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

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2answers
14 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
21 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 $u \in U$ ...
0
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2answers
61 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
votes
4answers
74 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
votes
2answers
18 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 $a_i=0$ ...
0
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1answer
33 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) = ...
4
votes
3answers
61 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
80 views

How to integrate this formula 3

$$\int_0^1 \left( \frac{2e^{2x}}{x}+\frac{1}{xe^x}-\frac{e^{2x}}{x^2}+\frac{1}{x^2e^x} \right) \, dx$$ I tried many times but still could not get it. Any help?
1
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0answers
24 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 ...
1
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0answers
32 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
18 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
22 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 ...
2
votes
2answers
124 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 ...
2
votes
1answer
44 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} ...
0
votes
1answer
25 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: ...
1
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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 = ...
0
votes
2answers
27 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 $(0,\infty)$? ...
1
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4answers
63 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: ...
2
votes
0answers
41 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
27 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
123 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
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3answers
49 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
82 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.
0
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1answer
30 views

Evaluate $\int_{-2}^{2}\int_{y^2-3}^{5-y^2}dxdy$ [duplicate]

In the black I evaluated the integral and I got 64/3, now I need to evaluate the same integral with $\color{red}{dydx}$ .in the $\color{blue}{\text{blue}}$ color is my attempt, I don't think that my ...
5
votes
3answers
139 views

What is the domain of $f(x)=\frac{1}{x}-\frac{1}{x}$?

Question: What is the domain of $f(x)=\frac{1}{x}-\frac{1}{x}$? Does the function have a removable discontinuity at $x=0$? My attempt: My first intuition told me that it was $\mathbb R$, since we ...
0
votes
3answers
26 views

Exponential Growth and Decay Question: A Bacteria Culture Grows with Constant Relative Growth Rate.

A Bacteria Culture Grows with Constant Relative Growth Rate. The bacteria count was 400 after 2 hours and 25,600 after 6 hours. a) What is the relative growth rate? Express your answer as a ...
0
votes
4answers
38 views

How to find perpendicular vectors in 3D

Find all values of a such that the vector $q = \langle 2, a, –2\rangle$ is perpendicular to the vector $p = \langle –3, a, 5 \rangle$.
1
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3answers
75 views

How to prove $f(x)=\sqrt{x+2\sqrt{2x-4}}+\sqrt{x-2\sqrt{2x-4}}$ is not differentiable at $x=4$?

How to prove $f(x)=\sqrt{x+2\sqrt{2x-4}}+\sqrt{x-2\sqrt{2x-4}}$ is not differentiable at $x=4$ ? Please let me know the fastest method you know of for such type of problems. Is there any way other ...
0
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2answers
42 views

What's the meaning of the $R(f(x),g(x))$ in $\int R(f(x),g(x))?$

Usually when I'm reading about integration, there is a notation for integrals on some forms, for example: $$\int R(\sin(x),\cos(x)) \;dx$$ Obviously I've deduced that this represents functions that ...
1
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0answers
34 views

Evaluate a limit of a sequence

Polya and Szego's book Problems and Theorems in Analysis contains a question concerning the evaluation of a sequence in general: $\lim\limits_{n\rightarrow\infty}\sqrt{a_1 + \sqrt{a_2 + \sqrt{a_3 + ...
0
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2answers
49 views

A question about Idempotent functions [on hold]

some functions are such that $f\circ f(x)=f(x)$ like these 1) $$f(x)=x \implies f\circ f(x)=x=f(x)\\$$ 2)$$f(x)=\lvert x\rvert \implies f\circ f(x)=\lVert x\rVert=\lvert x\rvert=f(x)\\$$ 3) ...
3
votes
2answers
30 views

Distance between a plane and set of points

Suppose $m$ data points belonging to a class represented by matrix $A$. Therefore, the size of matrix $A$ is $m\times n$. In addition, suppose $w\cdot x + b=0$ be equation of a plane in ...
3
votes
5answers
151 views

How to integrate $\int \frac{4}{x\sqrt{x^2-1}}dx$

In order to solve the following integral: $$\int \frac{4}{x\sqrt{x^2-1}}dx$$ I tried different things such as getting $u = x^2 + 1$, $u=x^2$ but it seems that it does not work. I also tried moving ...
1
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2answers
31 views

Minimum/Maximum Question (Calculus)

I have been solving questions about min/max & Lagrange methods. Now I got stuck with this excercise without any clue of how to solve: * Find the maximum of f(x,y) on the curve L: $f(x,y) = \frac ...
0
votes
1answer
51 views

How to prove the limit exists for function of two variables?

