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

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2
votes
1answer
50 views

Substitution question $\int_{-1}^{1}\frac{1}{(1+x^2)^2}\,\mathrm{d}x\ne\int_{-1}^{1}\frac{-t^2}{(1+t^2)^2}\,\mathrm{d}t$

$$\int_{-1}^{1}\frac{1}{(1+x^2)^2}\,\mathrm{d}x=\frac{1}{2}+\frac{\pi}{4}$$ Using the substitution $x=\dfrac{1}{t}$ we get ...
2
votes
2answers
89 views

Integration with trigonometric substitution

I have been stuck trying to figure out an integration problem involving trigonometric substitution. $$ \int \frac{1}{x^2\sqrt{x^2 + 9}}dx $$ So I substituted $$ x = 3\tan\theta $$ $$ dx = ...
0
votes
1answer
65 views

Is it possible that $f$ is differentiable?

Let $$f=\left\{\begin{matrix} 0,0 \leq x <1\\ 1,x=1 \end{matrix}\right.$$ This function is not continuous at $x=1$.Is it possible that $f$ is differentiable?
1
vote
1answer
25 views

Animal jumping equally likely to the left and right.

We have an animal that starts at the point zero and jumps equally likely to the left(-1) and right(+1). After 2k jumps, where $k \in \mathbb{N}$ ,it arrives back again at the point $0$. The question ...
1
vote
1answer
50 views

what about $\lim\limits_{x\to0}-\frac{\sin x}x=$?

we all know that: $\lim\limits_{x\to0}\frac{\sin x}x=1$ so what is the negative $\lim\limits_{x\to0}-\frac{\sin x}x=$? i am trying to prove what about $\lim\limits_{x\to0}\frac{x^2\sin \frac ...
0
votes
1answer
56 views

LaPlace Transform of a step function

Consider the function $f(t)=\begin{cases} 0 \;\;\;\;\;\;\;\;\;\;\;t<\pi \\t-\pi \;\;\;\;\; \pi\leq t \leq2\pi \\ 0 \;\;\;\;\;\;\;\;\;\;\; t\leq 2\pi \end{cases}$ Find the laplace ...
1
vote
1answer
35 views

how to prove $\int_{0}^{l_1}f(x)\,dx \leq \int_{0}^{l_2}g(x)\,dx$?

if $$ \int_{0}^{l_1} f(x)x\,dx = \int_{0}^{l_2}g(x)x\,dx $$ where $l_1>l_2, f(x) \ge0, g(x)\ge 0 ,f(l_1)=g(l_2)=0,f(0)=g(0) $ is $\int_{0}^{l_1}f(x)\,dx \leq \int_{0}^{l_2}g(x)\,dx$ true? If it ...
0
votes
3answers
31 views

Convergence / absolute convergence of alternating infinite series?

How can it be shown that $\sum _{n=0}^{\infty }{\frac { \left( -1 \right) ^{n}}{\sqrt [3]{n+1}}}$ is convergent and /or absolute convergent?
5
votes
1answer
164 views

Perturbation theory PDEs

I have the solution of a PDE of the form: $$ \Delta \Psi(r,\theta, \phi) = k \Psi(r,\theta,\phi)$$ on a set $\mathbb{R}^3 \backslash B(0,R)$. Hence, the actual solution is known there! Regarding ...
0
votes
1answer
44 views

Determine the point $a$ such that the given function is invertible in the vicinity of $a$

I try to understand the Inverse Function theorem, but I'm still got some confusion. Here is the function. $f(x,y,z)=(x+e^y,y+e^z,z+e^x )$ It’s easy to see that $f$ is defined on $\mathbb R^3$. Now ...
1
vote
3answers
281 views

Find complicated Taylor Series

According to some software, the power series of the expression, $$\frac{1}{2} \sqrt{-1+\sqrt{1+8 x}}$$ around $x=0$ is $$\sqrt{x}-x^{3/2}+\mathcal{O}(x^{5/2}).$$ When I try to do it I find that I ...
2
votes
1answer
88 views

calculate the flux of the vector field

Let $f(x,y)=\frac{1}{x^2+y^2}$, calculate the flux of the vector field ${\rm Grad}\ (f)$ across the boundary of the region $S$ bounded between the circles centered at the origin and radius $1$ and $2$ ...
0
votes
1answer
31 views

