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

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1
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0answers
39 views

The volume is to be found

Find the volume of $A=\{(x,y,z) \in \mathbb{R}^3: 2x^2+3y^2 \leq z \leq 4+2x+3y\}$ I know we are to solve it by using triple integral...
0
votes
0answers
31 views

How to solve this recursive integral?

$$f(p)= \int_a^\infty\frac{\exp(\iota k\dot p)}{k^2 + f(k)} dk$$ I thought of solving it like if I guess $f(k)$ equals a number then after solving the integral it should be itself.
2
votes
1answer
50 views

$a_n$ diverge $\nRightarrow a^2_n - a_n + 1$ diverges

Let $a_n$ be divergent sequence. Then a sequence $a^2_n - a_n + 1$ diverges. I have difficulties with finding out a counterexample. Could you help me?
1
vote
1answer
29 views

Aftermath of Cauchy's mean value theorem

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 ...
0
votes
5answers
77 views

$ \int \frac{1}{(x-a)(x+b)} dx $

Could you please explain how to integrate this integral: $$ \int \frac{1}{(x-a)(x+b)} dx $$
4
votes
3answers
476 views

Does a closed form exist for this summation?

How do I calculate $$\sum_{k=1}^\infty \frac{k\sin(kx)}{1+k^2}$$ for $0<x<2\pi$? Wolfram alpha can't calculate it, but the sum surely converges.
1
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4answers
51 views

$\lim_{n \rightarrow \infty} f_n(x) = n^2 \left( 1- \cos \frac{x^3 - 1}{n} \right)$

Let $$f_n(x) = n^2 \left( 1- \cos \frac{x^3 - 1}{n} \right)$$ Let M be the set of x s.t. $\lim_{n \rightarrow \infty} f_n(x)$ exists. For each $x \in M$ let $f(x) = \lim_{n \rightarrow ...
1
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1answer
29 views

Continuity of series $\sum_{n=0}^\infty \frac {x^n sin(nx)} {n!}$?

Let $$ S(x) = \sum_{n=0}^\infty \frac {x^n sin(nx)} {n!}~~,~~ S_k(x) = \sum_{n=0}^k \frac {x^n sin(nx)} {n!}$$ $$ \left |S(x) - S_k(x) \right| = \left | \sum_{n=k}^\infty \frac {x^n sin(nx)} {n!} ...
1
vote
0answers
26 views

Conjunctive Normal Form representation/ First Order Logic.

in my research problem, I need to represent three types of three types of relationships between the variables x,y as the following:: " y Cooperates with x" relationship: means if there is two ...
-2
votes
1answer
56 views

Calculus question about finding the length [closed]

A rectangular plot of ground has two adjacent sides along highways 40 and 60. In the plot is a small lake, one end of which is 256ft from highway 40 and 108 ft from highway 60. Find the length of the ...
3
votes
1answer
117 views

Compute $\int_1^e \frac{dx}{x(x+(\ln x)^2)}$

My friend asked me how to integrate the following: $$\int_1^e \frac{dx}{x(x+(\ln x)^2)}$$ How am I going to solve this?Any help is greatly appreciated. Thanks.
4
votes
2answers
305 views

True/ False differential equation

Are the statements in Problems 46-54 true or false? If $F(x)$ is an antiderivative of $f(x)$, then $y=F(x)$ is a solution to the differential equation $\frac{dy}{dx}=f(x)$. If $y=F(x)$ is a solution ...
1
vote
1answer
52 views

How to find the value of $c$ using the mean value theorem?

So I'm doing Mean Value theorem homework which states $$f'(c)=\frac{f(b)-f(a)}{b-a}$$ I have $f(x)=e^{\frac{-x}{2}}$ over the interal [0,12]. Using the mean value theorem I ...
1
vote
1answer
92 views

How Can I figure out when cosine = $\frac{2}{\pi}$?

So I'm doing Mean Value theorem homework which states $$f'(c)=\frac{f(b)-f(a)}{b-a}$$ So I am trying to find $c$ for $f(x)=\sin x$ over the interval $[0,\frac{\pi}{2}]$. So using the Mean Value ...
2
votes
2answers
60 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 ...
2
votes
1answer
127 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
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0answers
23 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} ...
6
votes
6answers
164 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 ...
-1
votes
0answers
30 views

Find the center of mass of a region with uniform density [closed]

A region on the graph is bound by the lines $y=x/2$, $y=0$, $x=2$ How can I calculate the center of the mass assuming a uniform density of "p" throughout the region?
0
votes
2answers
60 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
0
votes
3answers
110 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
29 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
votes
1answer
31 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 ...
-1
votes
1answer
26 views

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

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 ...
5
votes
7answers
108 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
2
votes
2answers
92 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
votes
1answer
14 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 ...
0
votes
2answers
56 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
votes
1answer
35 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.
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 = ...
1
vote
3answers
29 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
53 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 ...
2
votes
1answer
45 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 ...
2
votes
2answers
58 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.
1
vote
1answer
72 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, ...
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votes
2answers
51 views

Finding the root of an equation [closed]

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
0answers
30 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
votes
1answer
47 views

Maxima minima problem [closed]

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.
1
vote
1answer
44 views

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

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
0
votes
0answers
24 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 ...
1
vote
1answer
48 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 ...
1
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0answers
60 views

Proving a trigonometric inequality [closed]

Using inequality $$2(1-\cos x)\le x^{2},\quad \forall x\in[0,\pi/4],$$ prove that $$\sin(\tan x)\ge x,\quad\forall x\in[0,\pi/4].$$
1
vote
1answer
35 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: ...
0
votes
1answer
46 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: ...
0
votes
2answers
71 views

Finding maxima, minima of a function [closed]

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.
7
votes
2answers
162 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
votes
1answer
28 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 + ...
0
votes
3answers
60 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 ...
2
votes
3answers
42 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 ...
3
votes
1answer
58 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 ...