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

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0
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2answers
32 views

Compare the following numbers…

How can I compare this two numbers: $\tan \frac 7{10}$ and $1$? Hope your explanation will be as explicit as possible! Thank you very much!
2
votes
1answer
26 views

Convergent series multiplied by $n$

this one should be really simple... Let $\mathbb a_n$ be a sequence of strictly positive real numbers such that $\sum_{n\in\mathbb N}a_n$ is finite, i.e., the limit $\lim_{k\to\infty}\sum_{n=1}^ka_n$ ...
1
vote
0answers
29 views

Radius of convergence and uniqueness

I was reviewing some fundamental concepts of real analysis, and the following exercise popped up: Let there be a power series: $\sum\limits_{k=0}^\infty a_k \mid x - 3 \mid $ Such that for $\ x = ...
3
votes
1answer
77 views

$\lim_{x \to \infty} \frac{3+ 2 \sin (x)}{4+5x \sin(x)} $

A friend asked me to compute the following limit: Problem: Compute: $$ \lim_{x \to \infty} \frac{3+ 2 \sin(x)}{4+5 x \sin (x)}$$ I was clueless at first, so I gave it some thought and considered ...
1
vote
1answer
57 views

what key axioms are behind calculus

what key axioms make calculus correct? I know there are axioms for real numbers, are there any other important axioms behind calculus?
1
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1answer
10 views

Determing if a parametric curve is smooth

I have to determine whether the following curves are smooth or not and I'm having trouble with the following two functions: Consider $f(t) = (t^{2}-1,t^{2}+1)^{T}$ The solution states: $f'(t) = ...
1
vote
0answers
24 views

Using matrices to solve systems,2 questions, pre-calc. help? [on hold]

1.) Solve for a and b. With work. [a b] [2 0] = [8 -8] .................................................................................................................................... ......[1 ...
3
votes
2answers
47 views

Uniform continuity of $f(x)=(1-\cos(x))/\sin(x)$ on the interval $(0,1)$

I got this question: Prove that the function $f(x)=\frac{1-\cos(x)}{\sin(x)}$ is uniformly continuous on the interval $(0,1)$ I tried to prove it directly using the definition of uniform continuity ...
1
vote
1answer
24 views

Find speed of an aircraft flying towards an observer

An aircraft is flying towards an observer at an altitude of $2000$ m. When the angle of observation, $z$, (between the ground and the aircraft) is 30 degrees, the rate of change of this angle is $2$ ...
5
votes
3answers
74 views

Limit of $x \ln^2|x|$ when $x\to 0$

I want to evaluate this limit :$$\lim_{x\to 0}x\ln^2|x|$$ I wanted to use L'Hôpital's rule for this: $\lim\limits_{x\to0-}\frac{\ln^2|x|}{\tfrac{1}{x}}$, but I don't know how to differentiate the ...
1
vote
1answer
66 views

Convergence of $\sum^\infty_{n=1}\frac{a_n}{1+a_n^2}$

We have a positive series $\displaystyle\sum^\infty_{n=1}a_n$. is the following series converge or diverge ?$$\displaystyle\sum^\infty_{n=1}\frac{a_n}{1+a_n^2}$$ ...
3
votes
2answers
43 views

Convergence of $\sum^\infty_{n=1}\frac{a_n}{1+n^2a_n}$

We have a positive series $\displaystyle\sum^\infty_{n=1}a_n$. is the following series converge or diverge ?$$\displaystyle\sum^\infty_{n=1}\frac{a_n}{1+n^2a_n}$$ Suppose ...
3
votes
1answer
61 views

An easy question on number theory

Let $p$ be an odd prime. Is there any positive integer $k>1$ such that $p^k-1$ be a power of 2, that is $p^k-1=2^{\alpha}$ for some $\alpha\in \mathbb{N}$?
1
vote
2answers
29 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 ...
1
vote
2answers
35 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.
0
votes
1answer
22 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 ...
0
votes
3answers
26 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 $\sum_{k=0}^{\infty}\frac{4^k-3^k}{5^k}$
3
votes
1answer
111 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 ...
1
vote
1answer
29 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 ...
2
votes
0answers
16 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?
1
vote
1answer
11 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 ...
1
vote
1answer
23 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 ...
3
votes
5answers
94 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 ...
0
votes
2answers
35 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 ...
0
votes
1answer
22 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 ...
1
vote
0answers
23 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 ...
0
votes
2answers
20 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 $D_{1,2}f(0,0)$ ...
0
votes
1answer
34 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: $\operatorname{supp}(f) = \overline{\{f\neq 0\}}$. But this is strange, as $f$ is ...
1
vote
2answers
63 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 ...
0
votes
0answers
25 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
votes
1answer
20 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.
0
votes
1answer
24 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 ...
-2
votes
0answers
35 views

Properties of Supremum

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 ...
6
votes
3answers
136 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 ...
1
vote
4answers
57 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
0
votes
1answer
29 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 ...
0
votes
2answers
60 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 ...
1
vote
3answers
44 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, ...
2
votes
1answer
26 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 ...
15
votes
11answers
405 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 ...
1
vote
4answers
62 views

A limit tend to infinite! Need a little help…

Can someone help me solve this limit? $$\lim_{t \to \infty} \frac{t-t\sqrt{t}}{2t^{3/2}+3t-5} $$
1
vote
1answer
60 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 ...
1
vote
2answers
32 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). ...
-1
votes
0answers
22 views

Proof for a Z transform [on hold]

Proof for Z transform. $Z$-transform of the below function. $$2|c||p|^n\cos(n\cdot\text{Arg}(p)+\text{Arg}(c))$$ $c,p\in\mathbb{C}$, is $$\frac{cz}{z-p}+\frac{c^*z}{z-p^*}$$ I would like to know ...
1
vote
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} ...
7
votes
1answer
66 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 ...
2
votes
1answer
24 views

Solving $\operatorname{ctg} x=x/b$

I have no problems finding first solution (both: $b \to 0$ and $b \to \infty$). My solutions on photos. I got stuck trying to find solution when $x \to \infty$. As I think, solution for $x$ will have ...
1
vote
2answers
46 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
votes
1answer
38 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 ...