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

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2
votes
1answer
64 views

Evaluate if $f_{_n}$ converge uniformly or not

We have $f_n:[1,2]\to \mathbb{R},\:f_n(x)=\frac{x^n}{x^n+1}$ and we have to see if the convergence is uniform or not. From what I understand we need to prove that $\lim _{n\to \infty } ...
0
votes
0answers
17 views

Fining the angular bounds of a triple integral function

This problem requires the taking of a triple integral over a region. I believe it's most useful to convert to cylindrical coordinates, which I did. However, I could not find the theta bounds due to ...
1
vote
3answers
94 views

Evaluate the definite integral $ \int_{\pi/6}^{\pi/2} \frac{\cos(x)}{\sin^{5/7}(x)}\, dx$

Evaluate the integral: $\displaystyle \int_{\pi/6}^{\pi/2} \frac{\cos(x)}{\sin^{5/7}(x)}\, dx$ (using substitution) Here's my attempt at solution: u = $\sin^5(x)$ $du = 5\sin^4(x) \cdot \cos(x) ...
0
votes
2answers
75 views

Evaluate the indefinite integral $\int \frac{1}{x^2} \sin\left(\frac{6}{x}\right) \cos\left(\frac{6}{x}\right) \, dx $

Evaluate the indefinite integral: $\displaystyle \int \frac{1}{x^2} \sin\left(\frac{6}{x}\right) \cos\left(\frac{6}{x}\right) \, dx $ (using substitution) The answer is: $\frac {1}{24} ...
1
vote
1answer
26 views

Green's theorem in divergence form and its line integral?

$$ \int_C F \times da $$ $$ k\iint_R \operatorname{div} F \ dx \, dy $$ Hi Let $F$ be two-dimensional vector field. State a definition for the vector-valued line integral so that your definition ...
4
votes
0answers
50 views

Trigonometric integral of $f(x)=(x^2)(\sin(x^2))$. [duplicate]

I've tried with the chain rule and $u$-subtitution ($u=\sqrt{x}$) but I get nothing. Can you help me please? $$\int (sqrt{x})(\sin(x)) \ dx$$
2
votes
3answers
99 views

Evaluate $\int \frac{\sec(11 x) \tan(11 x)}{\sqrt{\sec(11 x)}} \, dx $

Evaluate the indefinite integral: $\displaystyle \int \frac{\sec(11 x) \tan(11 x)}{\sqrt{\sec(11 x)}} \, dx $ (using substitution) The answer is: $\frac{2}{11} \sqrt{sec(11 x)} + C$ I don't get ...
1
vote
1answer
26 views

If I take partial derivatives of function using chain rule, how do I remove variable from the partials in the equation?

I have the function $$f(tx,ty)$$ and I want to take the partial derivative of this with respect to $t$. So set $x'=xt$ and $y'=yt$. I applied chain rule and got $$\frac{\partial f}{\partial t} = ...
2
votes
3answers
94 views

Area of a parallelogram with three dimensional vectors

There is a parallelogram that has the vertices 0, a, b, and a+b, all of which are three dimensional vectors. a = \begin{pmatrix} 2 \\ -6 \\ 5 \end{pmatrix}b = \begin{pmatrix} -1 \\ -2 \\ 0 ...
1
vote
0answers
27 views

3D Fourier Transform - Angle between $\mathbf{k}$ and $\mathbf{r}$

The definition of the Fourier transform for three dimensions is $$\mathcal{F}[f(\mathbf{r})](\mathbf{k})=\int e^{-i\mathbf{k}\cdot \mathbf{r}}f(\mathbf{r})\,d^3 r$$ If the function $f(\mathbf{r})$ ...
1
vote
1answer
37 views

$K$ is a region in $\mathbb{R}^2$ where the area is $5$

Say that $K$ is a region in $\mathbb{R}^2$ where the area is $5$. Let B = \begin{pmatrix} 3 & 8 \\ 4 & 6 \end{pmatrix} Find the area of the region B$K$. Any starting hints? Is it possible ...
3
votes
2answers
101 views

Can someone give me a counterexample to understand why this definition of limit is wrong?