Properties of a function $f=f(x,y)$ with maximum in $(x_{0},y_{0})$

I wanted to ask about the following statement. Let $f=f(x,y)$, $f$ is maximum at ($x_{0},y_{0}$). Show that: 1)$\frac{\partial f}{\partial x}(x_{0},y_{0})=0$ y $\frac{\partial f}{\partial ...
1
vote
1answer
41 views

Difference equation problem with trigonometric functions

Hello, I am trying to solve this. I believe (a) yields $y_{n+1}=arccot(0.5(cot(y_n)-tan(y_n)))$ For (b) onwards I am unsure how to tackle this question. I think that (b) might be a difference ...
1
vote
1answer
45 views

Addition corresponds to convolution and subtraction?

We know that if two random variables have proper densities, than the density of the sum of them is given by the convolution. But what can we say about the difference of two random variables? $X-Y$ ...
2
votes
4answers
66 views

limit with $\arctan$ and $\ln$

I have to find the limit and want ask about a hint: $$\lim_{x \to \infty} \frac{\frac{1}{2}\pi-\text{tan}^{-1}x}{\ln\left(1+\frac{1}{x^2}\right)}$$ I dont have idea what to do. Maple show me that ...
8
votes
3answers
339 views

Continuous decreasing function has a fixed point

Let $f$ be continuous and decreasing everywhere on $\mathbb{R}$. Show that: 1) $f$ has a unique fixed point 2) $f\circ f$ has either an infinite number of fixed points or an odd number of ...
1
vote
4answers
126 views

Find the limit as $x$ tends towards $\frac{\pi}{4}$

In looking at the corresponding graph and differentiating it after reducing it to a different form, I know the that limit is equal to $2$ but I am unsure as to how I can show this algebraically. Any ...
1
vote
1answer
80 views

derivative of a definite integral with base e

$$\frac{d}{dx} \int_3^{x^2} e^{t^3} dt$$ I can sorta figure out how to solve problems like this, if it was an indefinite integral...
1
vote
2answers
60 views

Check if series is convergent or divergent

$$\sum\limits_{n=1}^\infty\dfrac{n(2n+1)}{({3n^{7}+ln(n)})^{0.5}}$$ For this question, i would like to use comparison test. is it right to use $$\frac{2n^2}{3n^{3.5}}$$ for comparison?
0
votes
1answer
91 views

Alternating series test question

$$\sum\limits_{n=1}^\infty\dfrac{(-1)^{n}}{n^{2/n}}$$ I applied alternating series test. I tried to check if it fulfills the condition that it goes to zero. $\frac{2}{n}$ goes to $0$ . However $n^0$ ...
0
votes
2answers
42 views

Alternating series test

The series is as shown $$\sum_{n=1}^\infty\frac{(-1)^n\ln^2 n}{n^{0.5}}$$ By l'Hopital rule it proves that this goes to 0 and thus fulfill its first condition $$\frac{\ln^2 n}{n^{0.5}}$$ I ...
5
votes
1answer
67 views

Improper Integral $\int_0^2 \frac{1}{\sqrt{x}} \ \text{d}x$

I have to find the value of: $$\int_0^2 \dfrac{1}{\sqrt{x}} \ \text{d}x$$ Here is my work so far: $$\int_0^2 \dfrac{1}{\sqrt{x}} \ \text{d}x$$ $$=\int_0^2 x^{-1/2} \ \text{d}x$$ ...
3
votes
2answers
93 views

Domain and range of a function.

