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

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0
votes
1answer
33 views

Why $\int_a^b f(x) dx=\int_0^\infty g(t) dt \text{ with } t=\frac{x-a}{b-x}$?

I'm reading Nahin's: Inside Interesting Integrals. I don't get why he can write: $$\int_a^b f(x) dx=\int_0^\infty g(t) dt \quad \quad \quad \quad \quad \quad \quad \quad \text{with } ...
0
votes
2answers
45 views

Find the derivative of y^2=x as a function of y.

Find the derivative of $$y^2=x$$ as a function of y. i have found for the function of x, it will be $$\pm\frac{1}{2\sqrt x}$$ however for the function of y will be $$\frac{dy}{dx}=2y$$ ? it looks ...
0
votes
2answers
41 views

implicit differentiation, can't find my mistake

an equation $y^3+x^3=3xy$ need to find $\frac{dy}{dx}$ $$(y^3)' = 3y^2\frac{dy}{dx} $$ $$(x^3)' = 3x^2$$ $$-(3xy)' = -(xy+3x\frac{dy}{dx})$$ $$\frac{dy}{dx}(3y^2-3x)=xy-3x^2$$ therefore, ...
3
votes
4answers
176 views

Help finding the $\lim\limits_{x \to \infty} \frac{\sqrt[3]{x} - \sqrt[5]{x}}{\sqrt[3]{x} + \sqrt[5]{x}}$

I need help finding the $$\lim_{x \to \infty} \frac{\sqrt[3]{x} - \sqrt[5]{x}}{\sqrt[3]{x} + \sqrt[5]{x}}$$ I did the following: $$\begin{align*} \lim_{x \to \infty} \frac{\sqrt[3]{x} - ...
2
votes
5answers
78 views

Evaluate $\lim_{x\to 0}\frac{\sqrt[m]{\cos\alpha x}-\sqrt[m]{\cos\beta x}}{\sin^2x},m\in \mathbb{N}$

Evaluate $$\lim_{x\to 0}\frac{\sqrt[m]{\cos\alpha x}-\sqrt[m]{\cos\beta x}}{\sin^2x},m\in \mathbb{N}$$ I used L'Hospital's rule, but that didn't work. Could Taylor series be used? I don't know ...
0
votes
2answers
78 views

Comparing the greatest values of two functions (Derivatives)

I've tried doing this task, and for this kind of task I should be using derivatives. When I done all the calculus, everything I got were some weird result which I do not know how to compare. Task ...
1
vote
1answer
49 views

Explanation about a problem

I couldn't get the idea behind well defined. What does it mean?
0
votes
1answer
25 views

A continuity result

Suppose (i) $f:R^n_+\to R$ and (ii) $f(x)=f(\alpha x)$, for all $\alpha>0$ and (iii) For any $x,y\in R^n_+$, if $x_n\to x^*$, $y_n\to x^{*}$, we have $\lim f(x_n)=\lim f(y_n)$. Could I claim ...
2
votes
2answers
57 views

Max/min problem - show that diagonal is least when rectangle is a square.

A rectangle has constant area, show that the length of a diagonal is least when the rectangle is a square. $$Area (a) = xy$$ $$y=\dfrac{a}{x}$$ $$D^2 = x^2 + y^2$$ $$D^2 = x^2 + ...
3
votes
3answers
156 views

why is limit $\lim_{x\rightarrow 0}⌊\frac{\sin x}{x}⌋ = 0$?

I was evaluating the limit $$f(x) = \lim_{x\rightarrow 0} \left\lfloor\frac{\sin x}{x}\right\rfloor$$ and I substituted the equivalent infinitesimal $\sin(x) \sim x$, obtaining $f(x) = 1$. But on ...
2
votes
3answers
118 views

Advanced techniques needed to solve a difficult integral.

