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

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0
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0answers
40 views

Does this series violate the decreasing condition of the Integral Test for Convergence?

I'm working on the section involving the Integral Test for Convergence in my calculus II class right now, and I've run into a seeming conflict between the definition of the Integral Test, and the ...
5
votes
5answers
107 views

Prove that $\lim_{x\rightarrow 1}{\frac{x^n-1}{x-1}}=n$ for all integer n without L'Hôpital

Prove that $\lim_{x\rightarrow1}{\frac{x^n-1}{x-1}}=n$ for all integer n without L'Hôpital. Only things that can be used are epsilon-delta, squeeze theorem and stuff like $\lim_{x\rightarrow ...
0
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3answers
50 views

Trouble finding the derivative of an expression

I could use your help. I've spent over 20 minutes on this problem and my inability to solve it has my questioning my calculus skills. If someone could show me where I messed up and walk me through the ...
1
vote
0answers
26 views

A closed form for the series $\sum_{n=1}^{\infty} \frac{H_n^2-(\gamma + \ln n)^2}{n}$

I have found a closed form for the following new series involving non-linear harmonic numbers. Proposition. $$\sum_{n=1}^{\infty} \dfrac{H_n^2-(\gamma + \ln n)^2}{n} = ...
1
vote
2answers
56 views

Find $f(x)$ such that $f'(x)<0$ for all $x$. $f''(x)(|x|-1)>0$ and $\lim\limits_{x \to \pm\infty}f(x)=-x$

I am not sure if this problem belongs to this community, down vote if not, wont mind that
1
vote
0answers
13 views

Mellin transform of a shifted function

I have an application where it would be very useful to take the Mellin transform of a shifted function. Specifically \begin{equation} M(f(y-x))(y \rightarrow s) = \int_{y=0}^{\infty} ...
1
vote
0answers
30 views

How to fill in these steps to evaluate this Gaussian integral?

As a part of a much bigger problem, I came across this integral $$\int_{-\infty}^{\infty}\ln(|x|)\frac{1}{\sigma \sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}dx$$ which represents ...
0
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3answers
100 views

What do we lose by differentiating without using the rules of differential calculus?

I learned differential calculus and its rules (quocient, chain, etc) and I got curious about one thing: What do we lose by not using these rules when differentiating? Obviously I've noted some utility ...
1
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0answers
28 views

How to establish the equivalence of these two statements about integrals of step functions?

First Statement: Let $s$ be an arbitrary step function defined on the closed interval $[a, b]$. Then we have $$ \int_{ka}^{kb} s\left(\frac{x}{k}\right) \ dx = k \int_a^b s(x) \ dx $$ for every $k ...
0
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1answer
21 views

How to find the volume of revolution around a vertical line x

How can I evaluate the volume of a solid generated by the following lines using the washer method: $y=x$, $y=0$, $y=4$. Rotated about $x=5$. I have tried to find the outer radius of $5-x$ and the ...
3
votes
2answers
74 views

Elliptical Integrals

I was trying to figure out the length of the arc in a single cycle of a sinusoidal curve and I used the curve length formula to arrive at $$\int_0^{2\pi}\sqrt{1+\cos^2x}\ dx,$$ which I am fairly ...
5
votes
7answers
82 views

For $x>0$, Prove that $\dfrac{x}{1+x^2}<\tan^{-1}x < x$

Looking for an elegant way to do it. I know one way to do it, will post soon
6
votes
2answers
119 views

Integral: $\int_0^{\infty} \cos\left(\frac{a^2}{x^2}-b^2x^2\right)\,dx$for $a,b>0$

I tried this: $$\int_0^{\infty} \cos\left(\frac{a^2}{x^2}-b^2x^2\right)\,dx=\Re\left(\int_0^{\infty} e^{-ib^2x^2+ia^2/x^2}\,dx\right)=\Re\left(\int_0^{\infty} ...
0
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0answers
22 views

How to establish this equivalence for integrals of step functions?

