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

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4
votes
3answers
67 views

Let $f(x)$ be continuous on $[0,2]$, and differentiable on $(0,2)$ such that $0<f(1)<f(0)<f(2)$. Prove that $f'$ has a solution on $(0,2)$

Let $f(x)$ be continuous on $[0,2]$, and differentiable on $(0,2)$ such that $0<f(1)<f(0)<f(2)$. Prove that $f'$ has a solution on $(0,2)$. Here's a little crappy sketch: My attempt: ...
3
votes
3answers
151 views

If $x^2$ is not uniformly continuous then how does this theorem hold?

I am studying Spivak's calculus. The subject uniform continuity was in an Appendix and I thought it was dull comparing to the other subjects. So I turned to the math stackexchange for more clear and ...
5
votes
1answer
76 views

Question regarding $\int \tan(x) \sec^2(x) \,dx$

I am asked to find the following but unsure whether or not my solution is valid: $$\int \tan(x) \sec^2(x) \,dx$$ Setting $u=\tan(x)$ and $du=\sec^2(x)\,dx$: $$= \int u\,du$$ $$= \dfrac{u^2}{2}+C$$ ...
3
votes
2answers
56 views

What is the radius convergence of the series?

Find the radius convergence of $\sum_{n = 0}^{\infty} \frac{x^{2n+1}}{2n+1}$ Let's start with: $$f(x) = \sum_{n = 0}^{\infty} \frac{x^{2n+1}}{2n+1} = \int \sum_{n = 0}^{\infty} x^{2n} \,dx = ...
1
vote
3answers
761 views

Exponential decay word problem

How would I solve the following word problem. A radioactive substance weighed $n$ grams at time t=0. Today 5 years later the substance weight $m$ grams.How much will the substance weight 5 years from ...
-1
votes
2answers
140 views

How do I evaluate the integral of the Dirichlet function on $[0,1]$? [duplicate]

Define $f(x) = \left\{ \begin{array}{ll} 1 & \mbox{if } x \in \mathbb{Q} \\ 0 & \mbox{if } \notin \mathbb{Q} \end{array} \right.$ How to evaluate $\int_0^1 f(x)\,dx$ ? I have no ...
0
votes
1answer
79 views

Limit at infinity heine definition, true or false

True/False: Let $f:\mathbb{R}\to\mathbb{R}$ be continuous. For every non-negative sequence of reals $x_n$, which satisfies: $lim_{n\to\infty}x_n=\infty$ The sequence $f(x_n)$, has a ...
0
votes
2answers
58 views

Differentiate $f(x) = exp_{a}(x) $ from first principles

Differentiate $f(x) = exp_{a}(x) $ from first principles, for $ a > 0 $ (Recall that $ exp_{a}(x) = exp(x.ln(a)) $ Here is where I am so far: $ f'(x) = \lim\limits_{h \rightarrow 0} ...
0
votes
0answers
70 views

How to find range of integration for triple integrals

Evaluate the following triple integral \begin{align*} \iiint_R \sin(\pi y^3) dV \end{align*} over the pyramid with vertices $(0,0,0), (0,1,0), (1,1,0), (1,1,1)$ and $(0,1,1)$. I was trying to find ...
1
vote
1answer
54 views

prove that $\lim \limits_{n \rightarrow \infty} n \sum \limits_{j=1}^{n} \frac{cos(\frac{n}{j}) f(\frac{n}{j})}{j^2}$ exists and final.

$f$ is monotonically decreasing function such that $\lim \limits_{x \rightarrow\infty} f(x) =0$, prove that the following limit exists and final . $$\lim \limits_{n \rightarrow \infty} n \sum ...
0
votes
2answers
318 views

Putnam 1990 A1 Induction Help

A1. $(150,9,1,0,0,0,0,0,1,1,6,33)$ Let $$T_0=2,\quad T_1=3,\quad T_2=6,$$ and for $n\ge3$, $$T_n=(n+4)T_{n-1}-4nT_{n-2}+(4n-8)T_{n-3}.$$ The first few terms are ...
1
vote
1answer
59 views

Solve 2nd order ordinary differential equation by Laplace transforms and convolution of their inverse functions. (5.6-40)

Synopsis: Please check my work. I do not have a text "answers to odd problems" for reference as this is an "even" numbered problem. The following documents in good detail the steps taken to solve for ...
3
votes
1answer
83 views

How to integrate $\int\frac{\ln^5 x}xdx$

Can someone explain,why i can`t integrate this? $$\int\frac{\ln^5 x}xdx=\int\ln^5 x\ d(\ln x)$$ I`m really sorry for stupid question.
0
votes
0answers
43 views

How to integrate $\int_{\text{n}}^r \left(-s^b-s t+Y\right)^{-c} \, ds$?

