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

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0
votes
3answers
84 views

Show that: $\int_{0}^{\infty}{x^2e^{-x^2}}{dx} = \frac{1}{2}\int_{0}^{\infty}{e^{-x^2}}{dx}$

I am fully uncertain of how to approach this problem: Show that: $$\int_{0}^{\infty}{x^2e^{-x^2}}{dx} = \frac{1}{2}\int_{0}^{\infty}{e^{-x^2}}{dx}$$ We've just completed the section on improper ...
0
votes
2answers
69 views

How to calculate: $\lim \limits_{x \to 0^+} x \int_{x}^{1} $ $\frac{cost}{t^{\alpha}}dt$

How can I calculate: $\lim \limits_{x \to 0^+} x \int_{x}^{1} $ $\frac{cost}{t^{\alpha}}dt$ for rach $\alpha >0$, I tried to think about this as an improper integral and substituting ...
2
votes
2answers
96 views

Showing $s_n = \left(\frac{1}{2}\right)(s_{n-1} + s_{n-2})$ is Cauchy.

Let $(s_n)$ be a sequence defined as $s_1 = 1, s_2 = 2$ , and $s_n = \left(\frac{1}{2}\right)(s_{n-1} + s_{n-2})$. Prove that $(s_n)$ is Cauchy. I can see how it is convergent and Cauchy but not sure ...
1
vote
1answer
81 views

How do I show convergence of this sequence?

Given a sequence $\{a_n\}$ of positive real numbers such that $\sum\limits_{n=1}^\infty a_n<\infty$. Suppose that there exists $k\in \Bbb N$ such that $a_{n+k}\leq a_n, \,\,\,\forall n.$ Question: ...
2
votes
4answers
110 views

How to find $\int \frac{2}{e ^{-x}+1} dx$

How to integrate $\int \frac{2}{e ^{-x}+1}dx$ I tried by substitution, I required by substitution, thanks.
3
votes
1answer
84 views

How to solve this limit using laurent series?

$$\lim_{x\to\infty}\left(\left(\frac{x^2+5}{x+5}\right)^{3.7}+\left(\frac{x^3+5}{x+5}\right)^{1.6}\right)^{20/37}-\left(\left(x-5\right)^{3.7}+(x^2-5x+25)^{1.6}\right)^{20/37}=60$$ It is possible to ...
0
votes
2answers
40 views

Maximun vertical distance beween$ \frac{1}{\sqrt{x}} \:and\:\frac{1}{x\sqrt{x}}$

To calculate the maximun vertical distance beween$ \frac{1}{\sqrt{x}} \:and\:\frac{1}{x\sqrt{x}}$ at a point x=a, where a>1 I proceeded as follows: ...
3
votes
1answer
89 views

Can I calculate the scalar potential from the electric field using: $\phi = \int \nabla \phi = - \int \vec{E}$

If I have a relation between the electric field and the radius, can I calculate the relation between the scalar potential and the radius using: $\phi = \int \nabla \phi = - \int \vec{E}$? $$\nabla ...
2
votes
1answer
275 views

The limit of a product of functions equals the product of the limits: Is this proof rigorous?

All the proofs I've seen so for the limit of a product of functions equaling the product of the limits are based on the following: Let $f$ and $g$ be real or complex functions having the limits ...
6
votes
3answers
1k views

Finding the limit of $(1-\cos x)/x^2$

$$\lim _{x \to 0}{1-\cos x\over x^2}={2\sin^2\left(\frac{x}{2}\right)\over x^2}={\frac{2}{x^2}\cdot {\sin^2\left(\frac{x}{2}\right)\over \left(\frac{x}{2}\right)^2}}\cdot\left(\frac{x}{2}\right)^2$$ ...
1
vote
1answer
43 views

Simple question on maximizing a family of concave functions

Given $x \in [0,1]$, let $f_k(x)$ be concave function in $x$ for all $k= 0,1,...,N$. Is the following two maximization formulations equivalent? $$ \max_{0 \le k \le N} f_k(x) =?= \max \{f_0(x), ...
5
votes
2answers
160 views

Proving $e^x \sin x$ is not uniformly continuous on $[0,\infty)$

I have to prove that $e^x \sin x$ is not uniformly continuous on $[0,\infty)$ using the mean value theorem. I proved that $\sin x + \cos x \geq 1$ for all $x\in [2 \pi k, 2\pi k + \pi/2], ...
1
vote
1answer
104 views

How to evaluate Area of $B:= \{(x,y,z) \in A | z \le 1 \}$ with $A:=\{(x,y,z) \in \mathbb{R}^3 | x^2 + y^2 = z\} = \frac{\pi}{6}(5 \sqrt{5}-1) $?

