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

learn more… | top users | synonyms

0
votes
1answer
13 views

Show that if the set A is bounded then the set Bϵ(A) is bounded.

Let $A$ be a nonempty closed subset of $\mathbb{R}^{n}$. For $\epsilon>0$, $B_{\epsilon}(A)= \left \{ x\in \mathbb{R}: d(x,A)<\epsilon) \right \}$. a) Show that if the set $A$ is bounded then ...
2
votes
3answers
26 views

A not so hard basic calculus problem? But it appears to be very lengthy

Find the coordinates of the two points on the curve $y=4-x^2$ whose tangents pass through the point $(-1,7)$. My work: Let the two points be $(a,b)$ and $(c,d)$. And $\frac{dy}{dx}=-2x$, so the ...
3
votes
1answer
19 views

partial derivative of a sum 1

This is part of a much bigger question I am working on, but I dont understand how I would differentiate a finite sum of the form: $$\frac{\partial}{\partial \phi} \sum_{t=2}^{T} (y_t - \phi ...
2
votes
2answers
30 views

Computing the value of a series by telescoping cancellations vs. infinite limit of partial sums

$$\sum_{m=5}^\infty \frac{3}{m^2+3m+2}$$ Given this problem my first approach was to take the limit of partial sums. To my surprise this didn't work. Many expletives later I realized it was a ...
0
votes
1answer
26 views

Taylor Expansion Derivation

I have a (perhaps silly) question, but I am reviewing Taylor series approximation and looking at this slide. In the third panel, $f(a+h)$ is isolated but I am confused why $O(h^{2})$ does not have a ...
1
vote
3answers
71 views

Differentiation under the integral sign: Where is my mistake?

So I'm trying to find $\int_0^\infty \sin(x^2)\,dx$ by the method of differentiation under the integral sign. The idea is to use differentiation with respect to t on A(t) -- defined below -- and then ...
-2
votes
1answer
32 views

The limit goes to infinity and the integral goes to infinity [on hold]

I need help on the following limit and integral $$\lim _{ n\rightarrow \infty }{ \int _{ 0 }^{ \infty }{ \cos { { x }^{ n } } dx } } $$
0
votes
1answer
24 views

Critical point - relative minimum

Checking the function $f:\mathbb{R}^2 \rightarrow \mathbb{R}, (x, y) \rightarrow (y-3x^2)(y-x^2)$ we can take an idea for the difficulty of finding conditions that ensure that a critical point is a ...
2
votes
5answers
86 views

Determine whether $f(x)$ is increasing or decreasing

Let $f(x) = -x + (x^3/3!) + \sin(x)$ How do I determine if $f(x)$ is increasing or decreasing? I have already found the derivative of this function which is: $f'(x) = -1 + (x^2/2) + \cos(x)$ And I ...
4
votes
2answers
76 views

Finding $\int\frac{\sqrt{1-t^2}}{1+t^2}dt$

I wanted to find $\int\frac{\sqrt{1-t^2}}{1+t^2}dt$, so I substituted $t=\sin\theta$ and got $\int\frac{\cos^2\theta}{1+\sin^2\theta}d\theta$; but I'm not sure what the best way to proceed from here ...
1
vote
2answers
26 views

Help Regarding The Taylor Series Remainder Proof Understanding

I'm reading Mathematics: It's Content, Methods, and Meanings and I am in a chapter about Taylor Series. It made sense until I came across the remainder part of the theorem. In order to prove the ...
3
votes
3answers
117 views

is $dx$ greater than $\frac{dx}{2}$?

I wanted to ask if $dx$ is greater than $\frac{dx}{2}$? i will make conclusions i am sure they are wrong : a) if yes then why in integration we do not use smaller than $dx$ like its half ? b) if ...
0
votes
1answer
33 views

How can we continue to get the critical points?