I am looking to solve the following integral. $\int_{0}^{\infty} \frac{1-\cos(ax)}{x^2}e^{bx} dx$. I have made an attempt using the differentiation under the integral sign method and I got the ...
5
votes
2answers
130 views

A shirt with the imprint of a formula.

Before I began to study mathematics, a friend of mine bought me a shirt with the imprint of a formula. I did not know what these characters were and had no desire to think about it. Yesterday, I ...
2
votes
0answers
66 views

Example of a continuous non-lipschitz function with domain $[0,1]$ and co-domain $\mathbb R$

I would like an example of a function which is continuous with domain $[0,1]$ but is not Lipschitz continuous. Is this possible? I know a continuous function with domain $[0,1]$ is uniformly ...
0
votes
2answers
52 views

Evaluate $\sin^5 x$ using Imaginary part of $e^{\iota x}$

A few days ago somebody posted a problem on evaluating $\int sin^5(x) dx$. The answer posted by Jack D'Aurizio Sir used Complex Numbers to represent $\sin(x)$. This lead to me trying to think of a way ...
1
vote
2answers
28 views

A solution of a differential equation of first order in the large-variable limit

The differential equation reads: $ \dfrac{\partial R (t)}{\partial t} = \dfrac{c_2}{R^2} + \dfrac{c_3}{R^3} + O(R^{-4})$, Where $c2 > 0$ and $c3 > 0$, how to get the solution of the ...
12
votes
4answers
390 views

Solve trigonometric integral $\int_{-\pi/2}^{\pi/2} \frac{\sin^{2014}x}{\sin^{2014}x+\cos^{2014}x} dx $

Please help me to solve the following integral: $$\int_{-\pi/2}^{\pi/2} \frac{\sin^{2014}x}{\sin^{2014}x+\cos^{2014}x} dx$$ I have tried a lot, but no results. I only transformed this integral to the ...
8
votes
2answers
902 views

Axiomatic approach to differential calculus?

Normally we develop differential calculus by defining a derivative constructively as something related to a quotient (maybe an $\epsilon$-$\delta$-style limit of a quotient, or maybe the standard part ...
0
votes
2answers
134 views

Max/min problem, find max area of sector of circle.

A sector of a circle has fixed perimeter. For what central angle $θ$ (in radians) will the area be greatest? First I put together an equation for the perimiter of the sector being $p$ which I ...
2
votes
2answers
48 views

Prove the existence of minimal height of a convex polygon

Suppose we have a polygon in $\mathbb{R}^n$. Obviously, we can always trap the polygon into two parallel hyperplanes perpendicular to any given direction. Now the question is, how to prove the ...
-1
votes
3answers
56 views

What is the sum function $s(x)$?

What is the sum function $s(x)$ of $\Sigma_{n=1}^\infty\frac{2n-1}{2^n}x^{2n-2}$? Thanks for your help.
0
votes
1answer
40 views

function with two variables

find the value of the nexg limit: $$\lim_{(x,y) \to (0,0) }\frac{sin(x^3+y^3)}{x^2+y^2}$$ I have tried to use $$1\ge |sinx|$$ but it didn't work.
0
votes
1answer
49 views

Find the volume of the solid revolved around y-5

I am trying to setup this integral but I am having trouble figuring out the bounds. Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the ...
1
vote
1answer
62 views

Finding the limit that involves Fourier coefficients,

Given the function $f(x) = 1 - \dfrac{|x|}{\pi}$, I had computed its Fourier coefficients, using integration by parts and got: $$ a_n = \begin{cases} 0, & \text{for $n$ even}, \\[6pt] ...
2
votes
4answers
133 views

Integrate $\frac{1}{\sqrt{4-x^2}}$ [duplicate]

Evaluate $$\int\frac{1}{\sqrt{4-x^2}}dx$$ I had this question on my calc exam today, and I have no clue how it's done. I was trying to factor 4-x² to see if I could see any patterns but no luck. One ...
2
votes
1answer
47 views

Galois of successive polynomials in the series expansion $e^x=\sum_{n=0}^\infty\frac{x^n}{n!}$.

