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

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0
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0answers
10 views

follow-up question to Hake's theorem in Bartle's book

My question is based in here. Why is it that $b$ forces to be a tag of $[x_{m-1},b]$? I can't get the right trick. Can you please give me some hints? Thanks
0
votes
3answers
25 views

Values of $a$ for which $ f(x)=8 a x-a \sin6x -7x - \sin 5x $ increases

Please help me in this question: Find all the values of the parameter $a$ for which the function $f(x)= 8 a x-a \sin6 x -7 x - \sin 5 x $ increases and has no critical points for all real $x$. I ...
2
votes
3answers
64 views

Calculate an integral depending on n

Is there a way (simple or not) to calculate the following integral? $$\int_{-1}^{1} \sqrt[n]{1-x^n} dx$$ Thanks
2
votes
1answer
18 views

Can I interpret the exponential of the derivative operator, $e^D$, as infinite shift operators each shifting “infinitesimally”?

To better explain what I mean, and example can be very useful. Consider $e^{i\theta}$. We could express this using the series definition or the limit definition of $e^x$ instead: $$e^{i\theta} = ...
1
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3answers
40 views

How to solve this particular indetermination: $0*\infty$

The limit in question is: $$\lim_{x\to\infty} 2n\left(\sqrt{n^6+5n^2}-n^3\right)$$ By looking it up on wolfram alpha I found out the answer is 5 but I am not so sure how to arrive to it. I tried to ...
1
vote
3answers
56 views

taking the limit $\lim\limits_{n\rightarrow \infty} {\frac{(3^{n+1} + 4)(7^n-47)}{(7^{n+1}-47)(3^n +4)} }$

I need help with a guide on how i will deal with this kind of problem.. This a part of my solution in series convergence. I find it hard taking the limit of this: $$\lim_{n\rightarrow \infty} ...
1
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1answer
39 views

Spivak Calculus Ch. 19 #15

(a) Find $\int \sin^4 x\, dx$ in two different ways: first using the reduction formula and then using the formula for $\sin^2x$. (b) Combine your answers to obtain an impressive ...
2
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2answers
39 views

$\mathrm{d}f(x,t)$ this way $d\big(\,f(t,x)\big)=\frac{\partial f}{\partial t} \,dt+\frac{\partial f}{\partial x}\,dx$?

If $dX_t=a_t \,dt$ the next procedure is correct? $$\mathrm{d}\big(\,f(t,x)\big)=\frac{\partial f}{\partial t} dt+\frac{\partial f}{\partial x}dx=\frac{\partial f}{\partial t} dt+\frac{\partial ...
1
vote
0answers
15 views

finding Interior Points of a set

In the normed space $(\mathbb{R}^2, ||(x_1,x_2)||:=|x_1|+|x_2|)$ I want to find Interior Points of $$ \{ (x,1/n) ~~\big|~~ x\in \mathbb{R} \text{ and } n\in \mathbb{N} \}. $$ I guess that the ...
-3
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0answers
29 views

Change the subject of a formula [on hold]

$150 \cdot 10^6 = \dfrac{3pR^2}{4t^2}$ How do I find out what $t$ is, hence make it the subject of the equation. I think I know what the answer should be: $p=1.5 \cdot 10^6$ $R= 0.075$ ...
0
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0answers
16 views

$\int_{0}^{+\infty}\frac{x^2+x^3}{x^2\cdot \left | x-1 \right | \cdot \frac{3}{4} \left |x-4 \right |^{\frac{4}{3}}} dx$ convergence

Does $$\int_{0}^{+\infty}\frac{x^2+x^3}{x^2\cdot \left | x-1 \right | \cdot \frac{3}{4} \left |x-4 \right |^{\frac{4}{3}}} dx$$ converge? Domain of this integrand is $x \in \mathbb{R} : x\neq 0, ...
3
votes
3answers
214 views

double integral $\int_0^t \int_0^s \frac{\min(u,v)}{uv} \, dv \, du$

I want to calculate the double integral: $$\int_0^t \int_0^s \frac{\min(u,v)}{uv} \, dv \, du$$ I don't know how to o that even if it seems simple. Thanks in advance for your help
0
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0answers
32 views

some series about :On some strange summation formulas by R. William Gosper

I read the paper On some strange summation formulas by R. William Gosper and I looking the following series maple could sum any idea how to get it thanks $$\sum _{z=1}^{\infty } \frac{(-1)^z \cos ...
2
votes
2answers
55 views

For which $x\in\mathbb{R}$ is the series of general term $a_n = x^{n!}$ convergent?