First Statement: Let $s$ be an arbitrary step function defined on the closed interval $[a,b]$. Then we have $$\int_{a}^{b} s(x) \ dx = \int_{a+c}^{b+c} s(x-c) \ dx.$$ Second Statement: Let $s$ be ...
3
votes
0answers
79 views

${\mathfrak{I}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} \:\mathrm{d}x$ and $\int_{0}^{\pi/2} \frac{\log \cos x}{x^2}\:\mathrm{d}x$

I have found the following new result connecting two rational log-cosine integrals. Proposition. \begin{align} \displaystyle & {\mathfrak{I}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos ...
2
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2answers
54 views

Is this function $y=-\ln\left(1+\frac{\sin x -\cos x}{2}\right)$ convex?

Is this function convex for $x\in[0,\frac{\pi}{4}]$? $$y=-\ln\left(1+\frac{\sin x -\cos x}{2}\right)$$ without use the derivative.
7
votes
2answers
205 views

Proving that $ \displaystyle \gamma = \int_{0}^{1} \int_{0}^{1} \frac{x - 1}{(1 - x y) \log(x y)} ~ \mathrm{d}{x} ~ \mathrm{d}{y} $.

In 2005, J. Sondow found a surprising formula for the Euler-Mascheroni constant $ \gamma $. The formula is $$ \gamma = \int_{0}^{1} \int_{0}^{1} \frac{x - 1}{(1 - x y) \log(x y)} ~ \mathrm{d}{x} ~ ...
2
votes
2answers
51 views

Question about $(\epsilon,\sigma)$ definition of limit

I have a basic question about the $(\epsilon,\sigma)$ definition of limit. According to this definition, it is $\lim_{x \to c}f(x) = L$ if we have for each distance $|x-c| <\sigma$ we have an ...
0
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2answers
51 views

How to calculate integral $\int_{0}^{s}\frac{1+x^{a+1}}{x^a}dx$

Can you help me calculate this integral $\int_{0}^{s}\dfrac{1+x^{a+1}}{x^a}dx$, where $a>0$. And how to do it in matlab. I tried to do in matlab but there was error, maybe x.^a is not suitable in ...
0
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0answers
8 views

Question about Peano form of the remainder

Let $f(x)$ be a real-valued function defined on a closed interval [a, b], differentiable on the open interval (a, b) $n-1$ times. $x_0$ belongs to [a, b]. Suppose that we ...
1
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1answer
55 views

How to solve this graphing question?

$ \frac{|x-2|} {(x^2-4)}+\frac{(x-2)} {|x-2|} = b $ determine for which values of $b$ the equation has one and only solution. I tried sketching the graph, but was unable to do so accuratly...also, ...
0
votes
1answer
18 views

Volume of a parallelepiped depending on $\lambda$

I've got a relatively simple calculus problem here but it has an unknown variable that I am not sure how to deal with. Find the volume of the parallelepiped depending on $\lambda$ with; $a = ...
0
votes
1answer
31 views

Find Limit floor (sin x) / floor(x) as x approaches 0.

I am unable to evaluate this limit. The floor function is giving me trouble. Any help will be appreciated. And please edit it so that it looks readable.
1
vote
3answers
26 views

Growth restriction for nonnegative, continuous functions whose integrals on $\mathbb{R}$ are bounded

When we have a nonnegative, continuous function $f(x)$ whose integral over all real numbers $\mathbb{R}$ is bounded, like: $$\int_{-\infty}^{\infty}f(x)dx = A< \infty $$ with $A \in \mathbb{R}$ ...
4
votes
3answers
47 views

Radius of $\sum a_n b_n x^n$ via radii of $\sum a_n x^n$ and $\sum b_n x^n$

Series $\sum a_n x^n$ and $\sum b_n x^n$ have radii of convergence of 1 and 2, respectively. Then radius of convergence R of $\sum a_n b_n x^n$ is 2 1 $\geq 1$ $ \leq 2$ My ...
0
votes
0answers
56 views

Proving a trigonometric inequality [on hold]

Using inequality $$2(1-\cos x)\le x^{2},\quad \forall x\in[0,\pi/4],$$ prove that $$\sin(\tan x)\ge x,\quad\forall x\in[0,\pi/4].$$
0
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0answers
20 views

Show that the Fourier transform of f is in $L^p(R)$ for every $2 \leq p \leq \infty$.