I read in a French paper an assertion without proof that I had not known before and that really catches my attention: The Galois group over $\mathbb{Q}$ of the equation ...
1
vote
2answers
92 views

Meaning of a Dirac function

My question is about the $\delta$ function. It has following property: $$\int_{-\infty}^\infty f(x)\delta (x-t) \,\mathrm{d} x = f(t) $$ What's the meaning of the equation? Why not directly calculate ...
0
votes
0answers
15 views

From polynomial to Taylor formula properties

So if we have infinite polynomial in the form $f(x)$ =$a_{0}+ a_{1}.x ....$ I proved that it's rad of convergence say for example R whether is integrated or deintegraded we get new polynomial with the ...
3
votes
3answers
12k views

Non-Linear Transformation

Can someone explain to me in simple terms what a non-linear transformation is in maths? I know some single-variable calculus, but I read it has to do with multi-variable calculus, which I'm not ...
0
votes
2answers
83 views

Proof modification, $ \lim_{n\to\infty}(1+x/n)^n$

So I understand this is very similar to another question How to prove $\lim_{n \to \infty} (1+1/n)^n = e$?. But I want to show that for all $x\in\mathbb{R}$, $\lim_{n\to\infty}(1+x/n)^n$ exists. Can ...
0
votes
1answer
29 views

Why can we consider elements of a normed space $X$ as elements of a normed space $Y$, if there is an embedding between these spaces?

Let $(X,\left|\;\cdot\;\right|)$ and $(Y,\left\|\;\cdot\;\right\|)$ be normed spaces and $\iota :X\hookrightarrow Y$ be an embedding. Often when I read that such an embedding $\iota$ exists, I read ...
1
vote
1answer
234 views

Let $f\colon [0,1]\to\mathbb{R}$. Prove there exists $c$ such that $f(c)=c$

Let $f\colon [0,1]\to\mathbb{R}, 0<f(x)<1$ for all $x \in[0,1]$. $f(x)$ continuous. Prove there exists $c$ such that $f(c)=c$ My attempts: As $x \in[0,1]$ then $0\leq x \leq 1$ so $0 < x ...
0
votes
1answer
60 views

Use the Intermediate Value Theorem to show the equation

Use the intermediate value theorem to show that the equation, $ tan(x) = 2x $ has an infinite amount of real solutions. So far I have used the IVT to show that for $ f(x) = tan(x) $ in the interval ...
1
vote
1answer
40 views

Question about the Cauchy Product and how it is done

Lets say we have the following: $$ \sum_{k=0}^\infty z^k \sum_{j=0}^k \frac{1}{j!(k-j)!} B_{k-j}^f(x) \frac{d^{j}}{dx^{j}}[a_k(x)] $$ Would it be correct to say that: $$ \sum_{k=0}^\infty z^k ...
1
vote
2answers
69 views

Calculus - finite integration of $e^{y^3}$ in double integration

i have this problem that bugs me for 3 hours now. I searched the internet and did not find a solution to this specific problem which was asked in our final: $$\int_0^3 \;\int_{\sqrt{x/3}}^r ...
6
votes
1answer
104 views

Finding the area between $F(x) = 4\sin(x) + \cos(2x) -4x$ and the and $x$- and $y$-axes

I'm kind of stuck while studying for my AP Math final (11th grade). I need to find the area between $$F(x) = 4\sin(x) + \cos(2x) -4x$$ and the and $x$- and $y$-axes. So I did $$F(x) = 0$$ To ...
0
votes
1answer
67 views

convergence of a “nice ” subseries of a divergent series 2

$$S_0 := a_1+a_2+a_3+\cdots$$ is a divergent series of positive terms whose limit approaches 0. Obtain a subseries,called a first-stage nice subseries, $$S_1 := b_1+b_2+b_3+\cdots$$ from $S_0$ by ...
0
votes
1answer
194 views

two points with same tangent line

curve $y=\sin(x)−x/2+10$. The slopes of the tangent lines of this function are the same for any two points that are separated by a distance of $2\pi$. Find the two points $(x_0,f(x_0))$ and ...
4
votes
3answers
92 views