I firstly found the simplified form of $\frac{a_{n+1}}{a_n} = |x|\cdot|x^n|$ and used this to establish the end points $-1\lt x\lt 1$. I then tested the end points by finding the limit to infinity of ...
-2
votes
2answers
35 views

For which $x$ the inequality $ax+be^{x/2}>c$, where $a,b,c,x>0$ holds [on hold]

For which $x$ the inequality $ax+be^{\frac{x}{2}}>c$ where $a,b,c,x>0$ holds. Can someone help me for this. Thank you.
0
votes
1answer
48 views

How is the degree of a polynomial defined? $a_1+a_2x^2+\cdots+a_nx^{n-1}$ has degree $n$ or $n-1$?

I have this polynomial: $$a_1+a_2x^2+\cdots+a_nx^{n-1}$$ or: $$a_0+a_1x^2+\cdots+a_{n-1}x^{n-1}$$ What is degree of those polynomials? $n$ or $n-1$, I'm little bit confuse... Thank you!
0
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0answers
13 views

Ito Formula for Poisson Process: $d{X_t}=a_t dt +b_t dN_t$

Let $X_t$ solve the SDE $d{X_t}=a_t dt +b_t dN_t$, where $N_t$ is a Poisson Process. I want to demosntrate that in this case the Ito formula is the next one, but I dont know how to achieve it. ...
3
votes
1answer
36 views

At which points is the following function differentiable

The following function is a standard example for a function whose points of discontinuity are strange: $f(x) = \begin{cases}1/q& \mbox{ if } x=p/q \mbox{ and }p/q \mbox{ is a fully reduced ...
-1
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2answers
66 views

How to prove that $f_n(x)=\frac{nx}{1+n\sin(x)}$ does not converge uniformly on $[0, \pi/2]$? [duplicate]

If $f_n$ is a sequence of functions over $[0, \pi/2]$ given by $$f_n(x) = \frac {nx} {1+n\sin(x)},$$ then how would I go about proving that $f_n$ does not converge uniformly to a function $f$ on ...
3
votes
1answer
12 views

reducing a pde to a canonical form

I'm really struggling with this one and I can't seem to find what's wrong with my approach. I am given a PDE in the form $$U_{xx} + x y U_{yy} = 0,$$ and I am supposed to bring it to its canonical ...
10
votes
4answers
142 views

Prove that $\sinh(\cosh(x)) \geq \cosh(\sinh(x))$

Prove that $$\sinh(\cosh(x)) \geq \cosh(\sinh(x))$$ I tried to tackle this problem by integrating both lhs and rhs, in order to get two functions who show clearly that inequality holds. I've ...
3
votes
3answers
20 views

trying to prove the following convergence result

So, this is propably some standard result from integral calculus: Let $f:\mathbb{R} \mapsto \mathbb{R}$, $f \geq 0$ such that $\int^\infty_0 f < \infty$, and $|\frac{d}{dx} f| \leq C$ for all x ...
1
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0answers
33 views

How to Find the pointwise limit of $(f_n)$

For $x \in [0, \pi/2]$, if $$f_n(x) = \frac {nx} {1+n\sin(x)}$$ how do you find the pointwise limit of $(f_n)$ ?
0
votes
1answer
34 views

Ito Formula (Poisson basic process)

