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

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3
votes
3answers
143 views

How to find the sum $\sum_{n = 1}^{\infty}\left[n\sin\left(1 \over n\right)\right]^{n^{3}} $

$$\sum_{n = 1}^\infty {\left(n\sin\left(\frac{1}{n}\right)\right)^{n^3} } \,\,\sim \,\,\,\sum_{n = 1}^\infty { \left(n\left(\,\,\frac{1}{n}-\frac{1}{6n^3}+o\left (\frac{1}{n}\right)\,\, ...
1
vote
2answers
76 views

Convergence of $\displaystyle\int\frac{1}{\sqrt[3]{1-x^3}}\ dx$

Please help me to prove that this integral converges. $$\int_{0}^1 \frac{1}{\sqrt[3]{1-x^3}}\ dx $$ No ideas. Tried to find function which is bigger and converges, equivalent fun-s, but no result ...
4
votes
2answers
38 views

Proving inequalities about integral approximation

We can state that, with $n$ integer, $$\int_1^n \log x \ \mathrm{dx} \leq \sum_{m = 1}^n \log m$$ because the second is the area of $n$ rectangles with unity base, while the first is "just" the area ...
0
votes
1answer
68 views

Finding max/min through lagrangian

I am trying to solve this problem, but I am doing something wrong: $$f(x,y,z)=x^2-y^2,M=\{[x,y,z]\in\mathbb{R}^3:x^2+y^2+z^2=9,x+z\ge1\}$$ And let $g(x,y,z)=x^2+y^2+z^2-9$. Set M is closed and ...
2
votes
3answers
44 views

Find the solution of the initial value problem

Find the solution of the initial value problem: a) $x \, \dfrac{dy}{dx}-y=x+x^2, \ \ x>0 \ \text{ and } \ y(1)=2. $
1
vote
0answers
82 views

ODE with constraints

Given the ODE system $$\dot{x} = y \\ \dot{y} = \frac{1}{\alpha} (z - y)$$ where $\alpha > 0$ is a constant. How can I find a bound for $z$ depending on $x$ such that $\forall t ~x(t) \geq 0$ under ...
3
votes
2answers
33 views

Minimum of Integral, relation with area

Find the value of $a$ such that $F(a)$ is minimum, where $$F(a)=\int_{0}^{\pi/2} |\sin x - a\cos x| dx.$$ I want to differentiate the function but the absolute value prevented me from doing so... ...
1
vote
1answer
53 views

Area inside curve and ellipse using integrals.

i've done some progress with this one but i get stucked right before the integration part: $A)$ Use the right double integral and replacing $x=ar$cos$(\theta)$, $y=br$sin$(\theta)$ in order to find ...
0
votes
1answer
39 views

A limit with floor

I try to compute this limit (but without success). Any help will be welcome Let $p$ be an integer $\ge 2$ and $q>1, 0\le \varepsilon<1$ be reals. $$\lim_{\substack{n\in\mathbb ...
1
vote
1answer
44 views

A question about calculus

$\displaystyle\int_{ \mathbf Q} s^a \, dm $ I don't know how to start at all. Can anyone give me some hint how to calculate integral in this type?
0
votes
1answer
37 views

Calculus optimization quick question

A hotel fills only $120$ rooms when the price is $\$150$ per night for a room. When the price is decreased by $\$10$, it fills $16$ additional rooms. Find the price for maximum revenue. Ok frankly ...
0
votes
1answer
70 views

An equality from Fritz John's paper

Prove: $\frac{\rho}{(4\pi)^2}\int_{|\xi|=1}d\omega_\xi\int_{|\eta|=1}f(x^0+r\xi+\rho\eta)d\omega_\eta=\int_{|r-\rho|}^{r+\rho}\frac{\lambda}{8\pi ...
3
votes
1answer
28 views

Derivative of a Heaviside function?

If I have the following function: $\ y(x) = Ax + \frac{1}{2}Bx^2 + W U(x-\frac{1}{2}) $ where A,B,W are constants, and U(x - x0) is the Heaviside function. What would its derivative be? ...
0
votes
1answer
51 views

Finding the points of intersection of the circles [closed]

How can you find the points of intersection of the circles $x^2+y^2-2x-2y-2=0$ and $x^2+y^2+2x+2y-2=0$?
7
votes
4answers
438 views

Can you find this limit in a “nicer” way?