Could someone give me a counterexample to understand why this definition of $\lim_ {x\to a} f(x) = L$, does not work? $\forall \delta>0 \exists \varepsilon>0$ such that, if ...
-1
votes
1answer
47 views

Does the following converges?

Does the following converges? $$\int_{-\infty}^{+\infty}\frac{\sin x}{1+\cos^2 x}dx$$ Thanks ahead.
1
vote
2answers
31 views

Set up the integral for the volume found by rotating the region bounded by the curves $x=y+1$ and $x=-y^2+2y+3$ about the line $x=-1$

Set up the integral for the volume found by rotating the region bounded by the curves $x=y+1$ and $x=-y^2+2y+3$ about the line $x=-1$. Thanks! I don't know how to solve it about the line x=-1. I could ...
0
votes
1answer
55 views

Quotient rule for derivatives..am I making this to complicated

This is a straight forward question.. When I have something like 10/x (i.e basically whenever the numerator is just a number with no variables) and I need to take the derivative I go through the ...
0
votes
1answer
37 views

area of a bounded region

Find the area of the region bounded by $$f(x)=x^3+x^3+1$$ and $$g(x)=x^2 + x-1$$ I do know how to get the area of a bounded region, my problem now is that when I tried getting the graph of this region ...
0
votes
1answer
23 views

When is a series of sums the sum of the series?

In general, if $\Sigma_n (a_n+b_n)$ converges, then it may not be that $\Sigma_n a_n$ and $\Sigma_n b_n$ converge; for example, consider $\Sigma_n (1/n-1/n)$. If instead we know $\Sigma_n a_n$ and ...
0
votes
1answer
26 views

Integral Car Traffic Problem for $f(t) = 50*t*sin(\sqrt{t})$

So I have a question for this traffic function $f(t) =50t*sin(\sqrt{t})$ where f(t) is the rate at which cars pass through an intersection from noon(t=0) until 5pm (t=5). The question is asking to ...
2
votes
2answers
84 views

How do i evaluate the following integral?

Hi I was wondering if someone can help me evaluate the following integral. Show that if $-1 < x < 1$, then $$\int_{0}^{\pi} \frac{\log{(1+x\cos{y})}}{\cos{y}}dy= \pi \arcsin{x} $$ thank you ...
2
votes
2answers
86 views

Differential Equation: $\text dy/\text dx = x/y$

Consider the differential equation $\text dy/\text dx = x/y$ a) Write an equation for the line tangent to the solution curve that passes through the point $(1,2)$ Would it be correct to just use ...
2
votes
0answers
63 views

On a problem about Rolle's theorem

Let $f:[1,3]\to\mathbb R$ be a continuous function such that $\int_1^2 f(x)dx=2$, and $\int_1^3 f(x)dx=3$, then there exists a real number $c\in(2,3)$ such that $$ \int_1^c f(x)dx=cf(c) $$ Note. I ...
2
votes
3answers
304 views

How to express sum as triple summation

I am trying to express the following sequences as summations: $$ 1+2^2+3^2+4^4+5^4+6^4+7^4 $$ and $$ 1+(2+3)^2 + (4+5+6+7)^4 $$ as summations. I think they will likely be triple summations, so ...
0
votes
1answer
54 views

How do we prove $\int \frac{\ln(1+x)}{x}dx = -\sum_{k=1}^{\infty}\frac{(-x)^k}{k^2}$? [duplicate]

After working on the integral $\int_{0}^{1} \frac{\ln(1+x)}{x}dx$ for a couple of hours, I became convinced its antiderivative was not elementary. So I looked it up on Wolfram Alpha, and it found that ...
3
votes
2answers
67 views

Hint on how to find $\int \frac{x^2}{1+x^2}dx$

I am almost sure that this would have been asked before, but how can one find $$ \int \frac{x^2}{1+x^2} dx? $$ If I had a $x^2 - 1$ in the denominator, then I could factor into $(x-1)(x+1)$ and use ...
2
votes
0answers
128 views

Double Integral of an Exponential Function with an Absolute Value in the Numerator of the Exponent