Let, $N_t$ be a Poisson process and let $X_t$ solve the SDE $d{X_t}=a_t dt +J_t dN_t$. Then, Ito´s fórmula is: $$df(t,X_t)=(\frac{\partial f}{\partial t} + \frac{\partial f}{\partial x}a_t)dt + ...
-1
votes
0answers
84 views

Greek School Exams-Calculus problem [on hold]

This problem was posed yesterday - along with 3 others of lesser difficulty - on the Greek national exams for the 3rd grade of Lyceum. This is the final class that determines University success. The ...
1
vote
1answer
25 views

One-Dimensional Jump-Diffusion Ito’s Formula

Let, $N_t$ be a Poisson process and let $X_t$ solve the SDE $d{X_t}=a_t dt +J_t dN_t$. Then, what is the correct Ito´s fórmula: i)$df(t,X_t)=(\frac{\partial f}{\partial t} + \frac{\partial ...
0
votes
3answers
79 views

How do I calculate $ \int_{1}^{3} x/(2-x) \;\mathrm{d}x$

$ \int_{1}^{3} \frac{x}{2-x} \;\mathrm{d}x$ $ \int_{1}^{2} \frac{x}{2-x} \;\mathrm{d}x$ + $ \int_{2}^{3} \frac{x}{2-x} \;\mathrm{d}x$ $u = 2-x$ $\lim_{e\to0} \left[ \int_{-e}^{1} \frac{2-u}{u} ...
5
votes
2answers
70 views

Geometric intuition for derivatives of basic trig functions

I was inspired by this question to try and come up with geometric proofs for the derivatives of basic trig functions--basically, those that have simple representations on the unit circle ($\sin, \cos, ...
3
votes
2answers
272 views

Calculating the integral $\int_{1/3}^{3}\frac{\arctan(x)}{x^2-x+1}\;\mathrm{d}x$

Can somebody help me calculate the following integral: $$\int\limits_{1/3}^{3}\frac{\arctan(x)}{x^2-x+1}\;\mathrm{d}x$$ I have tried integration by parts, but I got stuck in it. Wolfram also didn't ...
0
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6answers
62 views

De Moivre's Theorem (Trigonometry)

How to prove that $\cos^4 \theta+\sin ^4\theta=\frac{1}{4}(\cos4\theta+3)$ by using De Moivre's Theorem? I know that $(\cos\theta+i\sin\theta)^n=\cos n\theta+i\sin n\theta$, but how to apply this ...
1
vote
2answers
67 views

How to find this limit $\lim\limits_{(x,y) \to (1,1)} \frac{y-x^4}{y^3-x^4}$ [on hold]

How would I find this limit? $$\lim_{(x,y) \to (1,1)} \frac{y-x^4}{y^3-x^4}$$
2
votes
3answers
53 views

Which function can be used for Substitution

Find the value of $$I=\int_{0}^{\frac{\pi}{2}}\left(\sin(x)-\cos(x)\right)\,\log(\sin(x))dx$$ Method $(1)$. I splitted up the Integral into two Integrals as $$I=I_1+I_2$$ where ...
0
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3answers
62 views

De Moivre's Theorem (Trigo)

Prove the trigo identity by using method based on De Moivre's Theorem. $\sin^6\theta=\frac{1}{32}(10-15\cos2\theta+6\cos4\theta-\cos6\theta)$ My attempt, Using $z-\frac{1}{z}=2i\sin \theta$ ...
2
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0answers
33 views

Continuous function on compact subset of $\mathbb R$ to itself has a fixed point.

Let $f:[a,b] \to [a,b]$ be continuous. Then $f$ has at least a fixed point. I read the following proof from Limaye book. Define $F(x)=f(x)-x.$ Since $a \leq f(x) \leq b,\ \quad F(a)\leq 0 \ \quad ...
3
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0answers
21 views

Identification of a function

I recently came across the following function $$\sum_{k=1}^\infty(\log(k))^n\frac{z^k}{k^n}$$ I found it while dealing with the polylogarithm function, $Li_n (z)$ (Notice that if instead of ...
2
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1answer
22 views

Extreme values of a two-variable continuous function on a connected closed domain