A service requires the dimensions of a rectangle box are such that the length plus twice the width plus twice the height do not exceed $274cm$ ($l+2w+2h \leq 274$). What is the maximum volume of the ...
2
votes
1answer
48 views

Please check my demonstration of de l'hopital's rule

I have demostrate the de l'hopital theorem but in some steps I'm not 100% sure; The theorem I demostrate is for: $\lim_{x\rightarrow a+} \frac{f'(x)}{g'(x)}=L \implies\lim_{x\rightarrow a+} ...
0
votes
2answers
25 views

$f(x,y)=x^3+3xy^2-2y^3$. Find all unit vectors, if any, such that $f_u(0,1)=\frac{6}{5}$

I think that I understand what the question wants me to do: $f(x,y)=x^3+3xy^2-2y^3$. Find all unit vectors, if any, such that $f_u(0,1)=\frac{6}{5}$ I worked out the partial derivatives: ...
1
vote
1answer
52 views

Proving that $f$ is differentiable at $0$

Let's consider the following function: $$f(x,y)=\begin{cases} (x^2+y^2)\sin\left(\dfrac{1}{x^2+y^2}\right) & \text{if }x^2+y^2\not=0 \\{}\\ 0 & \text{if }x=y=0 \end{cases}$$ I know that ...
5
votes
4answers
35 views

Derivative of $f(x) = \frac{\cos{(x^2 - 1)}}{2x}$

Find the derivative of the function $$f(x) = \frac{\cos{(x^2 - 1)}}{2x}$$ This is my step-by-step solution: $$f'(x) = \frac{-\sin{(x^2 - 1)}2x - 2\cos{(x^2 -1)}}{4x^2} = \frac{2x\sin{(1 - x^2)} - ...
0
votes
1answer
47 views

How to find the integral with $\sqrt [ 3 ]{ x } +\sqrt [ 4 ]{ x } $ in the denominator?

How to evaluate $$\int { \frac { 1 }{ \sqrt [ 3 ]{ x } +\sqrt [ 4 ]{ x } } } +\frac { \log { (1+\sqrt [ 6 ]{ x } ) } }{ \sqrt [ 3 ]{ x } +\sqrt { x } } dx$$ I'm not being able to make the right ...
-5
votes
0answers
30 views

calculus question ? please help [on hold]

A cell phone plan costs \$35 per month plus \$1.25 per minute for long distance calling. Find $c(x)$. if this was your plan and your long distance calls totaled 10 minutes this month, how much would ...
4
votes
4answers
50 views

Show that if $f$ is continuous at $a$ and $f(a)≠0$ then $f$ is nonzero in an open ball around $a$.

Here is the question I'm dealing with: Let $U$ be an open set of $\mathbb{R}^{n}$, $f:U\rightarrow\mathbb{R}^{n}$ a function and $a\in U$ a given point. Show that if $f$ is continuous at $a$ and ...
0
votes
4answers
37 views

Which function do we want to minimize?

A ray of light travels from the point $A$ to the point $B$ across the border between two materials. At the first material the speed is $v_1$ and at the second it is $v_2$. Show that the journey is ...
1
vote
3answers
44 views

Explicit form of this series expansion?

I am considering the following series expansion: $$f(k):=\sum_{n\geq 1} e^{-k n^2}$$ with $k>0$ a fixed parameter. Is there a possibility to either find a closed form expression for $f(k)$? Or at ...
3
votes
1answer
26 views

Two versions of the Inverse Function Theorem.

I first learned about the Inverse Function Theorem for $C^1$ functions in Rudin's Principles of Mathematical Analysis in the following form. Inverse Function Theorem ($C^1$ version): Suppose $E$ ...
-2
votes
0answers
19 views

Solid of revolution on the line x=2. [on hold]

I'm stuck on this. An area bounded by $y=x^2$ and the line $y=4$, rotated around the line $x=2$. I need the volume of the solid generated. Many thanks in advance.
-5
votes
1answer
33 views

$\int x(x+8)^8 dx$ , what is the value of that integral [on hold]

$\int x(x+8)^8 dx$ , what is the value of that integral ?
5
votes
4answers
99 views

