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

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7 views

An easy question on number theory

Let $p$ be an odd prime. Is there any positive integer $k$ such that $p^k-1$ be a power of 2, that is $p^k-1=2^{\alpha}$ for some $\alpha\in \mathbb{N}$?
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2answers
22 views

Find The range of the function

Find the range of the function $f(x)=\frac{1}{x+2 \cos x}$. I tried like this $-2<2 \cos x<2$ then $\frac{1}{x-2}<\frac{1}{x+2\cos x}<\frac{1}{x+2}$ and then can't find the range of ...
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2answers
25 views

Find The Minimum Value of the quantity

Find the minimum value of the quantity $$\frac{(a^2+3a+1)(b^2+3b+1)(c^2+3c+1)}{abc}$$,where $$a,b,c>0$$ and $$ a,b,c\in R $$are positive real numbers.
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1answer
19 views

Show that the image of a zero measure set is of zero measure

I saw a topic on the subject but I did not quite understand, and it was a bit old and I didn't want to resurrect it. I am going in the right direction, I just need a little nudge. let $f: \mathbb ...
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3answers
22 views

find the sum of series of $\sum_{k=0}^{\infty}\frac{4^k-3^k}{5^k}$ [on hold]

Find the sum of series $\sum_{k=0}^{\infty}\frac{4^k-3^k}{5^k}$
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1answer
54 views

How was this sequence discovered?

Let $N$ be a positive integer and consider the following rational sequence for $n \ge 0$: $$ a_{n+1} = \frac{N a_n + N}{a_n + N}, a_0 \in \Bbb{Q}. $$ If $-\sqrt{N} < a_0 < \sqrt{N}$, then ...
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1answer
24 views

find the maximum of the function F under the condition $ \sum_{i=1}^N x_i = 1$

Let F a function of $ \mathbb{R} ^N_+ \rightarrow \mathbb{R}$ defined as : $$F(x_1,..,x_N)= - \sum_{i=1}^N x_i log(x_i) , x_i \gt 0$$ How can i find the maximum of the function F under the ...
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0answers
12 views

On the uniform convergence of generalized integral

Is the integral $$ \int_{1}^{\infty} e^{-yx^2}\sin{y}dx.$$ uniformly convergent in $y \in [0,\infty]$? Why or why not?
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1answer
7 views

Related rates of change - concentric spheres

Two concentric spheres each have an initial volume of 0. Their radii are increasing at 3mm/s and 5mm/s respectively. Calculate the rate at which the volume between the spheres is changing after 4 ...
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0answers
11 views

Show that we can reorder mixed partials, if every partial is continuous

Suppose $f$ has all partial derivatives up to and including $k$ and all of these partials are continuous. Prove that if $\sigma$ is a permutation on $n$ letters (any reordering), then ...
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5answers
69 views

What are the best sites to get caught up on Calculus?

I'm going back to college this summer and will be taking engineering statistics and calculus based physics. I dropped out of college about 4 years ago and took calculus 1-3 before leaving. I'm ...
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2answers
28 views

Determine the derivative implicitly: $e^x + e^y - \frac{1}{2}x^4y^2= x$

I got to $e^x + e^y\cdot y' - 2x^3y^2 + 2y\cdot y' \cdot \frac{1}{2}x^4 = 1$ I'm not sure about the $\frac{1}{2}x^4y^2$ in the original problem and I feel I may have screwed that up in taking the ...
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1answer
14 views

Showing the Clairaut theorem in higher dimensions — partials commute

Suppose $f$ has all partial derivatives up to and including $k$ and all of these partials are continuous. Prove that if $\sigma$ is a permutation on $n$ letters (any reordering), then ...
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0answers
20 views

Sperner's Lemma/Intermediate Value Theorem - odd number of crossings counting multiplicity

Suppose $f:[0,1] \to \mathbb{R}$ is not just continuous, but also smooth. Let $f(0)<0$ and $f(1)>0$. Is it true that the graph of $f$ crosses the $x$-axis an odd number of times, counting ...
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2answers
16 views

Show that $f$ is everywhere differentiable and the partials commute

Take the function $$ f(x,y) = \begin{cases}\frac{x^3y -xy^3}{x^2+y^2} & (x,y) \neq (0,0) \\ 0 & (x,y) = (0,0) \end{cases}. $$ Show that it is everywhere differentiable and that $D_{1,2}f(0,0)$ ...
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0answers
20 views

Support of $L^p$ functions?

