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

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0
votes
1answer
152 views

Differentiation problem - airplane/observer question

An airplane is flying at a constant speed at a constant altitude of $3$ km in a straight line that will take it directly over an observer at a ground level. At a given instant the observer ...
1
vote
1answer
38 views

Finding the point where a function turns smaller then another

Sorry, couldn't explain better on the title. I mean, if you have a function for the income over time $I(t)$ and another one for costs $C(t)$ and you want to find out the time $t$ for which the profit ...
0
votes
1answer
152 views

Differential Calculus Problem - Sphere volume increasing (differentiation of algebraic functions)

The Air is pumped into a spherical ball which expands at a rate of 8cm^3 per second. Find the exact rate of increase of the radius of the ball when the radius is 2 cm. I have tried this question, ...
0
votes
4answers
126 views

How to calculete $\displaystyle \lim_{x \to 0} \frac{\sin 5x}{x}$

Please help me calculate $\displaystyle \lim_{x \to 0} \frac{\sin(5x)}{x}$, with out using L'Hôpital's rule or derivatives.
4
votes
4answers
78 views

To prove:$f(x)$ is continuous but not uniform continuous on $(0,+\infty)$.

where$$f(x)=\sum_{n=1}^{\infty}ne^{-nx}$$ Is there any good ways to deal with this kind of functions?
2
votes
1answer
55 views

Estimate for integral of sine to the power of $-(1+a)$ where $a>0$

I'm trying to solve or estimate this integral $$ I=\int\limits_{\arcsin{k}}^{\pi/2}\dfrac{1}{(\sin{x})^{1+a}}\mathrm{d}x, $$ where $0<k<1/2$ and $a>0$. The estimate should depend on $k$. I ...
1
vote
2answers
245 views

if $f_{n}(x)=f(f_{n-1}(x))$then $f_{10}(x)=x,x\in [0,1]$

Define the function $f:[0,1]\to[0,1]$ by the following. $$f(x)=\begin{cases} x+\dfrac{1}{2},&0\le x\le\dfrac{1}{2}\\ 2(1-x),&\dfrac{1}{2}<x\le 1. \end{cases}$$ Let $f_1(x)=f(x)$ and ...
1
vote
3answers
68 views

What's the fastest method to evaluate $\int \frac{1}{4-\sqrt{h}} dh$?

Question says it all. I believe there's a very simple method to evaluate it
6
votes
2answers
124 views

How to prove: $f(x)$ is differentiable on $(0,+\infty)$

The function $f(x)$ is defined on $(0,+\infty)$. We know $f'(1)$ exists and we have that $$\forall x,y \in(0,+\infty), \quad f(xy)=yf(x)+xf(y)$$ How to prove:$f(x)$ is differentiable on ...
3
votes
2answers
47 views

derivative after changing variable

I have just studied a lesson about derivative of a function but I still confuse in the following case. Suppose that I have a function: $$ f(x) = 2x^2 + 3x + 1$$ and I want to calculate ...
3
votes
2answers
75 views

How to do a change of variable here

How to do a change of variable here $\displaystyle\int_{0}^{2\pi}\int_{0}^{2\pi}2\left|\sin\frac{(\theta-\theta')}{2}\right|\ d\theta\ d\theta'$ If I replace $\theta-\theta'$ by $u$, double ...
2
votes
0answers
68 views

Is $\lim_{n \rightarrow \infty}\sum_{k=0}^{n} \frac{|(1-\frac{n p_n}{n})|^{n-k}- e^{- \lambda}|}{k!}=0$?

Does anybody know whether this limit here is zero, if we assume that $np_n \to \lambda$ for $n \to \infty$? $$ \mbox{So, do we have}\quad \lim_{n \to \infty}\sum_{k=0}^{n}{1 \over k!}\, ...
0
votes
2answers
41 views

Does this sequence converge to zero?

