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

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1
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5answers
178 views

Solve $e^x-1= 2x$ with numerical or analytical methods.

Find the value of x for which \begin{equation} e^x-1= 2x \end{equation} Some numerical or analytical methods are appreciated, thanks.
1
vote
1answer
29 views

Convergence of a subsequence .

If every subsequence of $x_n$ has a further subsequence which converges , is it true that the sequence is convergent? NOTE : This is not a duplicate ofthis . In this problem it is not given that the ...
1
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0answers
110 views
+50

Question on differentiate under integral

First we have the following theorem: Then we apply it to a concrete problem: Finally how to obtain the second rectangle?
0
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1answer
35 views

holomorphic function over the disk that is real on a closed curve must be constant

Let $f$ be holomorphic on $\{z\in \mathbb{C}\mid |z|\leq 3\}$ and real on the boundary of the square $\{z\in\mathbb{C}\mid Re(z)\leq1 \text{ and } Im(z)\leq 1 \}$. Prove $f$ is constant. How to ...
0
votes
0answers
56 views

Question on differentiation under the integral

How to obtain the red rectangle? Why factor n is disappeared?
1
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1answer
35 views

Solution to $u_t+\Delta^2u+\Delta u=0$

Suppose there exists a solution to *$u_t+\Delta^2u+\Delta u=0$ of the form $u(x,y,t)=c(t)e^{i\pi(x/4\pi+y/4\pi)}$. I need to find such a function $c(t)$. Plugging $u(x,y,t)$ into *, I got ...
-3
votes
0answers
33 views

Find the general solution to the first - order differential equation [closed]

(dy/dx)+(tanx)y=cosx ( dy(x))/( dx)+y(x) tan(x) = cos(x): Let mu(x) = e^( integral tan(x) dx) = sec(x). Multiply both sides by mu(x): sec(x) ( dy(x))/( dx)+(sec(x) tan(x)) y(x) = 1 Substitute ...
3
votes
2answers
69 views

Related Rates Galore!

Water from one tank is being drained into another tank at a rate of 3 m3/min. The first tank is an inverted circular cone with height 3 m and radius 2 m. The second tank is a circular cylindrical tank ...
1
vote
2answers
58 views

$\delta-\epsilon$ proof for the limit $\lim\limits_{x\to3^-}\sqrt{3-x}=0$. [closed]

I have no idea how to start in this question since it is a left hand limit. Thanks for any help! :)
2
votes
0answers
99 views

Definite trigonometric integral

This question is motivated by Iterative Mean, Covariance Algorithm Convergence: Is there a closed form for the integral $$ \int_0^{2 \pi} ...
20
votes
5answers
4k views

Why is $\sin(d\Phi) = d\Phi$ where $d\Phi$ is very small?

I haven't touched Physics and Math (especially continuous Math) for a long time, so please bear with me. In essence, I'm going over a few Physics lectures, one which tries to calculate the Force ...
5
votes
2answers
157 views

What is the closed form for $S=\displaystyle\sum_{n=1}^{\infty} \dfrac{\sin ({n})}{n!}$?

How do we find the following sum (closed form)? $$S=\displaystyle\sum_{n=1}^{\infty} \dfrac{\sin ({n})}{n!}$$ Edit What about $$S=2\left(\displaystyle\sum_{n=1}^{\infty} \dfrac{\sin ...
1
vote
2answers
38 views

Jordan measure zero discontinuities a necessary condition for integrability

The following theorem is well known: Theorem: A function $f: [a,b] \to \mathbb R$ is Riemann integrable if and only if its set of discontinuities has Lebesgue measure zero. Now if we change ...
0
votes
3answers
43 views

Trouble with Logarithmic Differentiation

Hey guys I'm trying to find the derivative of this equation using logarithmic differentiation but I'm having some trouble. Wolfram Alpha is giving me different answers and I'm having difficulty ...
0
votes
2answers
56 views

How to find the value of this summation?