I'm trying to show that: $$\lim\limits_{n\to\infty}{n(\sqrt[n]{n}-1)} = \infty$$ From what I've tried now, all I end up with is basically rewriting the left term as: ...
0
votes
1answer
35 views

How do I solve this second order partial derivative?

The answer I got was by first finding the partial derivative with respect to r which I found to be 2r/(r^2 + s) and then I squared it and got 4r^2/(r^2 + s)^2 I plugged this into wolfram alpha and it ...
4
votes
3answers
60 views

Limit evaluation method inconsistency?

I'm having trouble understanding what really happens when we evaluate the limit of a certain function f(x) as x approaches a certain value. For ex, if we have lim x-->2 $\frac{x^2 + x -6}{(x-2)}$ we ...
0
votes
0answers
32 views

Integrate over components of the unit vector

if I have a vector field$X: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ and $\phi:\mathbb{R}^3 \rightarrow\mathbb{R}$ such that $X(r, \theta, \phi) := \phi(r,\theta,\phi) e_j(\theta,\phi)$, where $e_j$ is ...
1
vote
1answer
46 views

Limit of vector-valued function is equal to the limit of its components

Let $f: \Bbb R^m \to \Bbb R^n$. Express $f(x)$ in terms of components: $$f(x)=(f_1(x), f_2(x), ... , f_n(x))$$ I need to prove that $f$ is continuous at $a$ if and only if each $f_i$ is continuous ...
1
vote
1answer
54 views

Show that $\int_{\mathbb R^n}e^{|x|^{-n}}dx=$ Volume of n-sphere

I'm preparing for a calculus exam, I'd like help in solving this question. Let $x \in \mathbb R^n$, $|x|={(x_1^2+x_2^2+...+x_n^2)^{\frac{1}{n}}}$, Show that $$\int_{\mathbb R^n} e^{|x|^{-n}}dx$$ is ...
1
vote
1answer
44 views

Which of the following functions on R are uniformly continous?

$a)\frac {1}{x^2+1} $ $b)\cos^3x$ $c)\frac {x^2}{x^2+2} $ $d)x\sin x$ $a)|\frac{1}{x^2+1}-\frac{1}{y^2+1}|\leq|\frac{|x-y|(|x|+|y|)}{(1+x^2)(1+y^2)}|\leq ...
1
vote
1answer
65 views

Partial derivatives-Why does this stand?

In my notes there is the following: $$u_{\xi \eta}=0 \Rightarrow \left\{\begin{matrix} u_{\xi}=0 \Rightarrow u=g(\eta)\\ u_{\eta}=0 \Rightarrow u=f(\xi) \end{matrix}\right.$$ I haven't understood ...
0
votes
1answer
45 views

Analogue of differentiation for sequences?

I remember learning (2 semester calculus for engineers) about all the below ones, but nothing that fits in place of the question mark. Is there anything nontrivial? ...
14
votes
3answers
2k views

Is this question too easy or am I getting it wrong?

In my homework, I am asked to find the limit $$\lim\limits_{x\to0}{\frac{x}{e^x}}$$ But obviously, you could just substitute $x = 0$: $$\lim\limits_{x\to0}{\frac{x}{e^x}} = ...
1
vote
2answers
99 views

Oops, $y^2=2x^2+C$ is not the same as $y=\sqrt{2x^2}+C$

Oops, $y^2=2x^2+C$ is not the same as $y=\sqrt{2x^2}+C$ I almost slipped and just assumed $\sqrt{C}=C$ but when you take the square root of both sides, you are really ending up with ...
1
vote
1answer
32 views

Maximization with constraints

How can I find $\lambda_H$ and $\lambda_T$ such that $$\max_{0 \leq \lambda_H , \ \lambda_T \ \leq 1 }\left\{\frac{4.6575342 \times 10^{-4}}{2.1722965 \times 10^{-4} + \lambda_H},\frac{1.0958904 ...
0
votes
1answer
35 views

Verify if the function $f(x,y)$ is differentiable in $(x,y) = (0,0)$ and $(x,y) \ne (0,0)$

Verify if the function $f(x,y)$ is differentiable in $(x,y) = (0,0)$ and $(x,y) \ne (0,0)$ $$f(x,y) = \begin{cases} 2xy (\frac{x^2 - y^2}{x^+ y^2}) & x^2 + y^2 \ne 0 \\[4pt] 0 & other ...
0
votes
1answer
52 views

Are these derivatives correct?