Find the domain and range of the function $$f(x)=\frac{1}{\sqrt{[\cos x]-[\sin x]}}$$ Where [] denotes the greatest integer function. I started as $[\cos x]-[\sin x]\gt0$ $\implies \cos ...
0
votes
1answer
28 views

Calculus question, $f(x,y)=\frac{xy}{x+y}$

Let $f(x,y)=\frac{xy}{x+y}$. Find a vector $u$ for which $D_u f(3,4)=0$. I found the gradient of the function $f(x,y)$. Is the next step solving the gradient of $f(3,4)$ multiplied by $u = 0$?
1
vote
1answer
29 views

prove that a function is not bounded

Prove that $$2x\sin{1\over x^2}-{2\over x}\cos{1\over x^2}$$ is not bounded when $x\to 0$ I tried to find two sequences that converges to $\infty$ and $-\infty$ but I can´t; I also derived the ...
-1
votes
1answer
49 views

Series using comparison test

The series is as shown $$\sum\limits_{n=1}^\infty\dfrac{\tan(1/n)}{n^{0.5}}$$ Using P-series, 0.5<1 therefore it is divergent $$\frac{1}{n^{0.5}}$$ It appears that i can't use P-series What test ...
0
votes
2answers
40 views

Evaluate ratio test for series

$$\sum_{n=1}^\infty\frac{(n!)^4}{(4n)!}$$ Applying ratio test $$\frac{\frac{(n!)^4}{(4n)!}}{\frac{((n+1)!)^4}{(4n+4)!}}$$ Simplification step 1 ...
0
votes
1answer
63 views

Proving series if it is convergent

I have a series below. i did test of divergence and it shows 0. next i proceed with the below comparision test using Comparison test -> $$\frac{1}{{n}^{3/2}}$$ Since $\frac{3}{2} > 1$, it ...
0
votes
1answer
40 views

convergence of series with positive terms

Hi! I am working on some calc2 convergence of series with positive terms online homework and I am having a particularly difficult time with this one problem. I solved the integral to be ...
0
votes
1answer
49 views

Directly integrable, separable and linear?

I have an assignment that asks whether an ODE is directly integrable, separable and/or linear. This is a modified version of the given question: $\frac{da}{db} = |a|,a(0) = 0$ $$-------------$$ ...
1
vote
1answer
37 views

Find a the minimum number $t$

Find the number $t$ such that area below $y=\frac{e^x}{x}$ for $x>0$ and above $[t,t+1]$ is a minimum. I looked at it as an optimization problem, and did the following ...
3
votes
1answer
76 views

What is a 'basic region' in calculus?

What is a 'basic region' in calculus? My friend told me that a basic region is defined as a connected set in which the total boundary consists of a finite number of continuous curve of the form ...
1
vote
0answers
43 views

'Proof' of backwards movement on a prolate cycloid?

I was looking at this web page: http://www.animations.physics.unsw.edu.au/jw/rolling.htm and had a hard time understanding the prolate cycloid bit at the bottom. Using the example from that website, ...
2
votes
2answers
81 views

Deriving limits when n goes to infinity

How do i evaluate the limit below? $$\lim_{n\to\infty}\frac{1}{1-x^{n+1}}=\left\{\begin{array}\\ &\text{if}&|x|<1\\ &\text{if}&|x|>1 \end{array}\right.$$
1
vote
0answers
70 views

How to apply the Green's Theorem in this case

Consider the region S bounded between the square with corners at the points (4,4),(-4,4),(-4,-4) and (4,-4) (oriented counterclockwise), and the circle of radius 2 centered at the origin (oriented ...
2
votes
1answer
88 views

Differentiate $P_{x_n}(z) = \prod_{i=1}^n\frac{1+z+z^2+…+z^{i-1}}{i}$ twice to calculate the variance of involutions.