I have following problem: Let $$A:=\{(x,y,z) \in \mathbb{R}^3 | x^2 + y^2 = z\} \\ B:= \{(x,y,z) \in A | z \le 1 \}. $$ Compute the area $\mu_2(B) $. First, I thought $\mu_2(B) $ would just be the ...
1
vote
2answers
98 views

does epsilon-delta definition of limit presuppose that the function is defined everywhere at (x-delta,x+delta)?

definition of limit says that we can choose a delta etc... f(x)-Limit is smaller than epsilon. but the notation f(x) presuposses that f is defined at x. so does definition say we can find delta where ...
2
votes
1answer
68 views

Integrate $\int_0^1 \min(1, \sqrt{x^{-2}-1})dx$.

I'm searching for efficient methods to integrate the left side of the following equality: $$ \int_0^1 \min(1, \sqrt{x^{-2}-1})dx = \log(1+\sqrt{2}). $$ This is on page 105 of Probability and Random ...
0
votes
1answer
117 views

Ratio test, Root test, and Divergence test related.

(I) Ratio test: If the result is smaller than 1 then the sum is convergent, and if the sum is larger than 1 then the sum is divergent, and that got me thinking if negative infinity (smaller than 1) ...
1
vote
1answer
34 views

Substitution needed for calculating an integral

Can you find two functions: $\phi:(0,\infty) \longrightarrow (0,\infty) $, $f:(0,\infty)\longrightarrow \mathbb{R}$, with $\phi$ differentiable, such that $f(\phi(x))\phi'(x)=\frac{1}{x ...
2
votes
4answers
156 views

Prove that $\lim_{x\to 1}\left(4+x-3x^{3}\right)=2$.

I was inspired by this <link> on page 10-11, where it was shown that $$\lim_{x\to 1}\left(4+x-3x^{3}\right)=2.$$ I wonder if it's possible to show this way without choosing a good value of ...
1
vote
4answers
42 views

Max/Min problem

Find the maximum area of a rectangle DACB where C and B are two points on the graph of $y=\dfrac{8}{1+x^2}$ and A and D are the two corresponding points on the x axis. I've been at this for ...
2
votes
1answer
59 views

Solve $\exp(x)(5-x)=5$ by hand

Is there a way to solve this equation by hand? $\exp(x)(5-x)=5$ Solutions: $x_1=0$ $x_2= 4.96511$
1
vote
1answer
32 views

Green's Theorem for the vector field $v=\begin{bmatrix} 1&xy \end{bmatrix}^{T}$

Using Green's Theorem for the vector field $v=\begin{bmatrix} 1&xy \end{bmatrix}^{T}$ over the upper half of the circle with radius 1 and center $(3,3)$ which is split in two by the line y=x. ...
3
votes
4answers
109 views

Derivative of$ \sqrt{x^2}$

I found the Derivative of $\sqrt{x^2}$ to be 1 using simplifications and the power rule, but when I checked the answer, it was in fact $=\frac{x}{\left|x\right|}$ and not $=1$. What could have been ...
1
vote
5answers
76 views

$f(x)=2x-e^x<0, \forall x \in \mathbb{R}$

The question is quite simple, but I'm finding some trouble doing it... Prove that the function $f(x)=2x-e^x$ is negative, i.e., $f(x)<0, \forall x \in \mathbb{R}$. Thanks for the help.
0
votes
1answer
32 views

function with 2 variables

we have the next function: $$f(x,y)=\left\{\begin{matrix} \frac{xe^{-y^{2}}}{y} \ y \neq 0 \\ \ c \ \ \ \ y=0 \end{matrix}\right.$$ Is there c that f(x, y) is continuous function because of him?
0
votes
1answer
57 views

Proving that the derivative of an even function is odd: intermediate steps.