Given the map $E: \mathbb{R}^3 \times \mathbb{R} \rightarrow \mathbb{R}^3$, such that (notice that this excercise is taken from Physics(Electrodynamics)). $$E(r,t) = -\frac{1}{4 \pi \varepsilon_0} ...
0
votes
0answers
50 views

Laplace Transform of $\sin(t-3)$

I wanted to compute the Laplace Transform of $\sin(x-3)$ using the shift rule: $\mathcal{L}(f(t-a)) = e^{-as}\mathcal{L}\left(f(t)\right) \Rightarrow \mathcal{L}(\sin(t-3)) = e^{-3s}\mathcal{L}(\sin ...
1
vote
1answer
65 views

How to take a partial derivative

I want to check that I understand this correctly before heading off to the exam. If $z = \sin(xe^y)$ where $x = 3u^2 + uv$ and $y = u^3 - \ln(v)$ find $\displaystyle \frac{\partial z}{\partial u}$ ...
1
vote
1answer
24 views

will locally extremal continuous function be constant

Let $f:[0, 1]\rightarrow \mathbb{R}$ be continuous function. Let us suppose that each point on the segment is either local maximum or local minimum for this function. Is this function a constant?
0
votes
0answers
15 views

Asymptotics of an integral with respect to the arcsin density

Let $g(x) = 1 / \sqrt{1-x^2}$ on $(-2,2)$ and zero elsewhere. I want to calculate the integral $$ \int F(n g(x) (\mu -x)) g(x) dx $$ as $n$ tends to infinity. Here $\mu \in (-2,2)$ and $F$ is a ...
0
votes
1answer
36 views

will not-necessary continuous function be constant?

Given a real function on the segment $[0, 1]$ such that each point on the segment is either local maximum or local minimum for this function. Is this function a constant?
0
votes
2answers
157 views

How to show $\sqrt x$ is uniformly continuous at [0,1] and $[1,\infty )$

From the definition if we choose $\delta=\epsilon^2$ $|\sqrt x−\sqrt y|^2≤|\sqrt x−\sqrt y||\sqrt x+\sqrt y|=|x−y|<ϵ^2⟹|\sqrt x−\sqrt y|<ϵ.$ does this suffice both interval [0,1] and ...
1
vote
1answer
100 views

Evaluate $\int_{0}^{\infty}\sqrt{\frac{\sqrt{(a^2-y^2)^2+4y^2}+a^2-y^2}{(a^2-y^2)^2+4y^2}}dy=\sqrt{2}\pi$

Prove or disprove that$$\int_{0}^{\infty}\sqrt{\frac{\sqrt{(a^2-y^2)^2+4y^2}+a^2-y^2}{(a^2-y^2)^2+4y^2}}dy=\sqrt{2}\pi$$ for any $a>1$. I came across with this integral evaluating inverse ...
3
votes
2answers
130 views

Calculate the value of $\sum\limits _{n=1}^{\infty }\:\dfrac{n}{2^n}$ [closed]

In a previous question it is asked to represent $f(x)=\dfrac{x}{1-x^2}$ as a power series. It gave me $\displaystyle\sum _{n=1}^{\infty \:}x\left(2x^2-x^4\right)^{n-1}$. Then they ask to use the last ...
0
votes
1answer
47 views

Proving $\forall x\in \mathbb R$ and $ \forall A>0: \exists B>0 \ s.t \ \forall y\in \mathbb R, \ |y-x|\le A \Rightarrow |f(x)-f(y)|\le B$

Let $f:\mathbb R\to \mathbb R$ be a continuous function. Prove that $\forall x\in \mathbb R$ and $ \forall A>0: \exists B>0 \ s.t \ \forall y\in \mathbb R, \ |y-x|\le A \Rightarrow ...
1
vote
3answers
109 views

Alternative to epsilon-delta definition

Back in Multivariable Calculus, I remember my professor, when explaining why he decided to skip a rigorous definition of limit (the reason was that those of us that would continue in math would go ...
3
votes
2answers
61 views

Evaluate the integrals $\int \sin{x} \cot^2{x} \,dx$ and $\int \cos{x} \cot^2{x} \,dx$.