This is a question related to statistics, but my major concern relates to the setup and evaluation of integrals. So I decided this question was better suited for Mathematics Exchange than CV. I know ...
4
votes
3answers
153 views

Evaluate $\lim _{n\to \infty }\int_1^2\:\frac{x^n}{x^n+1}dx$

We have $$I_n=\int _1^2\:\frac{x^n}{x^n+1}dx$$ and we need to find $\lim _{n\to \infty }I_n$. Have any ideea how we can evaluate this limit?
0
votes
1answer
39 views

sum of a geometric series

check if the series converges and if so what is the sum $$\sum_{n=1}^\infty \ln(6^{\frac{1}{4^{n}}})$$ By the ratio test: $$\frac{\frac{1}{4^{n+1}}\cdot \ln{6}}{\frac{1}{4^{n}}\cdot ...
1
vote
5answers
30 views

How do I find the radius and interval of convergence of $\sum_{n=1}^\infty {(-1)^n(x+2)^n \over n} $

$$\sum_{n=1}^\infty {(-1)^n(x+2)^n \over n} $$ I used the ratio test to test for absolute convergence, but I'm sort of stuck on: $$n(x+2) \over n+1$$
4
votes
0answers
57 views

How to prove an extremum existence in problems, regarding calculus of variations

Let's consider a functional $S(y)=\int_{a}^{b}{f(x, y, y') \cdot dx}$. It's known that if the function that attains minumum or maximum to $y(x)$ does exists, then it can be got from the Euler-Lagrange ...
-1
votes
1answer
84 views

Check whether the integrand is continuous when evaluating improper integrals

In order to evaluate improper integrals, I need to know whether the integrand is continuous between the limits of the integral. For the lower and upper limits, I believe you find out if it's ...
2
votes
4answers
70 views

sum of a telescoping series

calculate the following $$\sum_{n=1}^{\infty} \ln \left ( \frac{\left (\frac{n+3}{n+1}\right ) ^{n+1}}{\left (\frac{n+2}{n}\right )^n} \right )$$ I have manage to written it as ...
0
votes
1answer
60 views

Find the integral: $\int \frac{( x-6)^2}{x^4}\mathrm{d} x$

Find the integral: $\int \frac{(x-6)^2}{x^4}\mathrm{d} x$ I have so far $\int (u)^2(u-6x)^{-4}\mathrm{d} x$ $u= x-6$ and $du=dx$ and $u-6=x$ Am I on the right track?
1
vote
3answers
58 views

Finding the derivative of an inverse function.

Let $f(x) = (-x^2)/(x^2+1)$. If g(x) is the inverse function of f(x) and f(1)=-1/2, what us g'(-1/2)? Can someone explain how to do the above problem as I am not even sure where to start. Would I ...
-1
votes
1answer
3k views

Multiplying logarithms of different bases [closed]

How do you multiply the following logs... $$\log_5(n) * \log_2(n)$$
0
votes
2answers
190 views

A Sequence That has No Upper Bound But Does Not Tend To Infinity

Let $a_n$ be a sequence which has no upper bound. Give an counterexample sequence for the statement $$\lim_{n\to\infty} a_n=\infty$$ Any hint?
1
vote
2answers
58 views

Find the integral: $\int x^{7/2} sec^2(2+x^{9/2}) \mathrm{d}x$

Find the integral: $\int x^{7/2} sec^2(2+x^{9/2}) \mathrm{d}x$ Can I multiply and distribute the $ \ x^{7/2}\ $ and $ \ sec^2 \ $ together. What is the strategy to solve this problem.
1
vote
1answer
152 views

Evaluating the sum of a partial geometric sequence using Sigma notation

I have a worksheet from my instructor with this problem on it, but the solution he has given is different from what I got, and I don't know why. I'm not sure how to input the Greek letter sigma, but ...
0
votes
1answer
28 views

Darboux sums inequality with relation to Sup|f'(x)|

Assuming f is continuous on [a,b] and differential on (a,b) and assuming f ' is bounded on (a,b) ; denote k = sup(|f '|) prove that, for all P a partition of [a,b]: 0 ≤ U(f,P) - L(f,P) ≤ k(b-a)Δ(P) ...
6
votes
5answers
61 views

Prove that $f : [-1, 1] \rightarrow \mathbb{R}$, $x \mapsto x^2 + 3x + 2$ is strictly increasing.