I wonder if there exists a continuous two-variable function on a connected closed domain which has only two extreme values, one of them is local maximum and another is local minimum but the local ...
4
votes
2answers
67 views

How do ideas in differential geometry expand upon ideas from introductory calculus

I just went through first year in mathematics and used Stewart's book for calculus. I am trying to self study differential manifold and I find many concepts such as chart, atlas very similar to that ...
0
votes
1answer
60 views

Solution for a differential equation

I am stuck in getting the solution for the following non-linear differential equation: \begin{equation*} x^2 + B\frac{dx}{dt} = A\sin(wt) \end{equation*} Is there any method to solve this kind of ...
-1
votes
0answers
37 views

The Coin-Exchange Problem (Application of the Residue Theorem) [on hold]

These day, I have met a problem about application of the Residue Theorem, see section 10.4 of enter link description here.Could anybody help me solve it? (The Coin-Exchange Problem) Suppose $a$ and ...
0
votes
2answers
33 views

First order differential equation with initial conditions

I solved the differential equation $$\frac{dy}{dx} = \frac{x}{x^2+1}$$ to get the general solution $$y = \frac{ln|x+1| +c}{2}$$ Im given the initial condition $$yy' − 2e^x = 0, y(0) = 3$$ but ...
0
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0answers
22 views

Integrating $\operatorname{Log}(z+2)$ along the unit circle [duplicate]

For the function $f(z) = \operatorname{Log}(z + 2)$, where we choose the principal branch of logarithm (namely, $−\pi < \operatorname{Arg}(z) < \pi$), and the contour $C := \{z \in ...
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votes
2answers
39 views

Being a fresher i wanted to understand this [on hold]

So this is Calculus two and i have been having diffculties with math and our teacher asked us to do this, and i just want to give up in math but i think it is important to try. What is given.. Compute ...
1
vote
1answer
32 views

Question about conditions for conservative field

Question about conditions for conservative field In common textbooks' discussions about conservative vector field. There is always two assumptions about the region concerned, namely the region is ...
3
votes
1answer
20 views

How to calculate the continuum limit of a discrete system?

The question is based on the following excerpt from the book "Symmetries and Integrability of Difference Equations" Link: Book Excerpt Consider the discrete equation ...
-3
votes
1answer
19 views

two ways to find these vectors ortho? [on hold]

What are two ways I can show $r'(t)$ is orthogonal to $r'(t)$? With $$r(t)=(f(t),g(t),h(t))$$, that is on a curve on a sphere I tried showing that the dot of the two were equal to $0$.
2
votes
2answers
70 views

Integrating a square's perimeter to get its area

I am trying to wrap my head around some integration applications. I went through the exercise of integrating the circumference of a circle, $2*\pi*r$, to get the area of a circle. I simply used the ...
0
votes
1answer
26 views

Help understanding the result of a formula

I need some help understand the middle section of this formula. $$OA^2 = (100-40)^2 + 50^2=10^2(61)\to OA = r = 10\sqrt{61} $$ and $$\sin(\angle OCB ) = \frac{30}{r} = \frac{3}{\sqrt{61}}, ...
1
vote
2answers
45 views

If $\frac{\partial F^i}{\partial x^j}=0$ on a connected open set, is $F$ constant?

Let $U$ be open in $\mathbb{R}^n$ and let $$F:U\to \mathbb{R}^m$$ be a smooth map, i.e. $F\in C^\infty(U)$. It is easy to prove that if $U$ is convex and $$\frac{\partial F^i}{\partial x^j}=0\tag{1}$$ ...
-8
votes
1answer
52 views

Brilliant formulaes [on hold]

Hey Brilliant mathematician, i am very honored for having your time. I need general Formulas on breaking down a number to a different and being able to derive that number back, my requirements is to ...
1
vote
1answer
49 views

How many real numbers satisfy the following

How many real numbers satisfy: $$\sin x=\frac{x}{100}$$ I don't know where to start it how to do this at all. Can someone please help me?