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

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10 views

How to prove that the line perpendicular to the radius is the tangent in the calculus sense?

Let $P=(p_1,p_2)$ be a point on an semicircle and $r$ be the line perpendicular to the radius $\overline{OP}$, like the picture below. Euclid showed (Book III, Proposition 16) that $r$ does not ...
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0answers
5 views

“Maximum point lies on a curve” implies tangential derivative is zero there.

Given a function $f:\mathbb{R}^2\to\mathbb{R}$, suppose that it has a local maximum at the point $(x_0,y_0)$. Let $\gamma$ be a curve passing through $(x_0,y_0)$. Does it follow that the directional ...
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0answers
76 views

How to calculate the integral?

How to calculate the following integral? $$\int_0^1\frac{\ln x}{x^2-x-1}\mathrm{d}x=\frac{\pi^2}{5\sqrt{5}}$$
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0answers
55 views

An Infinite series I

By decompising fractions one can show that \begin{align} \sum_{n=1}^{\infty} \frac{1}{n \, (n+1)^{2} \, (n+3)} = \frac{65}{72} - \frac{\zeta(2)}{2}. \end{align} The fraction can also be seen in the ...
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1answer
50 views

what is the limit of $(-2)^{1/(2n+1)}$ as $n\rightarrow\infty$?

what is the limit of $(-2)^{1/(2n+1)}$ as $n\in\mathbb{Z}, n\rightarrow\infty$? and what is the limit of $(-2)^{2/(2n+1)}$ as $n\in\mathbb{Z}, n\rightarrow\infty$?
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3answers
27 views

Modelling interest with differential equations (Interpretation)

I am having trouble interpreting the meaning of this differential equation model for interest on an account. The problem is as follows: Assume you have a bank account that grows at an annual ...
3
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1answer
62 views

Where did I make a mistake?

This is an excerpt from a dynamical systems paper: They provide a proof of this Lemma, and numerical simulations also show it should be true. It's clear the equilibrium point on each axis is ...
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2answers
26 views

finding the volume of the solid via disk or washer method

the question is: $y = 1/4x^2$, $x = 2$, $y = 0$; about the $y$-axis I tried to draw it out, but I can't figure this stuff out. The graphing is the hardest part for me because I don't know what to do ...
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1answer
21 views

Does the gradient function at a point give the direction of greatest increase and also perpendicular at the same time?

So say if I have a cone and took the gradient and then evaluated it at a point would this vector that points in the direction of greatest increase also be perpendicular and is this true for all ...
6
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3answers
138 views

About the integral $\int_{-1}^1 \frac{1}{\pi^2+(2 \operatorname{arctanh}(x))^2} \, dx=\frac{1}{6} $

Here is a question that naturally arose in the study of some specific integrals. I'm curious if for such integrals are known nice real analysis tools for calculating them (including here all possible ...
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0answers
13 views

Show Uniform convergence of a Serise

Im trying to show that the serise $ \sum_{1}^{infinite}\frac{x^2}{(1+x^{2})^{n}} $ is not Uniform converge in the domain $(-\infty$ to $ \infty )$ . I managed to show that the sum of the serise when $ ...
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0answers
21 views

General formula for sinusoidal taylor series centered at any a?

I understand that to find a taylor series centred at a particular a value you need to find a formula for the nth derivative, but this is tricky for cos(x) and sin(x). Is it possible to have a formula ...
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0answers
35 views

Evaluating the indefinite integral $\int\sqrt{\cos2x}\sin^32x\,dx$

I have tried to integrate the following indefinite integral but I'm not sure if I get the right answer. Please tell me if I'm wrong and if so, please indicate what went wrong. $$ ...
1
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1answer
35 views

Quadric equation-physics

$$\frac{[(\omega_0^2-\omega^2)-2i\omega\gamma]^2}{[(\omega_0^2-\omega^2)^2+4\gamma^2\omega^2]^2}=\frac{1}{[(\omega_0^2-\omega^2)^2+4\gamma^2\omega^2]}$$ I don't understand how can I get to that ...
2
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1answer
35 views

How to prove that $f(x)x - \int_{0}^{x}{f(t) \,dt} = \int_{f(0)}^{f(x)}{f^{-1}(t) \,dt},$ for all invertible functions.

A while ago, I found that: $$f(x)x - \int_{0}^{x}{f(t) \,dt} = \int_{f(0)}^{f(x)}{f^{-1}(t) \,dt}.$$ I managed to prove it for a few functions, and I believe that it may be the case for all ...
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0answers
20 views

Quadric equation for a physics problem

[($\omega_0^2$-$\omega^2$)-2i $\omega$ $\gamma$]$^2$=1. I don't understand how can I get to that solution '1'. Any hint will be very thankful.
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4answers
73 views

limit of an integral question

Let $f : [0, \infty) \to \Bbb R$ be bounded and continuous. Prove that $\lim \limits _{h \to \infty} h \int \limits _0 ^\infty e ^{-hx} f(x) \, d x = f(0)$. Our intuition was to use l'Hospital's rule ...
1
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2answers
47 views

If the area bounded by $y=x^2+2x-3$ and the line $y=kx+1$ is the least, find $k$ and the least area.

