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

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4
votes
2answers
175 views

How to integrate $\int \dfrac{x^{13}\ dx}{x^5 + 1}$

We get this problem from our teacher today. I only wish that it was $x^{14}$ in the numerator, so we can use substitution method: $$\int \dfrac{x^{13}\ dx}{x^5 + 1}$$ I cant find way to integrate ...
2
votes
2answers
64 views

How much velocity can a canister of fuel give a spaceship?

I've recently considered the issue of how much velocity a canister of fuel can provide a 'spaceship'. I assumed we could approximate a basic solution If we know the mass of the fuel $m$, the mass of ...
0
votes
1answer
72 views

Find the limit $\lim_{x\to 0} x^{-3}\int_0^{x^2}\sin{(\sqrt t)}dt$

I use the fundemental theorem of calculus $$ \displaystyle\lim_{x\to 0}\frac{\displaystyle\int_0^{x^2}\sin{(\sqrt t)}dt}{x^3}=\frac{F_{(x^2)}-F_{(0)}}{x^3}="\frac{0}{0}" $$ Than apply L'hopital rule ...
0
votes
1answer
11 views

cylindrical shells method

I am really struggling with this cylindrical shells problem. i really do struggle with the rotating around the y axis. Find the volume of the solid that results by revolving the region enclosed by ...
9
votes
2answers
140 views

Evaluating $\int_0^{\infty} {\frac{\sin{x}\sin{2x}\sin{3x}\cdots\sin{nx}\sin{n^2x}}{x^{n+1}}}\ dx$

How to calculate $$ \int_{0}^{\infty}{\sin\left(x\right)\sin\left(2x\right)\sin\left(3x\right)\ldots \sin\left(nx\right)\sin\left(n^{2}x\right) \over x^{n + 1}}\,\mathrm{d}x $$ I believe that we ...
0
votes
0answers
33 views

Which one of the following answers is correct.

The question is logx/x and both x approach to zero , If we use L hospital rule it comes out to be positive infinity . Case -2 If we draw the graph of log x as x approach to zero it approach (...
-4
votes
0answers
12 views

Volume of the solid generated by revolving the given region

Can someone help me with this problem? Find the volume of the solid generated by revolving the given region about the $y-$axis: $y=4x^2, y=8x.$ (Round your answer to 3 decimal places). I got $.458$ ...
1
vote
2answers
37 views

Why does changing the center of a geometric power series change the interval of convergence?

I know that the interval of convergence of the geometric power series $$\sum_{n=0}^\infty x^n=\frac{1}{1-x}$$ is $(-1,1)$. Why is it that if I do the following manipulation $$\frac{1}{1-x}=\frac{1}{...
-1
votes
1answer
61 views

Does any technical definition of embedding accept a “non-injective” function as opposed to only “injective”?

Embedding is defined to be a one-to-one structure preserving mapping. My question is if the one-to-one condition is really critical. Like if linear mappings from high-dimensional space to low-...
2
votes
2answers
49 views

Representing $\ln(x)$ as a power series centered at $2$ without computing any derivatives

I am working through a calc book and one of the problems asks the above question. However, taylor and maclaurin series have not been introduced yet. In some worked examples, they leverage old series,...
2
votes
2answers
37 views

How to evaluate this infinite series arising from a CS problem? [duplicate]

$$\sum_{i=1}^{\infty}i\cdot((1-\frac{1}{N!})^{i-1}\cdot\frac{1}{N!})$$ where N is an integer and $N \geq 2$? I obtained this series from the following problem: Given a sequence of $N > 2$ unique ...
1
vote
3answers
39 views

Is the function $\frac { x-1 }{ \ln { { \left( x \right) }^{ 2 } } } $ continuous at$ x=0$?