Let $$f(x)=\sum_{n=1}^\infty \sqrt{n} \chi_{(\frac{1}{n+1},\frac{1}{n})}(x)$$. The Fourier transformation of f is $$\hat{f}(y)=\sum_{n=1}^\infty ...
1
vote
0answers
28 views

Solving indefinite integrals gives multiple answers. Are all those answers correct?

While solving problems on indefinite integrals many a times I get answers which are different from those given in my text book's answer keys page. I then verify my solution steps to ensure that even ...
18
votes
2answers
553 views

$\cos x\,$ is the only function satisfying $\,f(x)\,f(y)-f(x+y)=\sin x\,\sin y.$

I need to find all continuous functions $f$ which satisfy the functional equation $$ f(x)\,f(y)-f(x+y)=\sin x\,\sin y, $$ for all $x,y\in\mathbb R$. I can prove that ...
1
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0answers
24 views

numerical solution of integral equation

Consider the basic type of integral equation. In particular, a volterra integral equation of the first kind. That is, we have the following integral equation $$\int_a^xf(s)g(s,x)~ds=h(x)$$ where $h$ ...
0
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0answers
20 views

Solving an integral equation in general

I have an integral equation such that $$\int_t^Tf(s)g(s,t)~ds=h(t)$$ where $g$ and $h$ is given. we want to know function $f$ explicitly. As I know, this type of question is about the integral ...
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votes
2answers
49 views

Finding the root of an equation [on hold]

For $p \in [-1,1]$, show that the equation $4x^3 - 3x - p = 0$ has a unique root in the interval $[\frac{1}{2},1]$ and identify it.
1
vote
1answer
42 views

Using 4 step-rule $y = 2/ (4t - 3)^{2}$ [on hold]

I tried solving it. My answer is $-4/16t^{2} + 48t + 18$, if your answer is different kindly show how is it done too thanks
-1
votes
1answer
46 views

Maxima minima problem [on hold]

Let $f(x) = \sin^3(x) + a\sin^2(x)$, $-\pi/2 < x < \pi/2$, find the intervals in which $a$ should lie in order that f(x) has exactly one minimum and one maximum.
3
votes
1answer
37 views

How to calculate this expression?(multiplication)

How do I show that for any starting $n$? (I am not really sure it is $0$, but I think it is). $$\prod_{i=n}^\infty \left[1-\frac 1 i\right]=\prod_{i=n}^\infty \left[\frac{i-1}{i}\right]=0$$ I tried ...
1
vote
1answer
34 views

identity with gamma function

I used the program "Mathematica" to get a closed form to $\sum_{j=k}^{\infty}{\frac{a^j}{j!}}, \ \ a>0 $ (and real) and the solution I got is: ...
3
votes
3answers
131 views

I need help finding the critical values of this function.