Use the Probability Generating Function for Involutions: $P_{x_n}(z) = \prod_{i=1}^n\frac{1+z+z^2+...+z^{i-1}}{i}$ To Calculate the Variance of Involutions where: $Variance \space X_n = ...
4
votes
2answers
109 views

Evaluate $\int\frac{\sqrt {25 - x^2}}{ x^4}$

I'm pretty sure the method used is trig substitution. But I'm having trouble setting up and solving the problem.
2
votes
3answers
177 views

Compute this double integral

Compute the integral $$ \iint_{S}\sin\left(\,y - x \over x + y\,\right)\,{\rm d}A $$ where S is the trapezoidal region of the plane bounded by the lines $x + y = 1$, $x + y = 2$ and the coordiante ...
0
votes
2answers
73 views

Integrate $\int \frac{25 - x^2}{x^4} \operatorname d\!x$

I think we use trig substitution for this. But I'm not sure where to even begin with this $$\int \frac{25 - x^2}{x^4} \operatorname d\!x$$
3
votes
3answers
157 views

Evaluate limit $\lim_{n \to \infty } {1 \over n^{k + 1}}\left( {k! + {(k + 1)! \over 1!} + \cdots + {(k + n)! \over n!}} \right),k \in \mathbb{N}$

Evaluate the limit: $$\lim_{n \to \infty } {1 \over n^{k + 1}}\left( {k! + {(k + 1)! \over 1!} + \cdots + {(k + n)! \over n!}} \right),k \in \mathbb{N}$$ It looks like a classic Cesaro-Stolz problem, ...
1
vote
1answer
33 views

Julia sequence : $u_n$ is bounded if and only if $|u_k|\leq 2$ for all $k$.

Let $u_0\in \mathbb{C}$ and $c \in\mathbb{C}$ such that $|c| \leq 2$ and $u_{n+1}=u_n^2+c$. Show that $u_n$ is bounded if and only if $|u_k|\leq 2$ for all $k$. The first implication is ...
1
vote
1answer
72 views

Very interesting multivariable calculus question.

If $\displaystyle z = \frac{f(x-y)}{y}$, show that $\displaystyle z + y \frac{\partial z}{\partial x} + y \frac{\partial z}{\partial y} = 0$.
2
votes
3answers
69 views

Evaluate the integral: $\int\tan^5 (4x)\,\mathrm dx$.

We have to use a trig identity for this. I think it's $\tan^2 x = \sec^2 x - 1$. But I'm having difficulty setting up the integral.
0
votes
1answer
63 views

Using Green's theorem to reduce the integral

Let S be the region of the plain bounded by the graph of $x^2$-$y^2$=4 and the lines y=2 and y=-2 and F(x,y)=($\frac{-y}{x^2+y^2}$,$\frac{x}{x^2+y^2}$). Use Green's theorem to reduce the line integral ...
2
votes
1answer
65 views

What does this notation involving greater than sign mean?

Basically it says that $$|x|>1,2n|a_{n-1}|,\ldots,2n|a_0|$$ Does this mean that $$|x|>\max(1,2n|a_{n-1}|,\ldots,2n|a_0|)$$ It seems to be essential when proving that polynomial whose degree ...
2
votes
1answer
49 views

Evaluate the limit $\lim\limits_{n \to \infty } \root n \of {{a_k}{n^k} + {a_{k - 1}}{n^{k - 1}} + … + {a_0}} $

Evaluate the limit $\lim\limits_{n \to \infty } \root n \of {{a_k}{n^k} + {a_{k - 1}}{n^{k - 1}} + ... + {a_0}} $ $a_0,...a_k > 0$ Is the following right? for $n$ sufficiently large: $$1 ...
1
vote
1answer
51 views

Prove that $4x(x-5)^3 + (x-5)^4=5(x-5)^3(x-1)$

Given the expression $x(x-5)^4$ I need to differentiate it. Upon using the product rule followed by the chain rule I get this answer, $4x(x-5)^3 + (x-5)^4$. The answer in the back of the book is ...
0
votes
3answers
82 views

epsilon delta to prove $\lim_{x \rightarrow a} \frac{1}{f(x)}$

i was solving problems on my textbook.... and i became stuck. The question is: Let $a\in (- \infty , \infty ).$ Suppose $\lim_{x \rightarrow a} f(x)=L \neq 0$. Use the $\epsilon - \delta$ arguement ...
5
votes
4answers
321 views

Convergent or divergent

For homework (Calculus 2) I have to determine does this series converge or diverge and I don't know how to start: $$\sum\limits_{n=1}^{\infty} \dfrac {\ln(1+e^{-n})}{n}. $$