I would like to know whether the function shown in the title is continuous or not at $x=0$. This problem is disturbing since the function isn't defined at x=0, but the limit of the function as x ...
17
votes
10answers
2k views

What does it mean when dx is put on the start in an integral? [duplicate]

I have seen something like this before: $\int \frac{dx}{(e+1)^2}$. This is apparently another way to write $\int \frac{1}{(e+1)^2}dx$. However, considering this statement: $\int\frac{du}{(u-1)u^2} = \...
1
vote
2answers
44 views

Limit with Lambert-$W$ function

I have asked a similar question about this one particular limit: \begin{equation} A=\lim_{c\to 1}\exp\left[ -\left(\frac{1}{1-c}\right)\left(W_{0}\left[ B\left( 1+\frac{x}{rc}\right) \right]-W_{0}[B]\...
2
votes
1answer
43 views

What is $\lim_{t \rightarrow0} \frac{\Gamma(\alpha t)}{\Gamma(t)} (\Gamma$ is the Gamma function)?

What is $\lim_{t \rightarrow0} \frac{\Gamma(\alpha t)}{\Gamma(t)} (\Gamma$ is the Gamma function)? I took the numerator, used the change of coordinates $u = \alpha t$ and got that the limit was $\...
2
votes
0answers
33 views

How can I find the measure of $B=\{(x,y,z) \in\mathbb R^3| \; x^2+y^2+4z^2 \le3, \;x^2-y^2+4z^2\le1, \; z\ge 0\}$? [on hold]

$B=\{(x,y,z) \in\mathbb R^3| \; x^2+y^2+4z^2 \le3, \;x^2-y^2+4z^2\le1, \; z\ge 0\}$ The question is similar to that which I shared in another topic. Also here, the set is defined by an ellipsoid, ...
0
votes
2answers
23 views

volume of surface of revolution around y axis

Can anyone help walk me through this problem style? I have a lot of homework problems like this and I really want to understand how to do these problems. Find the volume of the solid generated by ...
4
votes
2answers
87 views

How can I solve this triple integral $\iiint_{B} y\;dxdydz$ on a defined set?

Calculate $$\iiint_{B} y\;dxdydz.$$ The set is $\;B=\{(x,y,z) \in \mathbb R^3$; $\; x^2+y^2+4z^2\le12$, $-x^2+y^2+4z^2\le6$, $y\ge 0 \}$. I know that B is defined by a real ellipsoid, an ...
0
votes
1answer
26 views

Test the uniform convergence of $x^n-x^{2n}$ in $[0,1]$

$$ f_n(x)=x^n-x^{2n} \\ f_n:[0,1]\rightarrow \mathbb R $$ I know that the function $x^n$ is not converging uniformally because the limiting function is not continuous (when x=1 there's a "step" in ...
2
votes
3answers
120 views

How to evaluate $\sum_{n=1}^{\infty}a_n$?

If $$a_{n}=1-\frac{1}{2}+\frac{1}{3}-\cdots +\frac{\left ( -1 \right )^{n-1}}{n}-\ln 2$$ then how to eveluate $$\sum_{n=1}^{\infty}a_n$$ does it converge?
0
votes
1answer
12 views

In Constrained Optimization, Restrict Domain to Open Set $A\subset\mathbb{R^N}$?

In constrained optimization and context of economics (e.g. utility function with quantity of goods as arguments subject to wealth), why do textbooks always restrict domain of the objective function ...
1
vote
4answers
32 views

Convergence and sum of series with exponents

So the question is how can I see if this series : $$\displaystyle\sum_{n=1}^{\infty} \frac{1}{(4-(-1)^n)^n}$$ converges and find its sum. So I would probably need to use the Leibnitz criterion for ...
1
vote
1answer
35 views

What is $\int_{|\vec x| = 1, z \geq 0}(x^2+y^2)^pz^q$ for $p,q \geq 0$?