Any hints on how to integrate the following? $\int_{\text{n}}^r \left(-s^b-s t+Y\right)^{-c} \, ds$
0
votes
2answers
76 views

Calculate $\lim_{x \to 0^+} x\lfloor 1/x\rfloor$

I need to find the following limit: $$ \lim\limits_{x \to 0^+} x \left\lfloor\frac{1}{x}\right\rfloor $$ Thanks!
9
votes
3answers
178 views

Limit of $x^2\sin\left(\ln\sqrt{\cos\frac{\pi}{x}}\right)$

Find $$\lim_{x\to\infty}x^2\sin\left(\ln\sqrt{\cos\frac{\pi}{x}}\right).$$ I tried substituting $x=1/t$ with $t$ approaching $0$ but the term inside the bracket is not giving me ideas on how to ...
0
votes
1answer
109 views

Circumference of the curve

We have a cardioid curve. The length of the curve is given form the relation $\int_a^b |z'(t)|dt$. How can we find the circumference of the cardioid curve???
2
votes
1answer
177 views

How do I solve $F = \nabla\times G$ for $G$?

Given the vector valued function $F(x,y,z) = (xz,-yz,y)$ find $G$ such that $F = \nabla\times G$ I let $G(x,y,z) = (G_1,G_2,G_3)$ and expanded $\nabla \times G$ then equated the components to $F$ but ...
0
votes
1answer
73 views

Finding limit using Riemann integral problem

How do I evaluate the following limits ? $$\lim_{n\to +\infty}\left(\frac{1}{n^2}\sum_{k=1}^n\left(2k-1\right) \sqrt[n]{e^{2k-1}}\right)$$ In this case I'm confused about function to integrate: ...
0
votes
2answers
56 views

Suppose {$a_n$} be a sequence of posetive real numbers such that $a_n\ge a_{n+1}$ for all $n\ge1$ and $\sum_{n=1}^{\infty}a_n <\infty$

Suppose {$a_n$} be a sequence of posetive real numbers such that $a_n\ge a_{n+1}$ for all $n\ge1$ and $\sum_{n=1}^{\infty}a_n <\infty$ then which of the following(s) is true: A. ...
0
votes
1answer
26 views

Prove Taylor expansion converges to $f(x)$ for all $x\in I$ fulfilling $|x-x_0|<{1\over M}$.

Is a single piece data incorrect? How can I know? Let $f\in C^{\infty}(I)$, and let $C,M >0$ such that $|f^{(n)}(x)|\le CM^n n!, \forall x\in I, n\in \Bbb{N}$. Prove that for all $x_0\in I$, ...
0
votes
2answers
57 views

Help with an improper integral!

I need some help with an indefinite integral problem (only the $2^{\textrm{nd}}$ part thou). Problem is as follows. Consider the function $f(x) = \dfrac{\ln\!\left(x\right)}{x^{p}}$, where $p>1$ ...
0
votes
3answers
199 views

limit of a sequence / riemann sums

i have a question about this limit. $\lim\limits_{n\to \infty } \ (\frac{1}{3n} + \frac{1}{3n+1} + \frac{1}{3n+2} + \dots + \frac{1}{4n} )$ I tried to calculate it using the squeeze theorem and ...
1
vote
1answer
215 views

What time did the snow start? Snow Plow

I am having trouble proceeding with the problem below. I have solved some stuff to a certain extent, but do not understand what to do from here. The problem statement is: One morning snow began to ...
-2
votes
1answer
52 views

Finding $f'(x)$ if $x\neq 0$ using definition of derivative

$$f(x) = \begin{cases} x^3 \sin (1/x),\quad x > 0 \\ x \sin x, \quad x\leq 0 \end{cases} $$ I'm not sure if I am doing this right but I applied the definition of derivative on $x^3 \sin ...
1
vote
2answers
99 views