Let $f: \mathbb{N}^2 \rightarrow \mathbb{R}$, such that $\forall k \in \mathbb{N}: \lim_{n \rightarrow \infty} f(n,k) = 0$. Is it then true, that $\sum_{k=0}^n \frac{f(n,k)}{k!}$ converges to zero ...
0
votes
1answer
16 views

Showing that the set $\displaystyle \{e^{2\pi i (mx+ny)}\}_{n,m=-\infty}^\infty$ is an orthonormal set

I have the following problem: Show that the set $\displaystyle \{e^{2\pi i (mx+ny)}\}_{n,m=-\infty}^\infty$ is an orthonormal set in $L^2(D)$, where $D$ is any square whose sides have length ...
0
votes
1answer
47 views

If a differentiable function has bounded derivative, Must it be that its derivative continuous?

I got this question: Let $f$ be a continuous function on the closed interval $[a,b]$ and differentiable on the open interval $(a,b)$, If $f'$ is bounded on $(a,b)$, Must it be the case that $f'$ is ...
2
votes
3answers
471 views

Differential Equation: Am I missing a trick?

I am trying to solve the following calculus problem: Show the function $$\displaystyle y(x)=\int_{0}^x \sin(x-t)f(t)dt$$ solves the differential equation $$y''+y=f(x)$$ I have put ...
1
vote
4answers
100 views

Finding the sum of series $\displaystyle\sum \limits_{n=1}^{\infty} (-1)^{n}\frac{n^2}{2^n}$

I have some problems in finding the values of series that follow this pattern: $$\sum \limits_{n=0}^{\infty} (-1)^{n}*..$$ For example: I have to find the value of this series $$\sum ...
0
votes
1answer
21 views

Directional Derivate for sums

I know the directional derivative as $D_uf(x)=\nabla f(x) . u$ But I do not know how this applies here?
0
votes
3answers
27 views

Show that scalar field f must be equal at two points…

If $\nabla f$ is proportional to $x\hat{i} + y\hat{j} + z\hat{k}$. The two points are $(0,0,a)$ and $(0,0,-a)$ So far what I have is that along the z-axis from the origin the function doesn't change ...
1
vote
1answer
76 views

Show that topologies are the same

I just read a proof where it was said that if for each element in the topology 2 we find an element in topology 1 that is contained in this set and vice versa, then they are the same. How do I see ...
3
votes
2answers
304 views

Difficult GRE Math question, an definite integral. [closed]

I would like to solve $\int_{0}^{\infty} \lfloor x\rfloor e^{-x} dx$ for its exact solution. This was on a previous GRE Mathematics exam.
0
votes
1answer
28 views

Releated rates involving trig and velocity

question is- A rocket is launched vertically and is tracked by a radar station located on the ground 5km from the launch pad. Suppose that the elevation angle θ of the line of sight to the rocket is ...
24
votes
4answers
2k views

Using Integration By Parts results in 0 = 1

I've run into a strange situation while trying to apply Integration By Parts, and I can't seem to come up with an explanation. I start with the following equation: $$\int \frac{1}{f} \frac{df}{dx} ...
1
vote
2answers
97 views

Why is there an “absolute value” and a norm in the Schwarz Inequality?

This really bothers me, and I'm not sure if it's just that I'm not understanding it correctly. For the moment, assume we are working in a vector space $V$ over $\mathbb{R}^n$. Let $x,y \in V$. We have ...
8
votes
3answers
229 views

Integral $\int_0^{\pi/4}\frac{dx}{{\sin 2x}\,(\tan^ax+\cot^ax)}=\frac{\pi}{8a}$

I am trying to prove this interesting integral$$ \mathcal{I}:=\int_0^{\pi/4}\frac{dx}{{\sin 2x}\,(\tan^ax+\cot^ax)}=\frac{\pi}{8a},\qquad \mathcal{Re}(a)\neq 0. $$ This result is breath taking but I ...
1
vote
3answers
64 views

Calculating limits using l'Hôpital's rule.