How to solve this summation? $$S=\displaystyle\sum_{n=1}^{\infty}\dfrac{(-1)^n}{n4^{4n+1}}$$ I tried it to convert it into a definite integral but wasn't successful. Help will be appreciated.
0
votes
0answers
33 views

find an adjoint operator for operator N [closed]

I read this in a paper how to find its adjoint?
1
vote
1answer
46 views

Prove that $\lim_{x \to -1^-}\frac{5}{(x+1)^3} = -\infty $ using the $\delta M$ definition of infinite limits

I am posting this for you guys to let me know whether it's wrong and/or give me any advice regarding the proof. Thank you. Given $ M < 0 $ we need $\delta > 0$ such that $ -1 -\delta< x ...
0
votes
1answer
34 views

By looking at a graph of a function, how do I find the maxima/minima of its curvature function. [closed]

All Ideas are appreciated. I could think of some intuitive ideas, but could not back them by solid clean reasoning. thanks, I will post If i find anything
3
votes
3answers
99 views

Trouble evaluating the sum involving logarithm

I was trying to solve this problem: Closed form for $\int_0^1\log\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\mathrm dx$ In the procedure I followed, I came across the following sum: ...
-3
votes
2answers
46 views

Left and right hand limit [closed]

I need to find $$\lim_{x\to 1} \{x\}$$ where $\{x\}$ is fractional part of $x$. I know how to find limit for polynomials. Here, I don't know how to find left and right hand limit of $1$. Could you ...
1
vote
2answers
49 views

Inequality with mean value theorem with convex function [closed]

Let $f:[a,b] \rightarrow \mathbb R$ be a convex function. Prove that $$ f \left( \dfrac{a+b}{2} \right) \leq \dfrac{1}{b-a} \int_a^b f \leq \dfrac{f(a)+f(b)}{2}.$$ Any hint on how to approach this?
5
votes
4answers
143 views

How find the maximum of the $x^3_{1}+x^3_{2}+x^3_{3}-x_{1}x_{2}x_{3}$

Let $$0\le x_{i}\le i,\, i=1,2,3$$ be real numbers. Find the maximum of the expression $$x^3_{1}+x^3_{2}+x^3_{3}-x_{1}x_{2}x_{3}$$ My idea: I guess $$x^3_{1}+x^3_{2}+x^3_{3}-x_{1}x_{2}x_{3}\le ...
1
vote
1answer
48 views

A definite integral involving $\exp(-a\cosh x)$

Wolfram alpha can not integrate the following integral: $$\int_0^\infty x^n \exp(-a\cosh x)dx$$ Thanks- Mike
0
votes
1answer
27 views

Show this integral operator is compact for various values of $\alpha$

I am having some problems evaluating a multivariable integral. This question is features in Stakgold's book Green's functions and boundary value problems. page 359. Consider the kernel for $a\leq ...
2
votes
1answer
19 views

differential in integration cancelled but variable endpoint is changed

In kinetic energy equation in wiki. I have difficulty problem how endpoint in integral change from t to v. This doesn't look like it is using substitution method. ...
0
votes
1answer
45 views

Find $\frac{dy}{dx}$ of $y=\sqrt{u}$

Find $\dfrac{dy}{dx}$ of $y=\sqrt{u}$, $u=7-x^2$ This is on my homework and I don't know what to do exactly. Steps would be helpful!
7
votes
3answers
296 views

Sum related to zeta function

I was trying to evaluate the following sum: $$\sum_{k=0}^{\infty} \frac{1}{(3k+1)^3}$$ W|A gives a nice closed form but I have zero idea about the steps involved to evaluate the sum. How to approach ...
1
vote
1answer
49 views

My understanding on Fubini thereom

Fubini thm says: but In fact the only way to compute the multiple integral is term by term so we always use the following: Is there any exception?
1
vote
1answer
18 views

parametric form of vector

I am having trouble understanding what the question is asking at this point, I have solved the first parts correctly and was wondering if I could get help as to how to solve x=x(t)
2
votes
1answer
79 views

Homework on basic inequalities.

Let $a_j$ be a sequence of positive reals. Show that (a) $\left(\sum_{j=1}^\infty a_j\right)^\theta \le \sum_{j=1}^\infty a_j^\theta$ for any $0\le\theta\le1$. (b) $\sum_{j=1}^\infty a_j^\theta \le ...
0
votes
1answer
26 views

How to compute the area of this set in the plane?

Let $f$ be a non-negative function which is defined, bounded, and integrable on a closed interval $[a,b]$, and let $$ S \colon= \{\ (x,y) \ | \ a \leq x \leq b, \ 0 \leq y < f(x) \ \}. $$ Then is ...
0
votes
2answers
89 views
+50

Compute $\int_{a-b}^{a+b} \chi_{(-t,t)}(y)dt$

Compute $\int_{a-b}^{a+b} \chi_{(-t,t)}(y)dt$. So if I create a number line marking a-b and a+b. If that the integral above has 5 different answers depending on where (-t,t) is located on the number ...
-2
votes
1answer
25 views

unit vector orthagonal to two vectors

How would I calculate the unit vectors here? What is the appropriate way to solve this? I do not completely understand which approach to take. Thanks!
2
votes
1answer
23 views

