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

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-2
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0answers
3 views

Does 0.9 recurring = 1?

is it just me, or are the generic users of this site 40 year old virgins, who class a good saturday night as a half pint of ginger beer and a crossword and whose internet history is sooooooooooo full ...
1
vote
1answer
15 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 ...
0
votes
0answers
10 views

Hard sum with harmonics numbers

Prove or disprove $S=\sum_{n=1}^{\infty}\frac{{H_n^{2}}{H_n^{(2)}}+3{H_n^{(4)}}}{n2^n}=\frac{25}{16}\zeta(5)+\frac{7}{8}\zeta(2)\zeta(3)$
1
vote
2answers
48 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. ...
1
vote
1answer
41 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 ...
1
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0answers
27 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
30 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
votes
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 ...
0
votes
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
vote
2answers
41 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
votes
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
vote
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
votes
4answers
62 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 $x=2\cos^2\theta+5\sin^2\theta$)
1
vote
2answers
43 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 ...
1
vote
3answers
44 views

Periodic Functions

How to prove that a function is periodic, and find its period? Say for $\sin(2x+3)$.
0
votes
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
votes
1answer
31 views

Expressing limit of sum definite integral

Evaluate limit by expressing it as a definite integral. ...
1
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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
56 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
30 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?
2
votes
2answers
44 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
votes
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
vote
2answers
24 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
votes
2answers
75 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
votes
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
vote
1answer
60 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
votes
5answers
55 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
49 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?
1
vote
3answers
155 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 ...
1
vote
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 ...
0
votes
2answers
63 views

Deducing if the series converges. [duplicate]

$$\displaystyle \sum\limits_{}^{} \dfrac{1}{k(ln(k)^2)}$$ Integral test $$\int_{} \frac {1}{u^2} du = \int u^{-2} du = \frac {-1}{u} = \frac{-1}{\ln k} +c$$
0
votes
2answers
27 views

Deduce if the series converges absolutely or conditionally.

$$\sum_{k=0}^\infty (-1)^k \frac{(3^k)(k!)^2}{(2k)!}$$ 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
votes
0answers
15 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)}→ ...
1
vote
2answers
47 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
votes
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!
2
votes
5answers
47 views

Find the center and radius of polar circle equation [on hold]

Find the radius and center of the circle $$r=2\cos \theta+3\sin \theta$$
-1
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2answers
37 views

Using Taylor series to find $\lim_{x\to 1}\frac{2-(x+3)^{1/2}}{x-1}$ [on hold]

How to find the limit $$\lim_{x\to 1}\frac{2-(x+3)^{1/2}}{x-1}$$ using the Taylor series? I have done the derivatives on function following the regular steps. But all I get is zero.
1
vote
4answers
54 views

Trig differentiation

Prove that there is a constant C such that $$ \arcsin{\frac{1-x}{1+x}} + 2\arctan (\sqrt{x}) = C $$ for all $x$ in a certain domain. What is the largest domain on which this identity is true? What ...
0
votes
0answers
45 views

Integration of a polynomial

I am facing a problem in finding the integral $$\int\frac{r^2}{-C r^3 + r^2 -2 M r +Q^2}\,dr$$ Here M, Q, and C are parameteres (to be fixed later). Could anybody Please help me in finding it? I ...
2
votes
2answers
83 views

Calculation of $\int_0^{\pi} \frac{\sin^2 x}{a^2+b^2-2ab \cos x} dx\;,$

Calculation of $\displaystyle \int_0^{\pi} \frac{\sin^2 x}{a^2+b^2-2ab \cos x} dx\;,$ given that $ a>b>0$ $\bf{My\; Try::}$ Let $\displaystyle I = \int_{0}^{\pi}\frac{\sin^2 ...
0
votes
3answers
59 views

How to solve a differential equation?

I'm trying to solve the system $$\frac{d^4x}{dt}+4x=0\quad ,\quad\frac{d^3x}{dt}+x=0$$. However, I don't know of any method of tackling such a problem. Can anyone please provide a route to a solution? ...
0
votes
1answer
18 views

Sequences with intervals

I'm trying to play a bit with sequences & intervals and I've got a few questions which I'm not sure about: Let $a_n$ be a sequence and $I=(a,b)$ interval such that {$a_n|n\in N$} densed in $I$ ...
0
votes
1answer
16 views

About restricting variables in an integrand, and also changing the look of an integrands.

So, in the last step of, many, integrands, Wolfram chooses to restrict the $x$-values, even if I didn't specify which values $x$ can take on. Take for example: $$\int\frac{dx}{x(x^2-1)^{3/2}} = ...
0
votes
2answers
29 views

infimum and supremum of subsets question

Let $B \subseteq \mathbb{R}_{+}$ such that B is non-empty. consider $B^{-1} = \left \{b^{-1} : b\in B \right \}$. Show that if $B^{-1}$ is unbounded from above, then $\inf\left(B\right)=0$ How can i ...
1
vote
2answers
44 views

Calculating semi axes from given tilted ellipse equation

Hopefully no duplicate of Ellipse $3x^2-x+6xy-3y+5y^2=0$: what are the semi-major and semi-minor axes, displacement of centre, and angle of incline? (see below) Let the following equation $$x^2 - ...
4
votes
1answer
85 views

How to find the maximum and minimum of $\dfrac{\sin x}{x^2+1}$?

How can we find the values of $x$ that give the maximum and minimum of $$\frac{\sin{x}}{x^2+1}$$ I took a lucky guess and found that $\dfrac\pi4$ was fairly close to giving the max, but how does one ...
5
votes
2answers
81 views

Evaluation of $\prod_{n=1}^\infty e\left(\frac{n}{n+1}\right)^{n}\sqrt{\frac{n}{n+1}}$

During my calculation I ended with the following product: $$P=\prod_{n=1}^\infty e\left(\frac{n}{n+1}\right)^{n}\sqrt{\frac{n}{n+1}}$$ I tried to express in term of series by taking the logarithm ...
0
votes
1answer
22 views

how to find a closed form expression for a power series

my question is how do i find a closed form expression for a function f(x) which the power series $\sum_{n=0}^\infty 2n(-7)^n x^{n+2}$ converges to and the value of x for which f(x) equals the given ...