I was shown a proof that when an even function is differentiable, its derivative is an odd function. It was proven as: $$\begin{align} f'(-x) &= \lim\limits_{h\to 0} \frac{f(-x+h) -f(-x)}{h}\\ ...
0
votes
1answer
82 views

Calculating $\int t^2/dt $? [closed]

I was solving $x\cdot(\log x)^2$. I was able to do this with by-parts method, but I wanted to know if I could solve this using substitution method as well, by assuming $\log x$ as t, thus $x = 1/dt$. ...
1
vote
1answer
34 views

Finding the radius of convergence of power series

$$f:(-\rho,\rho)\rightarrow R:x+(\sum_{k=3}^{\infty}\frac{(-1)^k*x^{(2k)}}{2k(2k-1)}$$ I tried using the ratio test but a friend of mine said that it only accounts to even numbers and not the odd ...
0
votes
1answer
239 views

Find the distance traveled along a straight line with velocity equation given.

I have a fairly simple question which I just can not figure out how to solve - this is the question: A particle travels along a straight line with its velocity at time 't' seconds given by 'v' m/s ...
2
votes
0answers
81 views

Finding a weight function with a specific property

I am looking for a (smooth, quickly decaying) function $w : [0,\infty) \rightarrow \mathbb{R}$ such that $$w(t) \cdot \int_{0}^{t} \frac{1}{w(2x)} dx $$ is absolutely integrable on $[0,\infty)$. ...
0
votes
3answers
105 views

Derive $y=x\cos^3(5x+1)$

How can I derive the function $y=x\cos^3(5x+1)$ ? I obtain this derivative $$y=\cos^3(5x+1)+\left(3\cos^2(5x+1)\cdot2\cos(5x+1)\cdot\sin(5x+1)\cdot5\right)x$$ Is it wrong? My book gave me another ...
1
vote
1answer
49 views

Derive $\cos x(\ln x-x)^2$

I have a doubt in deriving this function $\cos x(\ln x-x)^2$, deriving following the rules I obtain $$-\sin x(\ln x-x)^2+\cos x\left[2(\ln x-x)\left(\frac{1}{x}-1\right)\right]$$ Is it correct? I ...
0
votes
1answer
113 views

Limit of $\frac{1}{x} - \frac{1}{\sin{(x)}}$ [duplicate]

Prove, without using l'Hôpital's Rule, that $\lim\limits_{x \to 0}{\dfrac{1}{x} - \dfrac{1}{\sin{(x)}}} = 0$. I proved that there exists a $s >0$ such that $\forall x \in (-s,s)$ $\Rightarrow$ ...
2
votes
0answers
101 views

Length between two circles intersection area?

How do I know the (smallest) length of the intersection area between two circles of different sizes? We know both circles radii and the overlapping area. ...
3
votes
2answers
72 views

Calculating the radius of convergence of a series.

Let $d_n$ denote the number of divisors of $n^{50}$ then determine the radius of convergence of the series $\sum\limits_{n=1}^{\infty}d_nx^n$. So obviously we need to calculate the limit of ...
1
vote
2answers
127 views

Describe the curves followed by the raindrops in the following question

Problem: It is raining and rainwater is running off an ellipsoidal dome with equation $4x^2 + y^2 + 4z^2 = 16$, where z ≥ 0. Given that gravity will cause the raindrops to slide down the dome as ...
5
votes
3answers
166 views

Antiderivative of $\frac{1}{\ln(x)}$?

I was looking on wikipedia, and found that the following expression cannot be expressed in terms of elementary functions: $$\int\frac{1}{\ln(x)}\text{d}x$$ Although the function looks simple, why is ...
0
votes
0answers
35 views

Legendre Polynomial change the argument Cos(x ) and rodruigez formula

making the change x=Cos(x) in rodriguez formula of legendre polynomial we get $$P_k(\cos (x))=\frac{(-1)^k \sin ^{2 k}(x) \left(\frac{d}{d \cos (x)}\right)^k}{2^k k!}$$ how it is posible to make the ...
-1
votes
1answer
52 views