So $h(t)=t^{\frac{3}{4}}-7t^{\frac{1}{4}}$. So I need to set $h'(t)=0$. So for $h'(t)$ the fattest I've gotten to simplifying os $h'(t)=\frac{3}{4 \sqrt[4]{t}}-\frac{7}{4\sqrt[4]{t^3}}$ and that is as ...
0
votes
1answer
45 views

How would you evaluate $\int_0^1 \sqrt{2+e^{2t}+e^{-2t}}$dt

Alternatively, is there a better way to find the arc length of the vector function $\mathbf{r}(t)=\langle\sqrt2t,e^t,e^{-t}\rangle$ for $t\in[0,1]$? My work: ...
2
votes
3answers
40 views

Series question with logarithms

I want to know how to check the divergence of following sum: $\sum_{k=0}^\infty \frac{1}{\sqrt[n]{\log n}}$ I tried to use this result: $ \lim_{n \rightarrow \infty} \frac{1}{\sqrt[n]{\log n}}=1 ...
-1
votes
2answers
69 views

Finding maxima, minima of a function [on hold]

Let $p(x) = a_0 + a_1 x^2 + a_2 x^4 + ... + a_n x^{2n}$ be a polynomial in a real variable $x$ with $0 < a_0 < a_1 < \ldots < a_n$. Prove that the function $p(x)$ has only one minimum.
0
votes
1answer
25 views

How do I find the critical values to find the maximum of this function?

The total daily profit in dollars realized by the TKK Corporation in the manufacture and sale of x dozen recordable DVDs is given by the total profit function below. $$P(x) = −0.000001x^3 + 0.001x^2 + ...
1
vote
1answer
211 views

why area under curve or riemann sum equals to definite integral

i do get that Riemann sums is sum of infinite triangles with with infinitely small length. But definite integral is completely different you are taking anti derivative of f(x) at b and subtract anti ...
42
votes
9answers
2k views

Proving an alternating Euler sum: $\sum_{k=1}^{\infty} \frac{(-1)^{k+1} H_k}{k} = \frac{1}{2} \zeta(2) - \frac{1}{2} \log^2 2$

Let $$A(p,q) = \sum_{k=1}^{\infty} \frac{(-1)^{k+1}H^{(p)}_k}{k^q},$$ where $H^{(p)}_n = \sum_{i=1}^n i^{-p}$, the $n$th $p$-harmonic number. The $A(p,q)$'s are known as alternating Euler sums. ...
4
votes
3answers
77 views

Evaluate Gauss-like Integral

Evaluate Integral $$\int_0^\infty e^{-ay^{2}-\frac{b}{y^2}}dy $$ Where a and b are real and positive. This integral is eerily similar to the Gaussian integral $$\int_0^\infty e^{-\alpha x^2}dx = ...
3
votes
1answer
54 views

Typo in Spivak's explanation of limits in Calculus?

Here's what he says (including the preceding paragraph): "To show in general that f [(where f(x)=1/x)] approaches 1/a near a for any a we proceed in basically the same way, except that, again, we ...
0
votes
1answer
39 views

A question about properties of integrals

Suppose g is differentiable with $g'(x)<0$ for all $x<1$, and $g'(x)>0$ for all $x>1$, and suppose $g(1)=0$. Now let $G(x)=\int_0^x g(t)dt$. Prove that G(x) is an increasing function (this ...
1
vote
0answers
24 views

A calculus problem regarding mass

I am reviewing calculus and working on the following problem. In $\mathbb{R}^{3}$, the density function is given by $\mu(x, y, z) = |z|$ and if the region $R$ is given by $R : 2 \leq x^{2} + y^{2} + ...
0
votes
1answer
48 views

Which Riemann integrable functions have all lower sums equal?

From Spivak's Calculus, 4th edition, problem 13-11(d): Which (Riemann) integrable functions have the property that all lower sums are equal? (Bear in mind that one such function is $f(x)=0$ for ...
-1
votes
3answers
36 views

Proof using mean value theorem [on hold]

Show by using mean value theorem that $$\frac{b-a}{1+b^2} < \arctan(b) - \arctan(a) < \frac{b-a}{1+a^2},$$ where $b > a > 0.$
0
votes
2answers
34 views

Function has vertical tangent or vertical cusp?

Determine whether or not the graph of the function has a vertical tangent or vertical cusp at the indicated point c. $f(x) = (x+2)^7/3$ $c=-2$ I took the first derivative and chain rule and that got ...