I noticed something strange. If we look at a function $f \in L^p$, then this is an equivalence class. Hypothetically: $supp(f) = \overline{\{f\neq 0\}}$. But this is strange, as $f$ is only defined ...
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0answers
23 views

Finding a particular solution to the non-homogenous system

I have the following problem $\vec{x}^{'}(t)=\begin{pmatrix} 2 & -5\\1 & -2 \end{pmatrix}\vec{x} + \begin{pmatrix} \csc t\\ \sec t \end{pmatrix}$ Step 1) Find the Eigenvalues ...
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0answers
20 views

Riemann Integrating a Step Function

So I've been trying to prove a step function with countably infinite discontinuities is Riemann integrable using only properties of Riemann integration, no Lebesgue or gauge integration for example. ...
2
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1answer
12 views

Finding Tangent line for a Graph with the Natural Log

I'm really confused on how my professor did this problem. Any in depth explanation would be awesome. Thanks for your time.
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1answer
20 views

Limit of division by zero problem.

Find the limit as $x\rightarrow0$ of $1/x$. 1) infinity. 2) 1. 3) 0. 4) The limit doesn't exist. So I tried an experiment by plugging some values and I found that as I put small values, 1/that value ...
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0answers
13 views

Supremums involving functions and constants

I was wondering whether the following was correct? $|(\Delta t/2)u_{tt}|+|(ah/2)u_{xx}|\leq (\Delta t/2)\sup|u_{tt}|+(h/2)\sup|au_{xx}|=(1/2)(\Delta t + h) \sup|u_{tt}|\sup|au_{xx}|$ i.e. does there ...
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3answers
82 views

Challenge: Demonstrate a Contradiction in Leibniz' differential notation

I want to know if the Leibniz differential notation actually leads to contradictions - I am starting to think it does not. And just to eliminate the most commonly showcased 'difficulty': For the ...
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4answers
47 views

$\lim_{x \to 0}(x^2(1+2+3+\cdots+[\frac {1} {|x|}]))$ where [a] is largest integer not greater than a and |x| is absolute value of x

As x tends to 0, the first term $x^2$ tends to 0 while the second term tends to infinity. So is the limit undefined
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1answer
24 views

Finding/approximating 2 unknowns using one equation

I’m doing experimental data in a chemistry lab and I have faced this mathematical problem at a point of my work. Hope you guys can help me with that. What would be the best way to find two constants m ...
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2answers
54 views

Convergence of $\sum_{n=1}^\infty\frac {n^{n}}{e^nn!}$

Check the convergence of: $\displaystyle\sum_{n=1}^\infty\frac {n^{n}}{e^nn!}$ Using the root test I get: $\displaystyle\lim_{n \to\infty} \dfrac {n}{e\sqrt[n]{n!}}$ now I'm left with showing ...
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3answers
42 views

Why is it required to change variable to get the right answer for this question?

The question is this : $$\lim_{x\to-\infty} {\sqrt{x^2+x}+\cos x\over x+\sin x}$$ The solution is $-1$ and this seems to be only obtained from the change variable strategy, such as $t=-x$. However, ...
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0answers
16 views

Advanced Calculus – (Real Analysis) function f

Def. The statement that f is continuous means that f is continuous at each point in its domain. Def. if D is a subset of \mathbb{R} and f is real valued function with domain D then the statment that ...
15
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11answers
353 views

hand evaluate $\sqrt{e}$

I have seen this question many times as a example of provoking creativity. I wonder how many ways are there to evaluate $\sqrt{e}$ as accurately as possible. The obvious way I can think of is to use ...
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4answers
56 views

A limit tend to infinite! Need a little help…

Can someone help me solve this limit? $$\lim_{t \to \infty} \frac{t-t\sqrt{t}}{2t^{3/2}+3t-5} $$
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1answer
59 views

A Horrible looking limit

I have the following limit question: $$\lim_{x \rightarrow 1 }\frac {({\rm log} (1+x)-{\rm log}\space 2)(3\times4^{x-1}-3x)}{[(7+x)^{1/3}-(1+3x)^{1/2}]{\rm sin}\space \pi x}$$ This has the form ...
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2answers
30 views

$\frac {a_{n+1}}{a_n} \le \frac {b_{n+1}}{b_n}$ If $\sum_{n=1}^\infty b_n$ converges then $\sum_{n=1}^\infty a_n$ converges as well [duplicate]

We have two positive series: $\displaystyle\sum_{n=1}^\infty a_n$, $\displaystyle\sum_{n=1}^\infty b_n$ and we know that: $\frac {a_{n+1}}{a_n} \le \frac {b_{n+1}}{b_n}$ (from a certain index). ...
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0answers
18 views

Proof for a Z transform [on hold]

Proof for Z transform. Z transform of the below function. is . I would like to know the proof for this
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0answers
39 views

Convergence of $\sum_{n=3}^{\infty}\frac{1}{n\log n(\log\log n)^\alpha} $

Does the following series converge: $\displaystyle\sum_{n=3}^{\infty}\frac{1}{n\log n(\log\log n)^\alpha} $ and $\alpha>0$ ? Using Cauchy condensation test twice: $\begin{align} ...
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1answer
62 views

