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

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8 views

Infinitesimal Calculus / Analysis - supremum question

We're given A = {(6n^2-5)/(4-3n^2) : n>=2} Prove that A is bounded above and below. Find the supremem of A and prove it is the supremum. Thanks for your help.
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1answer
18 views

Find $\int_0^2 tan^{-1}\pi x-\tan^{-1}x\hspace{1mm}dx$

Find $\int_0^2 tan^{-1}\pi x-\tan^{-1}x\hspace{1mm}dx$ The hint is also given : Re-write the Integrand as an Integral I think we have to Re-write this single integral as a double integral and then ...
5
votes
1answer
32 views

$\forall\ x,y,z\in \mathbb{R}$ Show that: $|x+y|+|y+z|+|x+z|\leq |x+y+z|+|x|+|y|+|z|$

$\forall\ x,y,z\in \mathbb{R}$ Show that: $$|x+y|+|y+z|+|x+z|\leq |x+y+z|+|x|+|y|+|z|$$ i tired, i notice that $x,y,z$ plays a symmetrical role in the inequality notice also that ...
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0answers
20 views

Contradiction proof for a limit law $f(x) \le g(x)$

Suppose that $f(x) \le g(x)$ for all $x$. Prove that $\displaystyle \lim_{x \to a} f(x) \le \lim_{x \to a} g(x)$, provided these limits exist. I posted a similar question, but this is a different ...
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0answers
33 views

Evaluate $\int_0^{2\pi} \frac{d\theta}{\left(1+\beta \cos (\theta )\right)^2}$

I am trying to evaluate the integral $$\int_0^{2\pi} \frac{d\theta}{\left(1+\beta \cos \left(\theta\right)\right)^2}$$ via change of variables and applying Cauchy's Residue Theorem. Here is how I'm ...
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1answer
40 views

Integration (Cosine Function)

Ive been doing some integration study and ive been caught by this question. Anyone have any ideas? Thanks. Apologies on how the question is presented, im no quite sure how to do it properly yet. - ...
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0answers
11 views

How to rewrite a derivative w.r.t. tensor as w.r.t. vector

I'm stuck on a (probably very simple) problem I've come across. Take a function $f(A)$ where $A$ is a 2-tensor. Now suppose $A=vv^T$ for an $\mathbb{R}^n$ vector, $v$. I want to rewrite the object ...
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4answers
60 views

Show that $\lim_{x \to +\infty}f(x)+f'(x)=0 \Rightarrow \lim_{x \to +\infty} f(x)=0$

How to show that $\lim_{x \to +\infty}(f(x)+f'(x))=0 $ implies $\lim_{x \to +\infty} f(x)=0$?
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4answers
43 views

Evaluating $\lim_{x\rightarrow\infty}(x+1-x)$

$$\lim_{x\rightarrow\infty}(x+1-x)=\lim_{x\rightarrow\infty}1=1$$ can we do like this: $$\begin{align}\lim_{x\rightarrow\infty}(x+1-x)&=\lim_{x\rightarrow\infty}x(1+\frac{1}{x})-x\\ ...
2
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2answers
55 views

If $f(2x-1) = 4x$, find $f(x)$

How can I solve these kind of problems? Can someone explain, please? $f(2x-1) = 4x$, find $f(x)$
2
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2answers
32 views

finding the period of $\sin(2x+3)$

I tried to find the period of $\sin(2x+3)$; looking for $p>0$ such that $ \sin(2(x+p)+3)=\sin(2x+3),$ for all $ x \in R$ which means: $ \sin(2(x+p)+3) - \sin(2x+3)= 0 ,$ for all $ x \in R$ ...
2
votes
0answers
29 views

What are the connections between the three Mertens' theorem?