Find the shortest distance between $y=x+10$ and $y=6\sqrt{x}$

This is a Max/min problem, I'm trying calculate the shortest distance between the 2 using pythagoras theorem and diffrenciate it in order to calculate the mininmum of the Red line below: I'm having ...
6
votes
6answers
174 views

Does the series: $\sum_{n=1}^\infty \frac{2n+1}{n^2(n+1)^2}$ converge?

does $\sum_{n=1}^\infty \frac{2n+1}{n^2(n+1)^2}$ converge? I think yes, it does, because the $a_n$ in the series converges to zero. but I'm trying to prove this by the help of the fact that: ...
0
votes
1answer
57 views

Find the limit of right angles ( is it possible?) [closed]

Image updated My question is it possible to find the limit of right angles which is what the question seem to be asking?
1
vote
2answers
63 views

How to solve this sequence?

I have this sequence: $\sum_{n=1}^{\infty} \frac{n^2+n-1}{\sqrt{n^\alpha+n+3}}$ For which values of $\alpha$ does this converge? I first tried to separate into cases where $\alpha \gt 0$ etc and ...
10
votes
2answers
645 views

Why should the substitution be injective when integrating by substitution?

I made a silly mistake in evaluating some integral by using a non-injective $u$-substitution. But why should $u$-substitutions be injective in the first place? I reasoned in the following way: the ...
4
votes
1answer
131 views

Finding the closest point to a set of lines in 2D

I would need to write an algorithm to find the closest point to a set of lines. These lines are infinite and are not parallel between each other. Closest point means that point where the sum of the ...
3
votes
4answers
164 views

How to compute the derivative of $\sqrt{x}^{\sqrt{x}}$?

I know have the final answer and know I need to use the natural log but I'm confused about why that is. Could someone walk through it step by step?
1
vote
5answers
65 views

Evaluate the limit as $x$ approaches infinity

$$\lim_{x\to \infty } \frac{2x-\cos x}{3x+\cos x} $$ Anyone can guide me for this question? Appreciate your help . Thank you!
-1
votes
1answer
41 views

Is the function $f(x)$ positive in $[0,1]$ where $ a, b>0$

Is the function $$ f(x) = \frac {e^{-ax}}{ax} -\frac{e^{-abx}}{1- e^{-ax}}\left((1-x)\cos (\pi x)+\frac{3}{\pi} \sin(\pi x)\right)$$ positive in $[0,1]$ where $ a, b>0$
0
votes
1answer
69 views

Does the series: $\sum_{n=1}^\infty (-1)^n \lbrack {\sqrt\frac{n}{2}} \rbrack$ Converge?

Does the series: $\sum_{n=1}^\infty (-1)^n \lbrack {\sqrt\frac{n}{2}} \rbrack$ Converge ? Note: by the brackets I mean the floor function. I tried to substitute numbers and look at the members of ...
6
votes
4answers
257 views

Find $\lim_{n\to\infty} \sqrt[n]{n}$ [duplicate]

Find: $$\lim_{n\to\infty} \sqrt[n]{n}$$ How does one not having any intuition able to do this?
0
votes
2answers
64 views

$\int { \max { \left( 1,{ x }^{ 2 } \right) dx } }$

I am confused with this integral,can anybody help me,thanks beforehand $$\int { \max { \left( 1,{ x }^{ 2 } \right) dx } } $$
0
votes
0answers
20 views

Given a 2-variable function, prove that: $xz * z'_x-yz*z'_y = -\frac{1}{2}$

I'm kinda new to this material and was having a hard time solving this exercise, can anyone please show me the correct way of solving this? Let $z(x,y)$ be an equation that satisfies: $ ...