x^{x^{\ldots}}}$ on $[1,e^{\frac{1}{e}})$?

The title is fairly self explanatory: I have been trying to rigorously prove that $y(x)=x^{x^{x^{\ldots}}}$ is a strictly increasing function over the interval $[1,e^{\frac{1}{e}})$ for a while now, ...
1
vote
1answer
66 views

Coefficient of operator and how to do it

This question stems from this $$ \frac{1}{x+z}- \frac{1}{x} = \sum_{k=0}^\infty \frac{z^k}{k!}\frac{d^k}{dx^k}[\frac{1}{x}] $$ Now, i need to find the Bell Polynomial of $\frac{1}{x}$, $$ ...
1
vote
0answers
20 views

Let $A,B:V\to V$ positive definite operators in complex linear space with inner product $V$, $dimV<\infty$

Let $$A,B:V\to V$$ positive definite operators in complex linear space with inner product $$V$$, $$dimV<\infty$$ Show that $$log det(A\cdot B^{-1})=-\int_{0}^\infty tr(e^{-t\cdot A}-e^{-t\cdot ...
0
votes
1answer
1k views

Surface Area of a Steinmetz Solid

A Steinmetz solid is the intersection of two cylinders. My question is how to find the surface area of one with both cylinders of radius 1, without parameterizing the solid. I parameterized it; that ...
5
votes
1answer
330 views

prove the general arithmetic-geometric mean inequality

Prove that the general arithmetic-geometric mean inequality \begin{equation*} (a_{1}a_{2}...a_{n})^\frac{1}{n}\leq\frac{a_{1}+a_{2}+...+a_{n}}{n} \end{equation*} holds for all $a_{i}$ positive real ...
1
vote
3answers
51 views

How do I integrate this exponential + Bessel function term?

I would like to integrate this in my research: $\int_0^\infty s e^{i bs^2}J_0(a s)$, where a and b are both real and greater than zero. Integration by parts seems like the obvious first step, but ...
2
votes
2answers
35 views

Find Taylor expansion of $f(x)=\ln{1-x^2\over 1+x^2}$, and then find radius of convergence.

Find Taylor expansion of $f(x)=\ln{1-x^2\over 1+x^2}$, and then find radius of convergence. My problem here is the Taylor series. Computing the few first derivative is possible, but I can't seem to ...
1
vote
1answer
33 views

finding and proving where function is…

So I have this function: $ f(x) = \begin{cases} ( 2 \sqrt{-1-x}-1)^{\frac{1}{4^{-x}-16}} & \quad \text{if } x<{-2}\\ - \frac{\pi}{4}x & \quad \text{if } -2\leq x \leq 1 \\ \frac{\sin{(\pi ...
14
votes
1answer
255 views

Evaluating $\int_0^1 \frac{z \log ^2\left(\sqrt{z^2+1}-1\right)}{\sqrt{1-z^2}} \, dz$

What kind of real analysis tools would you employ for this integral? $$\int_0^1 \frac{z \log ^2\left(\sqrt{1+z^2}-1\right)}{\sqrt{1-z^2}} \, dz$$ EDIT: Here is a supplementary question, the cubic ...
1
vote
2answers
30 views

complex numbers equation, find all z…

So i have to find all $z\in \mathbb{C}$ that solve these two equations(separately) first: $\bar{z}+z=i(\bar{z}-z)$ second: $\bar{z}+z^n=i(\bar{z}-z^n), \forall n \in\mathbb{N}$ So basically, i ...
0
votes
0answers
24 views

How do I formulate a specific formula for a sequence?

I have three arrays, for instance s = [1:2], j = [1:20] and b = [1:8], and I am trying to build a single row. The problem that I actually have is that I need to find a formula f(s,j,b) such that ...
0
votes
1answer
27 views

How do I find unknown values of constants that make a function differentiable everywhere?