In number theory the three Mertens' theorems are the following. Mertens' $1$st theorem. For all $n\geq2$ $$\left\lvert\sum_{p\leqslant n} \frac{\ln p}{p} - \ln n\right\rvert \leq 2.$$ Mertens' ...
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0answers
18 views

Get summation inside the division

Consider I have the following summation: $$\sum_1^{18} \frac{M_i}{I_i}$$ Now I have the "problem" that I only know the total sum of all M's: $\sum_1^{18} M_i$. The single ones I can only ...
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0answers
26 views

Help with First Order Differential Equations

Solve the given the two equations: $ xdy + ydx = ydy $ and $ (y^2 + 1)dx +(2xy + 1)dy = 0 $ For the first, I can see that solving this with respect to $ dy/dx $ might be a bit tricky. However, ...
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2answers
33 views

How to prove a function has no local minima.?

Suppose we have a function $ f:\mathbb{R}^2 \to \mathbb{R}$, of class $C^2$ that satisfies: $3\frac{\partial^2f}{\partial x^2}(x,y)+4\frac{\partial^2f}{\partial y^2}(x,y)=-1$, for all $(x,y) \in ...
5
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0answers
27 views

Hard sum with harmonics numbers

Prove or disprove that $S=\displaystyle\sum_{n=1}^{\infty}\frac{{H_n^{2}}~{H_n^{(2)}}+3{H_n^{(4)}}}{n~2^n}=\frac{25}{16}\zeta(5)+\frac{7}{8}\zeta(2)\zeta(3)$.
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2answers
51 views

How can I justify that $\int_0^{+\infty} f(t) \sin(t) dt$ diverges when $f$ is a polynomial?

I have this integral, $\int_0^{+\infty} f(t) \sin(t) dt$ where $f$ is a polynomial. Integrating by parts as many times as the degree of the polynomial, we can see that this integral doesn't converge. ...
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1answer
54 views

$ 0 \le f(x) \le 1 $ for $ 0 \lt x < 1 \implies \int_0^x f(t)t ~dt \le x^2 $ for all $ x\in(0,1) $?

I have the following implication, and I need to determine whether it's true: $ 0 \le f(x) \le 1 $ for $ 0 \lt x < 1 \implies \int_0^x f(t)t ~dt \le x^2 $ for all $ x\in(0,1) $ I tried solving ...
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0answers
32 views

Multi-variate Taylor Series Expansion

I understand how to use Taylor series to expand basic functions. However, I am trying to work out how to expand Taylor series with more than one variable. So far I have the equation with the two ...
1
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0answers
35 views

$F(x) = \int_0^x f(t)~dt \implies F(1)=f(0)+\int_0^1(1-t)f'(t)~dt$?

f is differentiable and has a continuous derviative, and $F(x) = \int_0^x f(t)~dt$. Based on this assumption, I have the following statement which I need to determine whether it's true or false: ...
0
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1answer
22 views

How to use Cauchy sequence on $|a_{n+2}-a_{n+1}| \le q|a_{n+1}-a_n|$ [duplicate]

I want to apply Cauchy prinicpal on the following question: If exists constant $$0<q<1$$ such that $$|a_{n+2}-a_{n+1}| \le q|a_{n+1}-a_n|$$ for any $n$ then $a_n$ converges. Now if I just ...
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0answers
4 views

Definite integral on elliptic integral where modulus is function of variable

How to prove: $\int_{0}^{\frac{\pi }{2}} {\frac{\sin \theta}{\sqrt{Z^2+(R+h \tan \theta)^2}} K[k(\theta)]}d\theta=\frac{\pi }{2\sqrt{R^2 + (h+Z)^2}} $ where $ k(\theta)= \sqrt\frac{4Rh \tan ...
1
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2answers
46 views

f is even or odd, prove that f^2 is even

I need to verify whether a statement is correct or false. The statement is as following: If the function f is either odd or even, then the function f^2 is even. To my understanding, the statement is ...
0
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0answers
18 views

Integrals: Average(f)*Average(g)=Average(f*g) [on hold]

So I've got everything but question #3 here. I understand that it isn't simply (1/4)(1/4)=16. And also not (1/4)(1/4)(1/4)=1/64. But I can't think of what else it might be. It isn't discussed in the ...
1
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2answers
19 views

Do the partial derivatives of this piecewise constant function exist? If yes, how can I compute them?