What concept to use in the Application of Integral question? Please help me
1
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1answer
55 views

A question about differential function

If $f(x)=f'(x^{2})+2x$, then $f(1)=?$ and $f''(1)=?$ Sorry. I am going to check the original problem, and then i will update.
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1answer
27 views

Help with a problem regarding sequence divergence.

There are two forms of definition of sequence divergence. By negation of the sequence convergence we have A sequence $x_k$ diverges iff $∀x∈\Bbb{R}∃ϵ>0∀N∈\Bbb{N}∃k>N$ st. $|x_k-x|>ϵ$. ...
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0answers
52 views

Integral calculation question [duplicate]

Calculate the following integral: $\int \limits _0 ^\frac \pi 2 \ln (\sin x) \Bbb d x$. We used the substitution $x=2t$ and then used the identity $\sin 2t = 2 \sin t \cos t$ but now we're stuck. ...
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1answer
36 views

What does it mean to say $f(x) \sim g(x)$, i.e. $f(x)$ behaves like $g(x)$ when $x \to \infty$?

If $\lim_{x\to\infty}\frac{f(x)}{g(x)}=\infty$, then $f$ grows faster than $g$. Same if $\lim_{x\to\infty} \frac{g(x)}{f(x)} = 0$. Would $f$ behave like $g$ if $\lim_{x\to\infty}\frac{f(x)}{g(x)} = ...
0
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1answer
44 views

Function with increasing property.

Prove that $\frac{1}{2}(x+2)^{-3/2}-(\frac{1}{2}x+3)(x+3)^{-3/2}$ is increasing function for $x\ge4$. I tried it by taking its first derivative but by first derivative for me its difficult to say it ...
23
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2answers
2k views

Why does $ \int_0^1 \lceil { x\sin({1 \over x})} \rceil = 1 - \frac{\log(4)}{2\pi} $?

One time I was bored and played around a bit with integrals and wolfram alpha and tested the following integral: http://www.wolframalpha.com/input/?i=integral_0%5E1+ceil%28x*sin%281%2Fx%29%29 Note: ...
3
votes
3answers
78 views

Evaluate $\iint_{R}(x^2+y^2)dxdy$

$$\iint_{R}(x^2+y^2)dxdy$$ $$0\leq r\leq 2 \,\, ,\frac{\pi}{4}\leq \theta\leq\frac{3\pi}{4}$$ My attempt : Jacobian=r $$=\iint_{R}(x^2+y^2)dxdy$$ $$x:=r\cos \theta \,\,\,,y:=r\cos \theta$$ ...
1
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2answers
26 views

Improper integral convergence question

Prove that the following integral converges: We divided the integral to 2 integrals (one from 0 to 1/2 and the other from 1/2 to 1). We managed to prove that the integral from 1/2 to 1 converges ...
1
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2answers
31 views

Connection between Fréchet derivative and the directional derivative in finite euclidean space

In the lecture notes I am reading, the following statement is made: Let $U$ be an open subset of $R^n$, and define the function $e:U \to R$. $e$ is said to be differentiable if for every $u \in U$ ...
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2answers
74 views

Improper rational/trig integral comes out to $\pi/e$

During my studying to integration I find this integration. So I tried to prove but I got stuk. So I need help in this integration. $$\displaystyle\int_{-\infty}^{\infty} \frac{x \sin (x)}{1+x^2} ...
2
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4answers
150 views

How should I go about solving this definite integral?

The integral is: $$\int_{-1}^1\sqrt{4-x^2}dx$$ I'm having difficulty figuring out how to go about this. I attempted to use u-substitution, both by substituting $u$ for $\sqrt{4-x^2}$ entirely, and ...
2
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2answers
21 views

Convegence of $\sum_{i\in J}a_i$ implies that index set is countable

Let $J$ be a uncountable set and $\{a_i\}_{i\in J}$ be a set of non-negative real numbers. Prove that $\sum_{i\in J}a_i<\infty$ implies that there is a countable set $H\subset J$ such that $a_i=0$ ...
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1answer
38 views

How do I convert a sum to an algebraic expression?

Something something Riemann sum to integral is the most that I remember. I just don't remember how we did it or whether or not that would be the best method for doing it. Let $ \theta(n) = ...
0
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0answers
39 views

Monotonicity of a discrete fucntion

I am trying to show that the following function is increasing in $m$: $$ f(m)=\sum_{i=0}^{m-1}(1-x)x^{i}\frac{1-x_{1}^{m-i}}{1-x^{m+1}}, $$ where $m$ is a positive integer, and $x$ and $x_{1}$ are ...
4
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3answers
79 views

Challenging $\lim_{x \rightarrow 10} \frac{1}{\lfloor x \rfloor} = \frac{1}{10}$ for $\epsilon=\frac{1}{2}$.

