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

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0
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2answers
111 views
3
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0answers
240 views

Question on Moment of inertia about center of mass of a smooth plane curve.

This question is in continuation to this and motivated to answer this question. If I have an answer to it, then I can prove a special case of this question. Consider two smooth plane curves $C \equiv ...
7
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0answers
310 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 ...
4
votes
7answers
120 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
1
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5answers
88 views

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

Could you please explain how to integrate this integral: $$ \int \frac{1}{(x-a)(x+b)} dx $$
0
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1answer
56 views

How do they integrate this exponential?

Below, I tired to integrate te^(-j2pi*t) from 0 to 1. But am not getting what my professor got for n not equal to zero, which is also shown. I tried LIATE but am always getting something with an ...
1
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1answer
39 views

Solve this with CBS

How can you see the mínimum value of $ 1/x + 4/y + 9/z $ with x+y+z=1 using the CBS inquality? I have seen a proof of that that use trigonometric substitutions, but i don´t see as one-step the ...
1
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0answers
42 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...
5
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3answers
512 views

A difficult integral evaluation problem

How do I compute the integration for $a>0$, $$ \int_0^\pi \frac{x\sin x}{1-2a\cos x+a^2}dx? $$ I want to find a complex function and integrate by the residue theorem.
1
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1answer
74 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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1answer
59 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
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0answers
36 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
55 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?
3
votes
2answers
401 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 ...
0
votes
1answer
119 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
vote
4answers
60 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 ...
4
votes
2answers
124 views

Compute integral: $\int_{0}^{\pi/2}\log(a^2\sin^2 x+b^2\cos^2 x )dx$

This is a integral for a calculus exam, and I have no idea how to solve it. $$\int_0^{\frac{\pi}{2}} \log \big( a^2 \sin^{2}(x)+b^2 \cos^{2}(x) \big) \, \mathrm{d}x$$
1
vote
1answer
41 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!} ...
6
votes
6answers
294 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
5k views

Centroid of a region

$$y = x^3, x + y = 2, y = 0$$ I am suppose to find the centroid bounded by those curves. I have no idea how to do this, it isn't really explained well in my book and the places I have looked online ...
2
votes
2answers
136 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 ...
1
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1answer
60 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
96 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 ...
1
vote
0answers
36 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} ...
0
votes
1answer
418 views

Why does direct substitution work for limits?

I see a lot of calculus texts stating direct substitution is a form of evaluation for a limit. Maybe I'm missing something because, to me, direct substitution only shows the value of a function ...
0
votes
3answers
69 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 ...
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
125 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
35 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 ...
2
votes
2answers
71 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.
0
votes
2answers
57 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 ...
1
vote
1answer
74 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, ...
1
vote
3answers
42 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
70 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 ...
1
vote
0answers
58 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
72 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.
3
votes
1answer
44 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
43 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
157 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
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1answer
50 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
55 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 ...
0
votes
1answer
53 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 + ...
3
votes
1answer
114 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
52 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 ...
0
votes
1answer
96 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 ...
0
votes
2answers
254 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 ...
6
votes
0answers
112 views

Modern notational alternatives for the indefinite integral?

I like the Leibniz notation, and I think the reason it's survived for over 300 years and continued to be almost the only game in town is that in many respects it's a miracle of design. Nevertheless ...
0
votes
4answers
162 views

Prove: If function is defined and continuous on [a,b] …

If $f$ is defined and continuous on $[a,b]$ and $f$ has no less than two values in its range, then set of points at which $f$ takes the minimum and maximum values can not be open. PS It's quite ...
2
votes
4answers
94 views

Differentiating $\frac{te^{\tan t}}{ln(3t+1)}$?

I've tried to differentiate the following function: $$f(t)=\frac{te^{\tan (t)}}{ln(3t+1)}$$ But I am confused at what I should do (and perhaps I forgot some identities too), I've learned the ...
1
vote
1answer
123 views

Does calculus of variations have a close connection to Feynman's ''differentiation under the integral sign''?

Most of the calculus I've studied seems separate math problems in to "derivative" or differential applications and integral applications. The one exception seems to be "calculus of variations," which ...