Finding $\int\frac{1}{x^{11}+4x^6}dx$

I wanted to find out if there is an easy way to evaluate $\displaystyle\int\frac{1}{x^{11}+4x^6}dx$. I substituted $u=x^5$ and then used partial fractions, but maybe there is a simpler way to find ...
-1
votes
1answer
30 views

Do we have to use the Lagrange multipliers method? [on hold]

Draw a cylindrical container (with a lid), so as to contain $1$ liter of water, using a minimal amount of metal. Could you give me some hints how we could do that?? Do we have to use the Lagrange ...
0
votes
0answers
6 views

Deriving information about asymptotics from finitness of a limit

Let $f_1,f_2:\mathbb{R}\setminus\{0\}\to \mathbb{R^+}$ be two $C^1$ functions and $\alpha:\mathbb{R}\setminus \{0\}\to \mathbb{R}$ be a function from a Zygmund class (in particular it is Holder for ...
5
votes
2answers
38 views

Prove that a function does not have a limit when $x\rightarrow 0$

Let $$f(x)=\left\{ \begin{array}{l l} x+2 & \quad ,x\in\mathbb{Q}\\ 6-x & \quad ,x\notin\mathbb{Q} \end{array} \right.$$ then $$\lim_{x \to 0}f(x)$$ does not exist. By limit ...
2
votes
0answers
34 views

Closed form of an infinite series of integrals $\int_{0}^{\eta} \cos nt \cos t \sqrt{\cos^2 t - \cos^2 \eta}$

Let $$ I(n,\eta) = \int_{0}^{\eta} \cos nt \, \cos t \, \sqrt{\cos^2 t - \cos^2 \eta}\; dt $$ where it is known that $0 < \eta \leq \frac \pi 2$. Is it possible to evaluate $S$, the infinite ...
1
vote
4answers
106 views

The convergence of the series $\sum (-1)^n \frac{n}{n+1}$ and the value of its sum

This sum seems convergent, but how to find its precise value? $$\sum\limits_{n=1}^{\infty}{(-1)^{n+1} \frac{n}{n+1}} = \frac{1}{2}-\frac{2}{3}+\frac{3}{4}-\frac{4}{5}+...=-0.3068... $$ Any help ...
-2
votes
0answers
17 views

Surface Integral of cone [on hold]

How would i calculate the surface integral of this the radius of the cone being 28.25 Thank you
1
vote
1answer
71 views

Doubt on Calculus made easy exercise question Exercise 5 Question 10.

A body moves in such a way that the spaces described in the time $t$ from starting is given by $s = t^n$ ,where $n$ is a constant. Find the value of $n$ when the velocity is doubled from the ...
9
votes
1answer
85 views

Integral involving Clausen function ${\large\int}_0^{2\pi}\operatorname{Cl}_2(x)^2\,x^p\,dx$

Consider the Clausen function $\operatorname{Cl}_2(x)$ that can be defined for $0<x<2\pi$ in several equivalent ways: ...
0
votes
1answer
32 views

How to calculate the exponential of a matrix?

Can anyone please give me an example of how to calculate the exponential of a matrix given its Jordan canonical form?
3
votes
4answers
113 views

Line integral of a conservative field over a squre

I am trying to evaluate $$\oint _C \frac{-ydx+xdy}{x^2+y^2}$$ clockwise around the square with vertices (−1,−1), (−1,1), (1,1), and (1,−1). So from the question, ...
0
votes
0answers
24 views

$\lim$ and $\lim \sup$ and radius of convergence

I have a problem concerning the $\lim \sup$ of the sequence $a_n^{1/n},\,\ a_n>0$. Let's consider two cases: the sequence an is convergent in $\mathbb{R}$. For the sake of simplicity let's ...
1
vote
1answer
25 views

proof of laplacian $1/\rho$ in cylindrical coordinates at $\rho = 0$

In spherical coordinates, it can be proved that the laplacian of $r = \sqrt{x^2+y^2+z^2}$ at the origin is \begin{align} \nabla^2 \dfrac{1}{r} = -4\pi\delta(\vec{r}) \end{align} as demonstrated in ...
0
votes
0answers
46 views