Consider the (incorrect) claim that $$\lim_{x \rightarrow 10} \frac{1}{\lfloor x \rfloor} = \frac{1}{10}.$$ How might I find the largest $\delta$ such that I can challenge $\epsilon = 1/2$? Clearly ...
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2answers
83 views

How to integrate this formula 3 [on hold]

$$\int_0^1 \left( \frac{2e^{2x}}{x}+\frac{1}{xe^x}-\frac{e^{2x}}{x^2}+\frac{1}{x^2e^x} \right) \, dx$$ I tried many times but still could not get it. Any help?
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0answers
26 views

Remainder Estimate for Integral test

I have the following question, it is a fill in the blank type question, however when I submit my answer, the system which verifies it say it is incorrect. I believe I am right, so I was hoping for ...
4
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0answers
99 views

Infinite Product Representation of $\sin x$

I've recently taken interest in infinite products, and I'm having trouble with a proof I found in this PDF file: "Infinite Products and Elementary Functions": An intermediate step in finding an ...
2
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1answer
20 views

Algebra with differential operators (Alternative forms of the Laplacian in spherical coordinates)

Given is the following: $$\frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \,\frac{\partial f}{\partial r} \right) = \frac{\partial^2 f}{\partial r^2} + \frac{2}{r} \frac{\partial f}{\partial ...
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2answers
30 views

Runge Phenomena and Taylor Expansion

From The Weierstrass Approximation Theorem Vs The Runge's Phenomenon: We contrast this to polynomial interpolation: this is a specific method for generating a sequence of polynomials that ...
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3answers
195 views

Not the toughest integral, not the easiest one

Perhaps it's not amongst the toughest integrals, but it's interesting to try to find an elegant approach for the integral $$I_1=\int_0^1 \frac{\log (x)}{\sqrt{x (x+1)}} \, dx$$ $$=4 ...
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votes
1answer
48 views

How to integrate this formula

I need to compute $$\int_{0}^{1} \frac{2e^{2x}}{x} dx.$$ I try to use integral by parts, let $u=\frac{1}{x}$, $dv=2e^{2x}dx$, then $du= -\frac{1}{x^2}$, $v=e^{2x}$. Then $$\int_{0}^{1} ...
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1answer
32 views

Describing surfaces

I'm working on a problem that deals with describing surfaces given specific information (we're studying rectangular, cylindrical, cartesian, spherical coordinates). I am posed with the question: ...
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1answer
28 views

Exponential Derivative Word Problem

I am having problem with a world problem derivative application question. The number of parasites in the blood after $h$ hours medication is taken is given by the function $p = ...
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votes
2answers
28 views

Open cover with no finite subcovers for the set [0, ∞)

I am trying to find an open cover with no finite subcovers for the set $[0, \infty)$ I am thinking union from $n=1$ to $\infty$ of the sets $(0,n)$ Does this work or does this give me $(0,\infty)$? ...
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vote
4answers
78 views

Evaluate $\iint dy\,dx;\frac{\pi}{4}\leq\theta \leq\frac{3\pi}{4};0\leq r\leq2$

I need to evaluate $\displaystyle\iint \color{red}{dydx}\;\;\;,\frac{\pi}{4}\leq\theta \leq\frac{3\pi}{4}\;\;\;\;,0\leq r\leq2$ $\color{blue}{\text{without using polar coordinates}}$. My attempt: ...
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0answers
63 views

Can $\int_{0}^{1}\frac{x^{p}\ln^{q}(x+a)}{(x+a)^{b}}dx$ be expressed in a simple form?

I was browsing the book Irresistible Integrals and found this gem, at page 97, $$ \int_{0}^{1}x^{n}\ln^{k}(x)dx=\frac{(-1)^{k}k!}{(n+1)^{k+1}} $$ that resembles a previous question of mine here. So, ...
1
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1answer
28 views

Leibniz integral rule (singular)

Definte $I(\epsilon):=\int_{ \epsilon}^1\frac{\,\mathrm{d}x}{\sqrt{x-\epsilon}}$ for $\epsilon<0$ Want to show that ...
9
votes
7answers
132 views

Evaluating the indefinite integral $\int\sqrt{16-9x^2}\,dx$

I need to solve the integral below, but I just can't figure how. $$\int \sqrt{16-9x^2}\,dx$$ I have tried to replace $9x^2$ with $16\sin^2\theta$. I get to a point where I have the function ...
1
vote
3answers
58 views

Dealing with indeterminate forms of the $1^\infty $ kind

$$\lim\limits_{x→{\frac π{2}}^-}\left(\frac {2x}{\pi}\right)^ {\tan x}$$ and $$\lim\limits_{n→\infty} \left(1+ \frac {1}{n}\right)^n$$ could anyone provide some hints? how to start. (with ...
2
votes
1answer
84 views

What is the definition of a Critical Point?

I was reading a book on Calculus, by Michael Spivak. There they mention that points where the derivative is equal to zero are called critical points. They nowhere mention that where the derivative ...
0
votes
1answer
25 views

matrix multiplication manipulation

a,b $\in \mathbb{R^n}$ and C $\in \mathbb{R^{nxn}}$. I have $ab^TCab^TC$. I try to manipulate this multiplication into: $b^TCaab^TC$. I need help.