A question of rationality

This problem was asked to me by a friend and I simply have no idea about it. So I have not progressed a single bit. The problem is this: If $f :\mathbb{R}\to \mathbb{R}$ is an infinitely ...
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0answers
17 views

Solving ctg(x)=x/b

B is small parameter. I have no problems finding first solution(both: b->0 and b->infinity). My solutions on photos. I got stuck trying to find solution when x->infinity. As I think, solution for x ...
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2answers
40 views

Convergence of $\sum^\infty_{n=1}\frac {\sqrt[m]{n!}}{\sqrt[k]{(2n)!}}$

Does the following series converges ? $$\displaystyle\sum^\infty_{n=1}\frac {\sqrt[m]{n!}}{\sqrt[k]{(2n)!}} \ \text{for} \ \ k,m\in \mathbb N$$ I tried the ratio test: $ ...
0
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1answer
36 views

List of topics for basic calculus (1st,2nd,3rd semester)

I am an computer science student, currently studying in 2nd semester. Therefore my math courses are pretty weak. Although I "aced" them, I still feel I could use some extra basic calculus knowledge in ...
2
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1answer
52 views

$\sqrt[\large m]{(x+y)}\over \sqrt[\large k]{(x+y)}$ $=\sqrt[\large m-k]{(x+y)} $?

Is it always true that: $\sqrt[\large m]{(x+y)}\over \sqrt[\large k]{(x+y)}$ $=\sqrt[\large m-k]{(x+y)} $ where $m,k \in \mathbb N$ ? I tried it with a few numbers and it seems to work every time.
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0answers
33 views

Line integral - should I parametrize the square?

I have the following $1-\text{form}$ defined: $$\omega = \displaystyle\frac{2xy}{(1-x^2)^2+y^2}\mathrm{dx}+\displaystyle\frac{1-x^2}{(1-x^2)^2+y^2}\mathrm{dy}$$ I'd like to find ...
4
votes
1answer
98 views

Show $\lim\limits_{a \rightarrow + \infty} \int_0^{\infty} \frac{1}{1+y^2}e^{-ay} dy =0 $

Need to prove $\lim\limits_{a \rightarrow + \infty} \int_0^{\infty} \frac{1}{1+y^2}e^{-ay} dy =0 $ and $\lim\limits_{a \rightarrow + \infty} \int_0^{\infty} \frac{y}{1+y^2}e^{-ay} dy =0 $ Can ...
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6answers
44 views

General solution for squared trigonometry questions: $\cos^2 x = 1$

$\cos^2 x = 1$ How do you solve trig equations with a power? Unsure what to do with the square? I get this $\frac{1+\cos2x}2 =1$ $\cos2x =1$ $2x=2n\pi\pm0$ $x=n\pi$ but the answer says $\pm ...
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1answer
34 views

How does a sequence's convergency change finite sums?

What has been troubling me lately is that I cannot grasp how a finite series could ever diverge if a finite sequence that is divergent can only imply to a finite sum every time. Perhaps my main ...
0
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1answer
31 views

Green's theorem calculate work

Calculate $$ \oint_C \mathbf F \cdot d\mathbf r $$ where $$ \mathbf F(x,y)=\frac{2xy \, \mathbf i + (y^2+x^2)\, \mathbf j }{(x^2+y^2)^2} $$ and $C$ is any positively oriented simple closed curve ...
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0answers
26 views

smooth extensions with unique critical point

Let $f:B^{2}\rightarrow\mathbb{R}$ be a continuous function on the unit disk $B^{2}$ which is smooth in $B^{2}\backslash\{0\}$ and has no critical points there. May we find a smooth function ...
3
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1answer
37 views

Residue of $\frac{1}{(1-z)^3}$ at $z=1$

I know there is a singularity of $z=1$ but I am a bit confused on how to find the residue at that point since if we have that $f(z)=\frac{g(z)}{h(z)}$ with $g(z)=1$ and $h(z)=(1-z)^3$ then $g(z)$ has ...
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0answers
44 views

The Flat Function

I have to write an essay on the flat function $$\text{flat}(x) = \begin{cases} e^{-\frac{1}{x^2}} & \text{for } x \ne 0 \\ 0 & \text{for } x = 0 \end{cases}$$ and I want to prove ...
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1answer
32 views

Calculus Series Homework Help

I can't figure out these homework problems. I did figure out that 3 is convergent though so I don't need help on that one.
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1answer
19 views

If $A + B = \frac{\pi}{3} (A,B>0),$ Then the minimum value of sec A + sec B is?

If $A + B = \frac{\pi}{3} (A,B>0),$ Then the minimum value of sec A + sec B is? I know the condition for minima but here there are two simultaneous variables , how and with respect to what do I ...
2
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0answers
34 views

Green's Theorem

Hey guys I am having difficulties in problem 5. I thought I understood it, but I suppose I was mistaken. I will now explain what I planned to do to solve this problem and where I got stuck. So I ...