Smoothing corners of a function

I start with a the function (in blue) which has corners. What would be an efficient way to smooth out the corners(like I attempt below) so that the function becomes differentiable? By efficient, I ...
4
votes
2answers
73 views

Binomial expansion inequality

In a paper I am reading, there is a step that seems to come from the following inequality: $$(1+x)^\alpha \le 1+2^\alpha x,$$ where $0<x<1$. (Also, $3\le \alpha \le 9/2$ in the context of the ...
0
votes
3answers
33 views

magnitude of two vectors

How would I find the crossproduct if all I have is the point values?
0
votes
1answer
26 views

Derivative from Graph

Hi, I am trying to study a bit ahead for my calculus class next year and I came across this question. I was wondering how to find the derivative of the graph without the function. I figured I could ...
1
vote
8answers
114 views

Find the points on the parabola $\,y=x^2$ that are closest to the point $\,(0,8)$

Find the points on the parabola $y=x^2$ that are closest to the point $(0,8)$. Having trouble figuring out where to start.
0
votes
2answers
19 views

plane which passes through three points

I am confused as to how to answer this question because I don't understand how to incorporate the 12 into my answer. Any suggestions?
6
votes
5answers
179 views

An improper integral : $\int_{0}^\infty {\ln(a^2+x^2)\over{b^2+x^2}}dx$

How to evaluate the following improper integral:$$\int_{0}^\infty {\ln(a^2+x^2)\over{b^2+x^2}}dx,$$ where $a,b>0$. I tried to suppose $$f(a)=\int_0^\infty {\ln(a^2+x^2)\over{b^2+x^2}}dx,$$ based ...
1
vote
1answer
37 views

Show that $\lim_{a\to 0^+} \int \frac{e^{-|x|/a}}{2a}f(x)dx=f(0)$

I'm trying to show $\displaystyle \lim_{a\to 0^+} \int \frac{e^{-|x|/a}}{2a}f(x)dx=f(0)$ where $f$ is continuous with compact support. I've already shown that for any $a>0, ...
1
vote
2answers
57 views

Find a power series for function

I'm having some difficulty with this problem even while noting the hint. I expressed the function as $(1/2)\frac{1}{1-(-3x/2)}$ and then thought I would work with $1/2$ of the infinite sum of ...
0
votes
1answer
76 views

Surface Integral calc 3

I am having difficulty setting up this problem. I know the bounds must be 0 to pi/2 for both theta and phi but I am unsure as to how to calculate the integrand. I know it must be the double integral ...
0
votes
2answers
21 views

Determine the point(s), (if any), at which the graph of the function has a horizontal tangent.

Determine the point(s), (if any), at which the graph of the function has a horizontal tangent. $y(x)= x^4-500x+2$ So I know the first thing to do is find the derivative which is: $y'(x) = 4x^3-500$ ...
3
votes
2answers
50 views

The right procedure on difficult related rates problems

I'm pretty sure the sample problems my teacher gives to us violate some article of the Geneva convention. I'm in talks with my embassy about that, but in the mean time maybe you guys could look over ...
0
votes
1answer
44 views

How do I go about factoring this polynomial?

I am horrid at factoring and I have to find the inflection points of $ f(x)=x^2(x − 3)^3$. So I to find the inflection points I need to set $f'$ equal to $0$ So I have ...
1
vote
1answer
37 views

Why does secant method converge

Assume $f$ is continuous and twice differentiable on $[a,b]$ such that $f'(x)>0$ and $f''(x)>0$, $x \in [a,b]$. If $f(b)>0$ and $f(a)<0$ and I choose $x_0=a$,why are we gauraunteed ...
2
votes
4answers
66 views

Examples of Proofs of Limits

Are there any good examples of the proof for limits? Every proof I see has examples like L=4 and things like that, but I'm trying to find something that shows the idea of limits is valid. Thanks!
1
vote
3answers
29 views

Related Rates Ladder Problem with Angles

The problem is as follows: A 13-foot ladder leans against the side of a building, forming an angle θ with the ground. Given that the foot of the ladder is being pulled away from the building at the ...
1
vote
3answers
106 views

Evaluate $\int \frac{\tan^3x+\tan x}{\tan^3x+3 \tan^2x+2 \tan x+6} dx$

$$\int \frac{\tan^3x+\tan x}{\tan^3x+3 \tan^2x+2 \tan x+6} dx$$ My approaches so far has been using substitution with $\tan x = t$ and $\tan \frac x2 = t$ but the calculations has been harder than I ...