I have this function, and I need to find the values of $a, b, c$ and $d$ so that $f(x)$ will be differentiable everywhere. $$f(x)=\begin{cases} ax^2+b, & x<1 \\ cx+d, & 1\le x<3\\ ...
2
votes
3answers
41 views

Reasoning about numbers close to two other numbers $a,b$ (inequalities)

Let $a < b$ and $0 < \varepsilon < (b - a)$ and let $x, y \in \mathbb R$ be such that $$ | x - a | < \frac{(b - a) - \varepsilon}{2}, \qquad | y - b | < \frac{(b - a) - ...
0
votes
4answers
44 views

find coordinate on line at given distance from given coordinate

I got two coordinates of a straight line $(-2,-4)$ and $(3,4)$. How can i find a coordinate that lies on this line and is $5$ units away from the $(-2,-4)$ coordinate?
0
votes
0answers
44 views

Volume between $y = 1/x$, $y= 0$, $x=1$, $x=3$ about $y = -1$ (using shell method)

Pretty obvious here that disk method is easy and I got the right answer according to the book with it. However for the last hour I have been trying to use shell method and nothing seems to be working. ...
1
vote
1answer
47 views
+50

Prove that the difference of continuous and monotonically increasing functions has continuous variation

Let $G:[0,\infty)\to\mathbb{R}$ be continuous and $$V^1_t(G):=\sup\bigcup_{n\in\mathbb{N}}\left\{\sum_{i=0}^{n-1}\left|G_{t_{i+1}}-G_{t_i}\right|:0=t_0\le\cdots\le t_n=t\right\}$$ be the variation ...
0
votes
3answers
69 views

How do I derive the volume of this cup?

How do I derive the volume of this cup? It's been many years since I've taken calulus... So far I've started with the radius of the bottom and integrated that around the circle. Did I start it right? ...
4
votes
1answer
451 views

Can a curve be an asymptote?

$f(x)=x^3+\frac{3}{x-1}$ This was the question given to me.I replied that $f(x)$ will have only a single vertical asymptote of $x=1$. My teacher told that there'll be be two asymptotes.One is the ...
0
votes
1answer
1k views

proof of the second generalized mean value theorem for integrals

Let $f,g,g´$ be continous on $[a,b]$ and $g$ monotone on $[a,b]$; then there exist $c\in (a,b)$ so that $$\int_{a}^{b}f(x)g(x)dx=g(a)\int_{a}^{c}f(x)dx+g(b)\int_{c}^{b}f(x)dx$$ Ineed to apply the ...
3
votes
0answers
73 views

Calculate in closed form $\sum_{n=1}^{\infty} \frac{\arctan(1/n) H_n}{n}$

Playing with Taylor series is not helpful enough. What else would you try out? $$\sum_{n=1}^{\infty} \frac{\arctan(1/n) H_n}{n}$$ $$\approx 2.1496160413898356727147400526167103602143301206321$$ It's ...
2
votes
1answer
33 views

What exactly is “integrated form”?

I am reading on population growth and I see $\Delta N_t = (b - d)N_t \, \Delta t = mN_t \, \Delta t$ , where $m = b - d$. As $\Delta t \to 0$, this becomes $\dfrac{dN_t}{dt} = ...
1
vote
4answers
95 views

Solving $\int \frac{dx}{(x^2 + y^2 + z^2)^{\frac{3}{2}}}$

This was in an old exam in a physics for mathematicians class. I haven't had to deal with these kind of integrals for a while and can't think of a decent substitution. I asked my teacher about it and ...
0
votes
1answer
32 views

Prove that $I_n=\int_0^{\frac{\pi}{4}} \tan^{2n}(t)\, dt$ is convergent to $0$

We have $$I_n=\int_0^{\frac{\pi}{4}} \tan^{2n}(t)\, dt$$ and we need to show that it is convergent to $0$.
1
vote
0answers
69 views

Make this integral zero

Consider the integral $$\int_0^\infty x^nf(x)\,\mathrm{d}x$$ from this answer. The integral is zero for the following $n$ and $f(x)$: $n=4k$, $f(x)=e^{-x}x^{-1}\sin(x)$ $n=4k+1$, ...
0
votes
2answers
51 views

Find the formula for the inverse of $y=\frac{1-\sqrt x}{1+\sqrt x}$

$$y=\frac{1-\sqrt x}{1+\sqrt x}$$ Here is my work. Let me know where I went wrong. $$\begin{align} x=\frac{1-\sqrt y}{1+\sqrt y}\implies x&=\frac{1-\sqrt y}{1+\sqrt y}\cdot1+\sqrt y\\[5pt] ...
0
votes
0answers
91 views

Determine the minimum of $\int_0^\infty\left|x^3+ax^2+bx+c\right|e^{-x}dx$

This question appeared on a graduate preliminary exam in real analysis. Determine $$\min_{a,b,c\in\mathbb{R}} \int_0^\infty\left|x^3+ax^2+bx+c\right|e^{-x}dx.$$
2
votes
1answer
28 views

What is $\nabla Au$ for $A:\mathbb{R}^n\to\mathbb{R}^{n\times n}$ and $u:\mathbb{R}^n\to\mathbb{R}$?