Double Integration Working Help

Help I dont know how to approach this question, I have the answer but dont know how to write a detailed working process of obtaining it. It is supposed to find the surface area of a cone that is $z = ...
2
votes
1answer
13 views

Given a function $f$ defined in $R^2$. Let $F(r,\theta)=f(r\cos\theta,r\sin\theta).$ Verify a formula of the modulus of the gradient.

Given a function $f$ defined in $R^2$. Let $$F(r,\theta)=f(r\operatorname{cos}\theta,r\operatorname{sin}\theta).$$ Verify the formula $$|\nabla f(r\operatorname{cos}\theta, ...
1
vote
3answers
40 views

Mean Value Theorem to show inequalities about numbers

Show $\sqrt{65}-8=\frac{1}{2\sqrt{c}}$ for some $c \in (64,65)$, and hence show: $$8+\frac{1}{18}<\sqrt{65}<8+\frac{1}{16}$$ I managed to do the first part easily, but I don't know how ...
1
vote
2answers
37 views

prove using Lagrange multipliers that for $x,y>0,\space n\in \mathbb N,\space (\frac{x+y}2)^n \leq \frac{x^n+y^n}2 $

I have been asked to prove using Lagrange multipliers that for \begin{equation*} \space (\frac{x+y}2)^n \leq \frac{x^n+y^n}2,~x,y>0,~n\in \mathbb {N} \end{equation*} I am familiar with the ...
1
vote
1answer
47 views

Struggling to prove inequality

I've been given to following inequality to prove: (The hint given was not to evaluate the integral) \begin{equation*} \frac{1}{4} \leq \int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\frac{sin(x)}{x}dx\leq ...
0
votes
1answer
18 views

Calculating limit of a complicated function

I want a formal way to calculate limit of this expression (which I encountered in a physics course) as $T\to 0$ and as $T\to \infty$. Here $\beta=\frac{1}{k_B T}$. I have tried and found that for ...
2
votes
2answers
60 views

what will be the value of this integral

$$ \large{ \int^{\Large{\frac{\pi}{2}}}_{0} \left[ e^{\ln\left(\cos x \cdot \frac{d(\cos x)}{dx}\right)} \right]dx}$$ We know that $\large{a^{log_a(c)} = c}$. But in this question, the expression in ...
3
votes
2answers
35 views

Mean Value theorem: showing an equation has a solution

$g(x)$ is a continuous function on $[0,\pi]$ and differentiable on $(0,\pi)$ with $g(0)=g(\pi)$. I need to show that the equation $g'(x)+g(x)cos(x)=0$ has a solution in $(0,\pi)$. Attempt at ...
0
votes
1answer
22 views

differential equation and general solution

I have the following differential equation ; $$\frac{\partial z}{\partial t}+\alpha z\left(t\right)=y\left(t\right)$$ I tried to find the general solution by multiplying two sides by $e^{\alpha t}$ ...
1
vote
0answers
13 views

Theorem about number of crossing-points between a function and a line

Assume $f(x)$, with $x \in [a,b]$. Take $u$ so that $f(a)<u<f(b)$. By the Intermediate value theorem, we know that $f(x)$ crosses $u$ at least once. My question is, given some extra information ...
1
vote
1answer
59 views

Integration a trigonometric expression

How would you evaluate the following indefinite integral? $$ \int \frac {\ln{(x)} \cdot \cos{(x)}}{\sin^2 {(x)}} dx $$
1
vote
1answer
18 views

Applying the theorem of Lagrange multipliers

I have to fund the extremas of $f$ subject to the contraints, that are given: $$fx, y)=x-y, x^2-y^2=2$$ I have done the following: We use the theorem of Lagrange multipliers. The constraint is ...