Problem: Evaluate the indicated limit or explain why it does not exist: \begin{align*} \lim_{(x,y) \to (0,0)} \frac{x^2 y^2}{x^2 + y^4} \end{align*} The definition of limit my calculus textbook gives ...
1
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1answer
41 views

Integrals, intermediate value theorem question

f∈c[a,b] (f is continuous in [a,b]), prove: We tried to use the integral intermediate value theorem to try to prove it but we don't understand why the limit has to be the max and not any other value ...
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votes
3answers
66 views

Integral for cos [on hold]

$$\int_{-a}^{+a}A^2\cos^2\left(\frac{n\pi a}{2}x\right)dx$$ where $A, n, \pi,a$ are constants. I was thinking about trigonometric formula for $\cos^2x=\frac{1+\cos2x}{2}$
0
votes
1answer
27 views

Trigonometric equation [on hold]

I have the equation $B\sin(ka)=0$. I must find what value has ($ka$). I was thinking about the trigonometric formula $\sin x=a$ which includes $x=(-1)^k \arcsin(x)+k\pi$. My final result must be: ...
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1answer
27 views

Area surrounded by a curve

I would need help to calculate the area surrounded by a curve. The curve is given with the following polar coordinates: I know we need need to integrate with respect to r and theta but am stuck ...
1
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0answers
48 views

Counting the sum $\sum^{\infty}_{k=0} q^{k^{2}}$

Is it possible to obtain explicit form of the sum $\sum^{\infty}_{k=0} q^{k^{2}}$ (without using elliptic functions)? It is well known that $\sum^{\infty}_{k=0} q^{k} = \frac{1}{1-q}$ for all $q \in ...
0
votes
1answer
23 views

Why is $F'(x) = 2x·\tan(x^2)-\tan x$ if $F(x) = \int_{x}^{x^2}\tan u\, \mathrm du$?

Evaluate $F'(x)$ if $$F(x) = \int_{x}^{x^2}\tan u\, \mathrm du$$ I tried to do this by the change of variables formula and hence, $$F(x) = \int_{x}^{x^2}\tan u\, \mathrm du=\int_{\sqrt x}^{x}\tan ...
0
votes
1answer
25 views

$f$ is smooth and periodic function,$\exists \lambda$ such that $f^{(4)}=\lambda f$ prove:$\exists \lambda$ such that $\lambda= (\frac{2\pi n}{T})^4$

Given $f:\mathbb{R} \rightarrow \mathbb{C}$ is smooth and periodic with period $T>0$ and exists $\lambda \in \mathbb{C}$ such that $f^{(4)}(x)=\lambda f(x)$ for any $x \in \mathbb{R}$. prove: ...
3
votes
1answer
24 views

How to prove define integrate from f(sin x)

i need help for prove this problem , i dont have idea for this prove, i very appreciate your sugerences. $$ \int ^{\pi }_{0}xf(\sin x)\,dx = \int ^{\pi }_{0}\frac{\pi }{2} f(\sin x)\,dx $$
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1answer
43 views

Proving an equivalent definition of the $\lim_{x\to a}f(x)$ exists [duplicate]

Prove that the following statements are equivalent. (a) $\lim_{x\to a}f(x)$ exists (b) Given $\epsilon \gt 0$, there is a $\delta \gt 0$ such that if $0\lt |x-a| \lt \delta, 0\lt |y-a| \lt \delta$, ...
2
votes
5answers
99 views

how to prove that $\ln(1+x)< x$

I want to prove that: $\ln(x+1)< x$. My idea is to define: $f(x) = \ln(x+1) - x$, so: $f'(x) = \dfrac1{1+x} - 1 = \dfrac{-x}{1+x} < 0, \text{ for }x >0$. Which leads to $f(x)<f(0)$, ...
1
vote
2answers
32 views

Finding the derivative of a multivariable integral?

Okay so I'm asked to find $\frac{df}{dt}$ of $f(t)=\int_{1}^{t^2}{\frac{1}{s}e^{s^2t}ds}$. Letting $g(s)=\frac{1}{s}e^{s^2t}$ I get: $$f'(t)=G'(t^2) \cdot 2t - G'(1) \cdot 0 =g(t^2) \cdot 2t = ...
2
votes
4answers
76 views

Calculus Optimization problem [on hold]

Help with the problem?? The cost of building the pipeline is $\$3$ million per mile in the water, and $\$4$ million per mile on land. Hence, the cost of the pipeline depends on the location of point ...
5
votes
1answer
57 views

Derivation of Gradshteyn and Ryzhik integral 3.876.1 (in question)

In the Gradshteyn and Ryzhik Table of Integrals, the following integral appears (3.876.1, page 486 in the 8th edition): \begin{equation} \int_0^{\infty} \frac{\sin (p \sqrt{x^2 + a^2})}{\sqrt{x^2 + ...
1
vote
0answers
29 views

To construct a power series such that the radius of convergence of the power series $\sum_{n=0}^{\infty} a_n b_n x^n$ is $2R$.

Let $\sum_{n=0}^{\infty} a_n x^n$ is a power series with radius of convergence $R(>0)$. To construct a power series $\sum_{n=0}^{\infty} b_n x^n$, other than $\sum_{n=0}^{\infty} (\frac x2)^n$, ...