Prove that the limit of the following function of two variables is zero

I need to prove the following: $$\lim_{(x,y)\to (1 ,2)} \frac{x^2+2xy-6x-2y+5}{\sqrt{(x-1)^2+(y-2)^2}}=0$$ I've tried to solve it by substituting $y=mx$ but I can't get the solution that way. ...
0
votes
1answer
65 views

Continuity and partial differentiability of $f=\frac{x^2y}{x^4+y^2}$ at $(0,0)$

Let $f:\mathbb{R^2}\to\mathbb{R}$ be defined by $f(x,y) = \begin{cases} \dfrac{x^2y}{x^4+y^2}, & \text{if $(x,y)\neq(0,0)$}, \\ 0, & \text{if $(x,y)=(0,0)$} \end{cases}$ Which of the ...
0
votes
1answer
47 views

Show that f is one-one

I'm given this problem of: Show that $f$ is one-one $$f(x) = (\ln x - 1)(\ln x - 2), \quad 0 < x < 4$$ Now I know that to show it is one-one, I just have to differentiate the $f(x)$. But why ...
2
votes
1answer
32 views

Give the infimum of the image of a function defined over a weird real set.

If I got a real function f defined over the closed interval $[2,6]$ as $\begin{eqnarray}f(x) = \left\{\begin{array}{lr} 5 &x=2 \\9-x^2 &2<x<6 \\ 3 &x=6\end{array} ...
1
vote
1answer
81 views

“ If $f$ -$g$ are integrable over [a,b] then f and g are integrable over [a,b] ?”

If $f$ :[a,b] $ \rightarrow$ $\mathbb{R}$ and $g$ :[a,b] $ \rightarrow$ $\mathbb{R}$ are bounded functions. Can anyone please give a counterexample of the following statement : " If $f$ -$g$ are ...
0
votes
1answer
864 views

Solve 2nd order ordinary differential equation by Laplace transforms and convolution of their inverse functions. (5.6-39)

Synopsis: I cannot duplicate the answer in my text although it does appear very similar. This tells me that my method is correct but I am making another kind of error -- perhaps in my integration? ...
1
vote
0answers
33 views

Evaluating this limit - help

Let $g_n(x) = $ \begin{cases} \hfill x^n \sin(\frac{1}{x}) \hfill & \text{ if} \space x \neq 0 \\ \hfill 0 \hfill & \text{if} x= 0 \\ \end{cases} Then why is $g_2'(0) = ...
0
votes
0answers
94 views

gradient vector and Hessian matrix of an n dimensional vector function

So I'm confused on how to proceed on this problem. I need to find the gradient and Hessian for functions of n variables $$f^T f$$ where $f$ is a vector function depending on $\vec x$ and $∇f^T$ ...
0
votes
2answers
23 views

Solve for $x$; $\sin(2x)=\sin (x+\frac{\pi}{8}); 0< x<\frac{\pi}{2}$

I would appreciate if somebody could help me with the following problem: Question: Solve for $x$; $$\sin(2x)=\sin (x+\frac{\pi}{8}); 0< x<\frac{\pi}{2}$$ I tried but couldn’t get it that ...
0
votes
1answer
26 views

Two parallel planes

When two planes have the same perpendicular vector $\overrightarrow{v}$, then they are parallel, right?? We have that the two planes $Ax+By+Cz+D_1=0$ and $Ax+By+Cz+D_2=0$ are parallel, since they ...
0
votes
2answers
26 views

Bounding a normal integral from above

Show that $$\left|\int_{1}^{\sqrt{3}}\frac{e^{-x}\sin{x}}{x^2+1}dx\right|\leq \frac{\pi}{12e}$$ I tried it by first doing: $$\left|\int_{1}^{\sqrt{3}}\frac{e^{-x}\sin{x}}{x^2+1}dx\right|\leq ...
-2
votes
2answers
110 views

Which of the following is true

I was just handed this question involving the Intermediate Value Theorem and a continuous function. Now the problem says that we have $f\left(1\right)=-12$, $f\left(5\right)=7$, with $f$ being of ...
21
votes
3answers
836 views

Evaluating $\int_0^\pi\arctan\bigl(\frac{\ln\sin x}{x}\bigr)\mathrm{d}x$

I found the following integral as a by product of another one. It has a nice closed form. $$ \int_{0}^{\pi} \arctan\left(\ln\left(\sin x \right) \over x\right)\,{\rm d}x $$ Mathematica and ...
3
votes
1answer
75 views

Valid justification for algebraic manipulation of $\mathrm{d}y/\mathrm{d}x$?