What is $\int_{|\vec x = (x,y,z)| = 1, z \geq 0}(x^2+y^2)^pz^q$ for $p,q \geq 0$? A hint is to use the Gamma function. I plugged in spherical coordinates and got: $I = 2\pi \int \int r^{2p+q+2}sin^{...
0
votes
1answer
27 views

Show relation and linearity related to differentiable functions

I have problems solving the following exercise: (a) Let $n\in \mathbb N$, $a\in \mathbb R$ and $f:\mathbb R^n \backslash \{ 0 \} \to \mathbb R$ $\mathbb R$-differentiable. Show that the relation $$...
2
votes
0answers
77 views

Integrate $I=\int _0^{\infty }\:\frac{\cos\left(x\left(\frac{x^2-a^2}{x^2-b^2}\right)\right)}{x^2+c^2}dx$ [duplicate]

Integrate $$I=\int _0^{\infty }\:\frac{\cos\left(x\left(\frac{x^2-a^2}{x^2-b^2}\right)\right)}{x^2+c^2}dx$$ I tried and solved using contours but i wanted to compute through real methods. using ...
0
votes
0answers
18 views

Factoring out a variable from an unknown multivariable function

I have a data set that follows the behavior of a function f that depends on a lot of different variables. Let's call two of those variables $a$ and $b$. The specific behavior I'm interested in is $f(a)...
0
votes
1answer
28 views

$f:\mathbb{R^N}\rightarrow\mathbb{R}$ Definition of Partial Derivative Using Limit or Epsilon

Can someone share the exact definition of partial derivative for a function $f:\mathbb{R^N}\rightarrow\mathbb{R}$ in both limit language and epsilon-delta language? In particular, I have hard time ...
-1
votes
3answers
58 views

Is function $F(x)= 2x^2 -3x$ increasing or decreasing [on hold]

$F(x)= 2x^2 -3x$. find the range of $x$ to check whether the function the is strictly increasing and strictly decreasing.
4
votes
2answers
66 views

A function who's image is $\mathbb{R}$ on every interval

I was wondering if there exists a function $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfies the property that for every $a<b$, $Image(f|_{(a, b)})=\mathbb{R}$. I tried to show that there isn't ...
2
votes
0answers
15 views

Given $\Omega\in \Bbb{R}^n$ and $u\in C^2$ in neighborhood of $\bar{\Omega}$, show that $u=0$ in $\Omega$

I trying to solve the following question, but to no avail. Any help will be appreciated. Let $\Omega\in \Bbb{R}^n$ be smooth domain. Let $u\in C^2$ in the neighborhood of $\bar{\Omega}$, such that ...
5
votes
1answer
52 views

Prove the uniform convergence of $f_{1}(x)= \sqrt x , f_{n+1}(x)=\sqrt{x+f_n(x)}$ in $[0,\infty]$

As far as I understand most of these questions use the M-test, but I can't find a series that suffices.
-3
votes
0answers
12 views

extrema of a three variable function when determinant of Hessian matrix is zero [on hold]

Can anyone suggest how to find extremas of a three variable function if determinant of Hessian matrix is zero.
0
votes
1answer
17 views

“Hessian” differential equation

In my homework, I'm given the following problem: Let $f: \mathbb{R}^n \to \mathbb{R}$ be a twice differentiable function. For an $\alpha \geq 2$, let: $$f(\lambda x) = \lambda^a f(x)$$ for ...
0
votes
2answers
35 views

Computation of a complicated limit

Good morning to everyone! I don't know how to compute this type of limit... I got stuck at $arctan$. The limit is the following: $$ \lim _{x\to \infty }\left(\frac{\arctan \left(1-\cos \left(\frac{1}{...
0
votes
1answer
21 views

Sufficiently proving a sum converges

This might be a bit basic, but I'm pretty sure I'm wrong about this so I'd appreciate at least a confirmation that I'm wrong. The question is: Prove or disprove: If $\sum_{n=1}^{\infty} ...
4
votes
2answers
97 views

Evaluate $\lim_{n\to\infty}\left(\frac{3}{2}\cdot\frac{5}{3}\cdot\dots\frac{2n+1}{n+1}\right)^\frac{1}{n}$

What's the value of $$\lim_{n\to\infty}\left(\frac{3}{2}\cdot\frac{5}{3}\cdot\dots\frac{2n+1}{n+1}\right)^\frac{1}{n}?$$ I've tried the AM-GM inequality with no luck. Also tried to right the ...
-1
votes
0answers
11 views

Calculate sale price for a component [on hold]