Let $A:\mathbb{R}^n\to\mathbb{R}^{n\times n}$ and $u:\mathbb{R}^n\to\mathbb{R}$. How can we compute $\nabla Au$? I assume we need to apply some kind of product rule, but I wasn't able to figure out ...
0
votes
1answer
24 views

What is $\nabla\cdot A\nabla u$ for $A\in C^1(\mathbb{R}^n\to\mathbb{R}^{n\times n})$ and $u\in C^2(\mathbb{R}^n\to\mathbb{R})$?

Let $A\in C^1(\mathbb{R}^n\to\mathbb{R}^{n\times n})$ and $u\in C^2(\mathbb{R}^n\to\mathbb{R})$. How can we compute $\nabla\cdot A\nabla u$? I assume we need to apply some kind of product rule, but I ...
0
votes
1answer
85 views

If $f(0)=0$ and $f''\ge 0$, then $f(a+b)\ge f(a)+f(b)$

Given $\ f$ so $\ f''(x) \ge 0$ for every $\ x \ge 0$, also $\ f(0)=0$. Trying to show that if $\ a,b \ge 0 \Rightarrow f(a+b) \ge f(a) + f(b)$ Using Taylor I used $\ f(0)=0$ and got ...
0
votes
1answer
81 views

The limit of $\sin \lfloor x\rfloor/\lfloor x\rfloor$ as $x\to 0$

If $$f(x) = \begin{cases}\dfrac{\sin \lfloor x\rfloor}{\lfloor x\rfloor} &, \lfloor x \rfloor \neq 0 \\ \quad 0 &, \lfloor x\rfloor = 0. \end{cases}$$ Find limit of $f(x)$ when $x$ tends to ...
0
votes
0answers
44 views

Packing problem for circles [on hold]

Can the packing problem for big circle in which it has circles with equal radius be true if: the number of circles is even and every 2 circles are symmetric via the center of the big circle,there is ...
2
votes
0answers
23 views

Reducing multi-variable functions to a composition of 1- or 2-variable functions

There are some special functions of 3 or more complex variables that are analytic in some domain (a region in $\mathbb C^n$) with respect to each variable. To give some examples: the incomplete beta ...
1
vote
2answers
21 views

Finding the angle of inclination of a cone.

After my lecture on solving triple integrals with spherical coordinates, we defined $\phi$ as the angle of inclination from the positive z-axis such that $0\leq \phi \leq\pi$. What I don't understand ...
0
votes
1answer
33 views

Evaluate the derivatives by implicit differentiation.

Assume all letters represent constants, except for the independent and dependent variables occurring in the derivative. $da\over db$ $$c^2=a^2+b^2-2ab\cos(t)$$ to find derivative $2ab\cos(t)$ use ...
1
vote
1answer
54 views

Find the derivative of an integral with respect to the upper limit

Let $f(t,y): \mathbb R^2 \to \mathbb R$ be a continuous function of two variables and let $\phi:\mathbb R\to\mathbb R$ be a continuous function of one variable. Fix $a\in\mathbb R$. Compute ...
3
votes
7answers
144 views

The meaning of the symbol $\infty$ in Spivak's calculus book

Spivak in "Calculus" writes ... symbols of $\infty$ and $- \infty$ are purely suggestive: there is no number $``\infty"$ which satisfies $\infty \geq a$ for all numbers $a$. What is the meaning ...
2
votes
1answer
2k views

Contradicting Fubini's theorem

I have a function defined as follows: $f(x,y)= \dfrac{x^2-y^2}{\left(x^2+y^2\right)^2}$, if $(x,y)\neq (0, 0)$ and $f(x,y)=0$ if $(x,y)=(0,0)$. Now, $$\int_0^1\int_0^1 ...
0
votes
1answer
81 views

How do I integrate $\frac{\sin x+\cos x}{\sin^4 x+\cos^4 x}$ [duplicate]

How do I integrate $$\frac{\sin x+\cos x}{\sin^4 x+\cos^4 x}$$ ? Tried different ways including the tangent half-angle substitution (which seems to be disastrous).