Finding minimum and maximum of 2-variable function in a closed area

I have this function: $f(x,y)=2x^3+3x^2y-3y$ I need to find the absolute minimum and maximum points that are within the triangle that is set by: $y=\frac{1}{3}x, x=1, y=0$ I started like this: ...
2
votes
4answers
381 views

Using L'Hospital's Rule to evaluate limit to infinity

I'm given this problem and I'm not sure how to solve it. I was only ever given one example in class on using L'Hospital's rule like this, but it is very different from this particular problem. Can ...
0
votes
1answer
12 views

Calculating a term using a multilinear map

How to calculate the value of the term $\Delta u:=u_{xx}+u_{yy}+u_{zz}=\large\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}+\frac{\partial^2u}{\partial z^2}$ for the function $u$ ...
3
votes
2answers
128 views

Prove this limit $\lim \limits_{x\to\infty}f(x)=0$

I have this problem in real analysis. I think it needs integral factor or knowledge of ODE to prove, but not sure how to it. Here is the question: Let $f$ be a real valued continuous function on ...
8
votes
5answers
169 views

Why does $\int_0^{2\pi} (1+2\cos(x))/(5+4\cos(x))\,dx$ vanish?

The standard substitution $y=\tan(x/2)$ shows that $$ \int_0^{2\pi} \frac{1+2\cos(x)}{5+4\cos(x)}\,dx = 0. $$ What is the "real explanation" for this fact? My guess is that the "book proof" ...
2
votes
1answer
106 views

Let $a_n>0$ for $n \geq 1$ and let series: $\sum_{n=1}^{\infty}a_n$ diverge. Let $S_n=a_1+a_2+…+a_n > 1$ for $n \geq 1$

Prove that the series: $$\sum_{n=1}^{\infty}\frac{a_{n+1}}{S_n \ln S_n}$$ diverges and the series : $$\sum_{n=1}^{\infty}\frac{a_{n}}{S_n \ln^2 S_n}$$ converges. (Using the famous criteria I ...
2
votes
2answers
51 views

Evaluate $\Im \bigg (\dfrac{1}{100\times 2^{100}}(e^{2\iota x} -1)^{100}\bigg )$

I was trying to evaluate $$\int \sin(101x)\sin^{99}(x) dx$$ I managed to work it out to $$\Im \bigg (\dfrac{1}{100\times 2^{100}}(e^{2\iota x} -1)^{100}\bigg )$$ However, I got stuck over ...
0
votes
1answer
41 views

Integral and radius of convergence

I have never met such a problem with integral and expansion,how should I approach it? Using Maclaurin series to find the radius of convergence of $$f(x)=\int_{0}^{x} \ln{(t+\sqrt{t^2+1})} \, dt. $$
0
votes
1answer
82 views

finding the centroid of composite areas.

I've been staring at this problem for hours. It seems so straight forward. I'm asked to find the centroid of this figure shown. The y coordinate to be exaxt. The book gets 135mm but I don't ...
2
votes
2answers
128 views

A book to prepare for analysis?

I am taking Real Analysis in the Fall and the Professor recommended Spivak's Calculus to prepare for the course. Unfortunately, someone was faster than me and already borrowed the book from the ...
0
votes
1answer
66 views

Describing the bounds (upper and/or lower) sequence

For ${(-1/2)^n}$, the sequence will be $\{-1/2, 1/4, -1/8, 1/16, ...\}$ and I determined the the lower and upper bounds to be -1/2 and 1/4, respectively, by plotting a couple of points. But Is there a ...
0
votes
4answers
85 views

The sum of the series $\sum_{n=0}^\infty\left(\frac{4n+3}{5^n}\right)$

What is the sum of the series $\sum_{n=0}^\infty\left(\frac{4n+3}{5^n}\right)$ ? I got that the series converges and the sum seems to be $5$. When trying to explicitly get the sum, I tried to find the ...
0
votes
2answers
57 views

Radius of convergence of $\sum_{n=1}^{\infty} (3^n +(-5)^n) x^{7n}$

What is the radius of convergence of the power series $\sum_{n=1}^{\infty} (3^n +(-5)^n) x^{7n}$? I tried to find the limit of $\dfrac{a_{n+1}}{a_n}$ and I found it is $5$. Is that true, and does ...