After a long page of solving limits using l'Hôpital's rule only those 2 left that i cant manage to solve $$\lim\limits_{x\to0}{\sqrt {\cos x} - \sqrt[3]{\cos x}\over \sin^2 x }$$ ...
1
vote
1answer
27 views

Help Understanding Evaluation of Integral

Please help me to understand the evaluation of this integral. $$\int_0^1\int_u^{\mathrm{min(1,u+z)}} 2\;dv\;du$$ I know that the correct answer is $$ f(z) = \left\{ \begin{array}{lr} ...
0
votes
2answers
31 views

Lengths of Plane Curves - Calculus 2: $\sqrt{1-x^2} ; x=-\frac{1}{2} \to x=\frac{1}{2}$

$$ \sqrt{1-x^2} ; x=-\frac{1}{2} \to x=\frac{1}{2} $$ I am having problems setting this up. Taking the derivative of $\sqrt{1-x^2}$. Leaves me with: $$ \frac{1}{2}\left(1-x^2 ...
1
vote
1answer
33 views

How is this step completed?

User Did, did this step in his answer to my previous question: $$\sum_{k=0}^n{n\choose k}(zp)^kq^{n-k}=(q+pz)^n.$$ How is it done? Is it simply an identity, or something more?
1
vote
4answers
86 views

Integrating a term again and again

So if you have $f''(x) = 24x$ you know you want to integrate it, because it would look much better integrated, so now we have $f'(x) = 12x^2$, but it could still look better, so we integrate it to ...
3
votes
2answers
82 views

Is my proof of the continuity of $\frac{1}{x}$ correct?

I need to prove that $\frac{1}{x}$ is continuous, but my proof, as far as I did now, only applies to the interval $x\ge1$ wich is a progress. So, there's how I did: ...
0
votes
1answer
164 views

How plot a bifurcation diagram ? or show find bifurcation points

I have a function $rx(3-x^2)$ How do I find the points it bifurcates and what does it mean ? I know how to find fixed points and check them for stability, how can I use that to answer this question ...
0
votes
2answers
86 views

Prove that $A$ is an open set and $B$ is a closed set.

Suppose that $E \subseteq \mathbb R^m$. Let $$A=\{x \in \mathbb R^m: \rho (x, E) < r\}, B=\{x \in \mathbb R^m: \rho (x, E) \le r\}.$$ Prove that $A$ is an open set and $B$ is a closed set. Thanks ...
1
vote
1answer
242 views

How to prove uniform continuity problem!

A) $f(x)=x^3$ , give an example of an interval where $f$ is uniformly continuous and another where it is not. explain your choose of examples B) decide if $f(x)= \dfrac{1}{\sin x} - \dfrac{1}{x}$ is ...
0
votes
0answers
77 views

Symbol of self-adjoint pseudodifferential operator

It seems that the following result should hold, but I can't find it explicitly anywhere. If $A=A^*$ is a properly supported pseudodifferential operator, does this imply that ...
0
votes
2answers
80 views

did:$\lim_{x\to 0}\frac{\sin(x^2 + \frac{1}{x} )- \sin(\frac{1}{x}))}{x}$

Could you guys give me at least a hint at: $$\lim_{x\to 0}\frac{\sin(x^2 + \frac{1}{x}) - \sin(\frac{1}{x}))}{x}$$ ? I already tried expanding the $\sin(x^2 + \frac{1}{x})$ but got nothing. Also, ...
1
vote
1answer
41 views

How to evaluate a limit that involves matrices

I've stumbled upon this problem while I was browsing through the contents of an admission exam . I've struggled tremendously with this exercise and I've got no idea what do to next , it's eating me ...
2
votes
1answer
81 views

$\iiint \frac{1}{x^2+y^2+(z-2)^2}dA$ where $A=\{x^2+y^2+z^2 \leq 1\}$ check my answer!