Given this piecewise constant function $$ f(x,a,b,c,d,e) = \begin{cases} a, & x \lt d; \\ c, & d \le x \lt e; \\ b, & e \le x. \\ \end{cases} $$ do the partial derivatives ...
0
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4answers
63 views

Evaluating $\displaystyle \int\frac{1}{\sqrt{(x-2)(5-x)}}\,dx$ using trigonometric substitution [on hold]

Using Substitution Integral Method, compute $$\displaystyle \int\frac{1}{\sqrt{(x-2)(5-x)}}\,dx$$ (let $x=2\cos^2\theta+5\sin^2\theta$)
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2answers
54 views

Is my proof regarding continuity at irrationals correct?

Consider the Thomae's function $$f(x)=\begin{cases} 0 \text{ ; when } x \text{ is irrational} \\\frac 1 q \text{ ; for } x=\frac p q \text{ irreducible fraction}\end{cases}$$ In the following proof ...
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2answers
48 views

Periodic Functions

How to prove that a function is periodic, and find its period? Say for $\sin(2x+3)$.
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0answers
9 views

Recursive formula for Laguerre guassian integral?

The integral of interest is: $ I_{l, m} = \int_{u0}^{u1} u^{(l+1)/2} e^{-u/2} L_m^l(u) du $ where $L_m^l$ is the laguerre polynomial. What I'm interested in is getting some relation to lower order ...
0
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1answer
32 views

Expressing limit of sum definite integral

Evaluate limit by expressing it as a definite integral. ...
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5answers
50 views

$\varepsilon$ - $\delta$ proof for $\lim_{x \to 27}2x^{2/3}=18$

Construct a careful $\varepsilon$ - $\delta$ argument to show $$\lim_{x \to 27}2x^{2/3}=18$$ From the definition of a limit $$\forall \varepsilon > 0, \space \exists \delta >0 \space : ...
2
votes
4answers
57 views

Derivation of the integral

Evaluate $$\large\frac{d}{dx}\int_{0}^{\large\int_0^{e^x}{\cos (s)\,\mathrm ds}}\sec(t^2)\,\mathrm dt$$ I got the answer to be $$e^x\cdot\sec(\sin^2(e^x))\cdot \cos(e^x)$$ but do not know if ...
0
votes
1answer
31 views

Consider $f(x) = \frac{2x^3-1+\sin x}{x^2-3}$. Show that $f (x) < 2x$ for most negative values of $x$.

Consider $$f(x) = \frac{2x^3-1+\sin x}{x^2-3}$$ Show that $f (x) < 2x$ for most negative values of $x$. How do I start this/ what concepts does this questions test?
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2answers
47 views

Homework help. From spivak calculus book

Show that $f$ is convex on an interval if and only if for all $x$ and $y$ in the interval we have $$f(tx+(1-t)y)<tf(x)+(1-t)f(y), 0<t<1$$ The only thing I know is that we have to approach ...
0
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0answers
15 views

Maximum volume of an open box with a square base?

A box with a square base and an open top is to be made. You have 1200cm^2 of material to make it. What is the maximum volume the box could have? Here's what I did: 1200 = x^2+4xz; where x=length of ...
1
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2answers
25 views

Continuous increasing bounded function, derivative

Is it true that a differentiable (and hence continuous) increasing bounded function $f:\mathbb{R} \to \mathbb{R}$ has derivative $f'$ that must go to zero as $x \to \infty$. If it is, could someone ...
3
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2answers
76 views

$\frac{1}{x^2} \int xe^x dx$ without using integration by parts

On a test i just had, i needed to solve a differential equation which lead me to having to find the result of $$ \frac{1}{x^2}\int xe^x dx $$ I then attempted to do this integral without integration ...
2
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2answers
41 views