With a starting price of $202.02$ for one item. I would like know the formula to get the prices listed. It will never be perfect because of rounding errors in the values. So a few cents does not ...
0
votes
1answer
28 views

Expected number of same numbered balls in a box

I have two boxes: A,B. The boxes A contains $n_1$ red balls which numbered from $(1, \cdots, n_1)$. The box B includes $n_2$ green balls which also numbered from $(1, \cdots, n_2)$. Throw balls from ...
1
vote
1answer
32 views

curl and stokes application

I cannot fin the flux of $$F(x,y,z)=(y^2cos(xz),x^3e^{yz},-e^{-xyz})$$ through the portion of sphere $$\Sigma = \{x^2+y^2+(z-2)^2=8, z\ge0 \}$$ I think Stokes th. must be used, so in spherical ...
2
votes
1answer
79 views

How to prove this integral equality?

The question is prove that$$ \lim_{n \to \infty} n^2 \left( \int_a^bf(x) \, \mathrm{d}x - \frac{b-a}{n} \sum_{i=1}^{n} f(a+(2i-1) \frac{b-a}{2n} )\right)= \frac{ (b-a)^2 }{24}\left( f'(b)-f'(a)\right)....
0
votes
0answers
11 views

Direction of a gradient at maximizer on the boundary

Let $u \in C(\bar{B})$ where $B=B_1(0) \subset \mathbb{R}^n$ is the unit ball. Assume $u$ attains its maximum at $x_0 \in \partial{B}$ and $\nabla u(x_0) \neq 0$. What can we say about the direction ...
-5
votes
3answers
127 views

Which function of $x$, other than $x +c$, and Integral of ($\cos x)^2+(\sin x)^2$, and Integral of $e^{iPi}$ has derivative = 1. [on hold]

It is a simple question: Which function of x, other than x +c, and Integral of (cosx)^2+/(sinx)^2, and the integral of $-e^{i*Pi}$, has derivative =1. Think Kepler's third law is a constant. [an ...
0
votes
2answers
31 views

Compute the limit of the sequence given by bn =(1+(3.4/n))^n [on hold]

If a sequence $c_1$ , $c_2$ , $c_3$ ,... $c_{n-1}$, $c_n$, $c_{n+1}$,... has limit $K$ then the sequence $e^{c_1}$, $e^{c_2}$, $e^{c_3}$ , ... $e^{c_{n-1}}$, $e^{c_n}$, $e^{c_{n+1}}$,... has limit $...
0
votes
0answers
21 views

Calculate the division between total height by base's diameter.

One silo for grains storage was built in a form of a cylinder (floor and walls) with a hemispherical roof. The silo is design to have a certain volume $V$. Calculate the division between total height ...
1
vote
1answer
78 views

A very difficult integral involving integration by parts.

Some fellow students were talking in a room a while back and apparently they're calculus professor told them a random integral they wrote up was "unsolvable" at the calculus semester 2 level. The ...
1
vote
1answer
24 views

Applications of Extrema [on hold]

A company wishes to manufacture a box with a volume of 16 cubic feet that is open on top and is twice as long as it is wide. Find the width of the box that can be produced using the minimum amount of ...
0
votes
0answers
39 views

Limit of infinite series. [duplicate]

I want to ask about the property of limit. I know that if we have limit finitely many sum, then we can open it separately i.e., $$\lim\limits_{n \rightarrow \infty} \sum_{k=0}^{m} a_{n,k} = \sum_{k=0}...
1
vote
2answers
44 views

What does it mean to perform calculus upon functions of complex values? [on hold]

Complex numbers exist in a plane. This would lead me to believe that calculus views them as multivariate, but I am not real sure. How would one define a rate of change for a complex number valued ...
2
votes
2answers
48 views

For which values of $a$ does $d\ge ac\ln c\implies d\ge c\ln d$?

For which values of $a>0$ is it true that for all $c,d>0$, $\hspace{.2 in}d\ge ac\ln c\implies d\ge c\ln d$? I believe that this is true for $a\ge2$, (see Showing if $n \ge 2c\log(c)$ then $n\...