I would like someone to review my solution please, the original question is to calculate $\iiint \frac{1}{x^2+y^2+(z-2)^2}dA$ where $A=\{x^2+y^2+z^2 \leq 1\}$ What I did: First I changed variables ...
14
votes
6answers
595 views

A logarithmic integral $\int^1_0 \frac{\log\left(\frac{1+x}{1-x}\right)}{x\sqrt{1-x^2}}\,dx$

How to prove the following $$\int^1_0 \frac{\log\left(\frac{1+x}{1-x}\right)}{x\sqrt{1-x^2}}\,dx=\frac{\pi^2}{2}$$ I thought of separating the two integrals and use the beta or hypergeometric ...
1
vote
2answers
81 views

About $\int \csc(x) \, dx$

One of the suggested proofs that I found to the $\int \csc(x) \, dx$ start with, $$\int \csc(x) \, dx= \int \csc(x) \cdot \frac{\csc(x)- \cot(x)}{\csc(x)- \cot(x)} dx = \cdots$$ By graphing the ...
1
vote
0answers
52 views

Is $1-f(x)\to 0$ equivalent with $f(x)\to 1$.

Problem. I am trying to make a proof of contradiction and have this far shown that if my assumption is true, both $$\lim_{x\to 0}(1-f(x))=0\quad \text{and}\quad \lim_{x\to 0}(1+f(x))=0$$ must be true. ...
0
votes
1answer
33 views

Cauchy-Euler equation set up

I have the following 2nd ordered ODE, and I want to transform it into a cauchy-euler equation to be able to solve it. xy'' - 7xy' + 12y = 0 To be a Cauchy-Euler ...
1
vote
5answers
78 views

Calculate $\displaystyle \lim_{x \to \frac{3}{2}} \frac{2x^2-3x}{|2x-3|}$

I know that $\displaystyle \lim_{x \to \frac{3}{2}} \frac{2x^2-3x}{|2x-3|}$ does not exist, because the lateral limits are different and I also know that the absolute value on the denominator has ...
2
votes
3answers
55 views

Find $\lim _{ n\rightarrow \infty }{ \sum _{ k=1 }^{ n }{ \frac { \sqrt { k } }{ { n }^{ \frac { 3 }{ 2 } } } } } $

Need help find the limit of $\lim _{ n\rightarrow \infty }{ \sum _{ k=1 }^{ n }{ \frac { \sqrt { k } }{ { n }^{ \frac { 3 }{ 2 } } } } } $ Now my intuition is that using Stolz-Cesaro $\lim _{ ...
2
votes
0answers
62 views

How to prove or disprove this statement?

For $n\geq1$, $$\int_{0}^{\pi/2}(\theta \sin\theta)^{n+1}d\theta>\int_{0}^{\pi/2}(\theta \sin\theta)^{n}d\theta$$ . It is hard to find $\int_{0}^{\pi/2}(\theta \sin\theta)^{n}d\theta$ so I have no ...
6
votes
8answers
242 views

Evaluate $\int_0^1\frac{x\ln x}{(1+x^2)^2}\ dx$

$$\int_0^1\frac{x\ln x}{(1+x^2)^2}\ dx $$ Help me please. I don't know any ways of solution. Thank you.
1
vote
0answers
35 views

Solving Differential Equations to satisfy a condition

I have two differential equations $$\frac{dX}{dt} = -\frac{d\cdot N(0)\cdot X}{m+X}$$ and $$\frac{dY}{dt} = \frac{d\cdot N(0)\cdot X}{m+X}$$ with initial conditions $X(0) = X_0$ and $Y(0) = 0$. ...
1
vote
0answers
98 views

Arc length of an ellipse

An ellipse has parametric representation $x = a\cos t$, $y = b \sin t$ for $0 ≤ t ≤ 2\pi$. Can you write a formula for its total length? Do not waste your time trying to calculate it. The way I was ...
1
vote
2answers
48 views

Convergence of an infinite Riemann sum to an integral

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be smooth, bounded, uniformly continuous, and $|f(x)| \leq 1/|x|^{N}$ for any $N$. Then is it true that $$\frac{1}{n}\sum_{k = -\infty}^{\infty}f(k/n) ...
0
votes
1answer
46 views

prove that a function whose derivative is bounded also bounded

I got this problem: Let $f$ be a differentiable function on an open interval $(a,b)$ such that $f'$ (the derivative of $f$) is bounded on $(a,b)$ (meaning there exist $0<M$ such that $\forall ...