Can you please show how to evaluate the integrals $$\int \sin{x} \cot^2{x} \,dx$$ and $$\int \cos{x} \cot^2{x} \,dx.$$
0
votes
3answers
46 views

Proof that every open set is a union of neighborhoods.

Definition: A set $U \in \Bbb R^n$ is open if every point $x \in U$ has a neighborhood contained in the set.... Question: Show that every open set $U \in \Bbb R^n$ is a union of neighborhoods ...
2
votes
1answer
17 views

Minimization of two variables

(General) How can I find $$\min\limits_{\phi\leq x\leq \alpha , \ \beta \leq y \leq \delta}\{a+bx,c+dy\}$$ given values of $a,b,c,d,\phi, \alpha, \beta, \delta \in \mathbb{R}?$ (Specific) How can I ...
1
vote
2answers
40 views

Can we write $\int f(x)g(x)dx \leq C\int g(x)dx$ given that $\sup_x|f(x)|\leq C<+\infty$?

Can we write $\int f(x)g(x)dx \leq C\int g(x)dx$ given that $\sup_x|f(x)|\leq C<+\infty$? My answer is yes, but I'm confused with the following $\int f(x)dx \leq C\int dx$. What $\int dx$ means, is ...
1
vote
7answers
54 views

An example of Inequality

Could someone please show me the step by step solution to the following problem? $$ \frac{x^2+1}{x-1} \leq x $$ the anwer should be $-1 \leq x <1$. I'd like to know how to do this without the use ...
1
vote
0answers
27 views

Definition of standard functions [duplicate]

In many texts and books about calculus we see There are functions $f$ for which the anti-derivative cannot be expressed in terms of standard functions or there are many integrals that cannot ...
1
vote
2answers
38 views

implicit differentiation-trouble with algebra

I'm having trouble figuring out where to go with this implicit differentiation problem. Problem: Find $\frac{dy}{dx}$ given that $\sin(x)=e^{-y\cos(x)}$ Here is how I start: ...
6
votes
2answers
118 views

A function such that $f'(x)>0$, but not strictly monotone increasing.

This is Exercise 10.3.5 from Analysis Vol.1 by Terence Tao. Give an example of a subset $X \subset \mathbb{R}$ and a function $f: X \to \mathbb{R}$ which is differentiable on $X$, is such that ...
1
vote
3answers
89 views

Calculus: improper integration

If $\displaystyle L = \int_0^1 \dfrac{dx}{(1+x^8)} $, then what is the upper bound and lower bound for $L$. I tried with numerical methods. But not got the answer
2
votes
3answers
50 views

Show $\lim \limits_{x \rightarrow c_{-}} f(x) \neq \lim \limits_{x \rightarrow c_{+}} f(x)$ imply $f$ is discontinuous at $c$

How to show $\lim \limits_{x \rightarrow c_{-}} f(x) \neq \lim \limits_{x \rightarrow c_{+}} f(x)$ imply $f: \mathbb R \rightarrow \mathbb R$ is discontinuous at $c$ ? I know that $f$ cannot have ...
1
vote
0answers
32 views

If $f$ is a continuous function $f:[0,1]\to[0,1]$ then there exists $x_0\in [0,1]$ such that $f(x_0)=2\sin x_0$

Prove/disprove: if a continuous function $f:[0,1]\to[0,1]$ then there exists $x_0\in [0,1]$ such that $f(x_0)=2\sin x_0$. Define: $g(x)=f(x)-2\sin x$ $g(\frac {\pi} 4)=f(\frac {\pi} 4)-2\sin\frac ...
2
votes
4answers
66 views

Find the value of $a$ if $\int_0^\infty \frac{2x}{a}e^{\Large\frac{-x^2}{a}}\ dx=1 $

Find the value of $a$ if $$\int_0^\infty \frac{2x}{a}e^{\Large\frac{-x^2}{a}}\ dx=1 $$ I tried to use integration by parts but I didn't get a good response.