Prove that $f : [-1, 1] \rightarrow \mathbb{R}$, $x \mapsto x^2 + 3x + 2$ is strictly increasing. I do not have use derivatives, so I decided to apply the definition of being a strictly ...
3
votes
1answer
85 views

Are there only a few 'universally convergent' Taylor Series?

The Taylor series for $\sin(x)$, centered at any point, converges for all $x$. The Taylor series for $e^{x}$ and $\cos(x)$ do as well. Thus, taking an algebraic function of these (without division) ...
4
votes
2answers
49 views

Roots of unity, where $\omega^3 = 1, \omega \neq 1$.

Say that $\omega^3 = 1$ and $\omega \neq 1$. Find the value of $(1 - \omega + \omega^2)(1 + \omega - \omega^2)$. I'm not very good at the roots of unity. May I have a couple of hints to get started? ...
1
vote
5answers
41 views

Proving convergence/divergence via the ratio test

Consider the series $$\sum\limits_{k=1}^\infty \frac{-3^k\cdot k!}{k^k}$$ Using the ratio test, the expression $\frac{|a_{k+1}|}{|a_k|}$ is calculated as: $$\frac{3^{k+1}\cdot ...
0
votes
4answers
68 views

Difference between $\nabla T$ and $\nabla \cdot E$

Why is $\nabla T = (\frac{\delta T}{\delta x},\frac{\delta T}{\delta y},\frac{\delta T}{\delta z})$, but $\nabla \cdot E \neq (\frac{\delta E}{\delta x},\frac{\delta E}{\delta y},\frac{\delta ...
0
votes
2answers
77 views

Partial ordering of functions

Let $X$ be the set of all real-valued functions $x$ on the interval $[0,1]$ and let $x \leq y$ mean that $x(t) \leq y(t)$ for all $t \in [0,1]$. Does it define a partial ordering/ total ordering? Does ...
2
votes
2answers
56 views

For a strictly increasing function $f$, prove that $f(x)=cx, x\in\mathbb Q\implies f(x)=cx,x\in\mathbb R$.

For a strictly increasing function $f:\mathbb R\to\mathbb R$, prove: If $f(x)=cx$ for $x\in\mathbb Q$, then $f(x)=cx$ for $x\in\mathbb R$. I found this statement on M.SE. How can I prove ...
0
votes
3answers
53 views

Integral of $\cosh^3(x)$

What is the integral of $\cosh^3(x)$? And how exactly can I calculate it? I've tried setting $\cosh^3(x)=(\frac{e^x+e^{-x}}{2})^3$ but all I get in the end is one long fraction.
1
vote
4answers
54 views

Find the integral: $\int_0^{1/2} x \sin(\pi x^2)\,dx$

Evaluate the integral $$\int_0^{1/2} x \sin(\pi x^2)\,dx$$ I have: $-\cos (1/4) -1$.
1
vote
1answer
50 views

How calculate: $\lim \limits_{(x,y) \to (0,0)} \frac{ln(1+2x^2+4y^2)}{arctan(x^2+2y^2)} $?

How do I calculate: $\lim \limits_{(x,y) \to (0,0)} \frac{\ln(1+2x^2+4y^2)}{\arctan(x^2+2y^2)} $? Is there any certain path that is recommended to go through? I tried to use the inequality: $\ln(1+x) ...
2
votes
1answer
59 views

Integrate $\int_{0}^1 (1 + 4y^2)^{1/2} dy$ [duplicate]

$$\int_{0}^1 (1 + 4y^2)^{1/2} dy$$ So, how do I integrate this without the use of trigonometrical substitution? Can anybody give me a hint? Thank you!
2
votes
1answer
86 views

Trouble solving an integral

So I have been trying to solve this equation, The given answer is, I began by using substitution to change the integral. Substituting t back in where t is taken from 0 to infinity. ...