I've read many question/answer threads here on SE re: justification for the algebraic manipulation of $\mathrm{d}y/\mathrm{d}x$ in the standard formulation of calculus. I worked up my own shot at a ...
0
votes
2answers
109 views

Critical Points of $f(x) = \sin(3x)$ — Help!

The problem is asking me to find the Critical Points of : $$f(x) = \sin(3x)$$ on the closed interval: $$[\frac{-\pi}{4},\, \frac {\pi}{3}]$$ I know that $f'= 3\cos(3x)$. the problem I seem to be ...
7
votes
2answers
190 views

Asymptotics of $\int_{0}^{+\infty}\!\!\frac{dx}{\sinh^2(\epsilon \sqrt{x^2+1}) } $ for $\epsilon$ near $0$

How to find an asymptotic expansion, for $\epsilon$ near $0$, of the following integral $$ I(\epsilon):=\int_{0}^{+\infty}\frac 1{\sinh^2 (\epsilon \sqrt{x^2+1}) } {\rm d}x. $$ As $\epsilon ...
0
votes
1answer
91 views

Triangle in space

Using vector notation describe the triangle ( in space ) that has as vertices the origin and the endpoints of the vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. $$$$ The solution is the ...
0
votes
1answer
39 views

Determine con-/divergence of $\sum\limits_{n=1}^\infty \left[ \sqrt{n^3+1} - n^{\frac32} \right]$ by (limit) comparison test

Problem: Use the comparison test, or limit comparison test to determine whether the sum $\sum\limits_{n=1}^\infty \left[ \sqrt{n^3+1} - n^{\frac32} \right]$ converges or diverges. My attempt: Given ...
3
votes
4answers
118 views

Determine con-/divergence of $\sum\limits_{n=1}^{\infty}\frac{1+2+\cdots+n}{2^n}$

Problem: Determine if $\sum\limits_{n=1}^{\infty}\frac{1+2+\cdots+n}{2^n}$ converges or diverges. My attempt: I'm having a hard time with this one. Trying the ratio test, I'm unable to simplify ...
1
vote
3answers
87 views

Prove that the equation $x^{2}-x\sin(x)-\cos(x)=0$ has only one root in the closed interval $(0,\infty)$.

Here's the graph (http://www.wolframalpha.com/input/?i=%28x%5E2%29-xsenx-cosx%3D0). The part I'm having trouble with is proving that the root is unique. I can use the intermediate value theorem to ...
0
votes
1answer
253 views

How do you get this function in terms of x and y?

I'm trying to model an equation for a graph of distance from origin, per second. There was a question on the physics forum that I am trying to model in mathematical terms, but I'm stuck... Anyway, ...
0
votes
1answer
45 views

Finding a fixed point of a function satisfying $|f(x)-f(y)|\le K\cdot |x-y|$.

Let $f:[a,b]\to [a,b]$ fulfill $|f(x)-f(y)|\le K\cdot |x-y|$ $\forall x,y \in [a,b]$, $0<K<1$. Prove $\exists! z\in [a,b]$ such that $f(z)=z$. I am really lost here. I would appreciate your ...
3
votes
2answers
274 views

Integration by De Moivre Theorem

For example , integrate $\sin ^4(x) dx$. I solved this question by reduction formula which is fairly easy. But my senior said that it would be easier if you expand the $\sin ^4(x) $by using De Moivre ...
1
vote
2answers
41 views

Is every convergent sequence in $L^1$ dominated?

This looks very obvious, but I can't prove it, and google is not helping (I also can't find any explicit mention to this in my textbook). I want to prove that, given a measure space $\left(X,\mathcal ...
0
votes
0answers
76 views

Hessian Of Convolution's Quadratic Form

For the discrete inputs $\mathbf{x} \in \mathbb{C}^{M}$ and $\mathbf{y} \in \mathbb{C}^{N}$, I want to find the Hessian of $\Vert x \ast y \Vert_2^2$, where $\ast$ is the discrete convolution ...