Trouble Understanding Continuity Theorem

I am looking at Calculus on Manifolds by Michael Spivak, but there's a theorem that I don't quite understand. 1-8 Theorem. If $A \subset \mathbb{R}^n$, a function $f: A \to \mathbb{R}^m$ is ...
1
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1answer
61 views

Proof of $\lim_{x \to \infty}\tan x/x$ does not exist

There is an answer for this question as follows: If we approach to infinity with the sequence $a_{n}=n\pi$ then limit is zero, on the other hand if we approach with the sequence ...
0
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5answers
56 views

Composition of two functions is not commutative

I have been always shown that the composition of two functions is, in general, not commutative with a counterexample. But can you give a more general proof of this statement (that is to say, one that ...
2
votes
4answers
50 views

Integration by parts of $\cos(x)e^{-x}dx$

I do the integral but I end up getting the original $\cos(x)e^{-x}dx$ on both sides and canceling them out resulting in no solution. Can I get a step by step break down of how to solve?
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vote
3answers
159 views

Optimization problem?

Hi I was having trouble figuring out this question. Find the point on the circle $x^2 + y^2 = 1$ in the first quadrant where the tangent line to the circle encloses with the coordinate axes a ...
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0answers
30 views

Derivative question with series

I am having this question here. Find the 66th derivative of $\displaystyle \cos x^3$ Yes, the cube is on the $x$. The idea is to do it with series. I got an answer which I want to verify if it is ...
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2answers
67 views

Deducing if the series converges. [duplicate]

$$\displaystyle \sum\limits_{}^{} \dfrac{1}{k(ln(k)^2)}$$ Integral test $$\int_{} \frac {1}{u^2} du = \int u^{-2} du = \frac {-1}{u} = \frac{-1}{\ln k} +c$$
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2answers
31 views

Deduce if the series converges absolutely or conditionally.

$$\sum_{}^\ (-1)^k \frac{(3^k)(k!)^2}{(2k)!}$$ I start by using the absolute convergence test. This eliminates the -1: |1|^k = 1 Then I use the ratio test. $$\left|\frac{3^{k+1} ...
0
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0answers
18 views

How to prove whether the series problem converges or diverges?

$Σ$ $(-1)^k$ $ \frac{k^2+3k}{k^3+k+2)}$ I use the absolute value theorem for this problem. Then I use the a limit comparison test on it. $\frac{k^2+3k}{k^3+k+2}*1/k =\frac {k(k^2+3k)}{(k^3+k+2)}→ ...
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2answers
48 views

Finding the exact amount of a sigma problem?

$$\begin{align*} \sum_{k=1}^\infty \frac{1}{k(k+1)} \end{align*}$$ This is a telescoping series; therefore I use partial fractions to solve. $\int_{1}^{∞} (1)/(k+1) $ = $ ((A/k)+(B)/(k+1))$ A= 1 ...
1
vote
4answers
56 views

without using l'hopital rule

Can someone give me please some guidance hoe to solve the following limit, without using L'Hopital rule? $$\lim\limits_{n \to \infty } \frac{n}{\ln\left(\frac{3n}{5}\right)}$$ Thanks a lot!
0
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2answers
24 views

Cauchy sequence in practice

Let $a_n$ be a sequence. Based on Cauchy can I say that if $|a_{n+2}-a_{n+1}| < |a_{n+1}-a_n|$ then $a_n$ converges? The reason behind this is that $a_{n+1}$ is just a small offset of $a_n$ and ...
0
votes
5answers
48 views

Consider the following limit: $\lim_{n \to \infty } \frac{\ln(1+n)-\ln(n^{2})}{\sin(1/n)}$

Can someone give me some guidance on where to begin with the following limit? $$ \lim_{n \to \infty } \frac{\ln(1+n)-\ln(n^{2})}{\sin(1/n)} $$ Thanks!