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

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0answers
8 views
-1
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2answers
25 views

Convergence of $\sum_{n=1}^{\infty}(1-n\sin\frac{1}{n})$

Can someone help me to understand how to find out if this series absolutely convergent and regular converges: $$\sum_{n=1}^{\infty}(1-n\sin\tfrac{1}{n})$$
0
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2answers
34 views

Simple calculus question (limits)

So I have to calculate the following limit $$\lim_{u\downarrow 1}\frac{\frac{2u}{3}-\frac{2}{3u^2}}{2\sqrt{\frac{u^2}{3}-1+\frac{2}{3u}}}.$$ I tried to use L'Hopitals rule, but it doesnt work it ...
0
votes
2answers
38 views

What is the derivative of $\arcsin(x/4)$?

I tried it and got $\frac{1}{4\sqrt{1-\frac{x^2}{16}}}$ But WolframAlpha is saying that the correct answer is $\frac{1}{\sqrt{16-x^2}}$ What did I do wrong, and what is the correct way of solving ...
1
vote
1answer
35 views

Approximate the value of the integral with an error less than $ 10^{-3}$

Approximate the value of the integral with an error less than $ 10^{-3}$ [Do not add the numbers in the sum!] $$\large \int_0^1 \sin (x^2)dx$$ So this is what i have tried and am stuck from there. ...
1
vote
1answer
23 views

Series convergence and Big O

I am trying to prove that if there exists $\theta \in \mathbb{R}$ such that $f(n) = \mathcal{O}(n^{\theta})$, then $\sum\limits_{n=1}^\infty \frac{f(n)}{n^s}$ converges. Intuitively it makes sense ...
-4
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1answer
31 views

Find the nth derivative of $x/(x^2 +1)(x+2)$

Find the nth derivative of $\dfrac{x}{(x^2 +1)(x+2)}$, Pls show me the step by step solution. I got the partial fraction decomposition as $\dfrac{2x+1}{5(x^2 +1)} + \dfrac{2}{5(x+2)}$. Can't figure ...
0
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0answers
12 views

Calculating the surface integral of a vector field.

We have a vector field, $\vec{F}$, defined by $F=\nabla \wedge A$ where A is $$A = \left(\begin{matrix} yz^2\\ -3xy \\ x^3y^3 \end{matrix}\right)$$ I get $F$ to be $$F= \left(\begin{matrix} 3x^3y^2\\ ...
2
votes
1answer
38 views

Application Stokes's Theorem

I am a bit unsure the way Stoke's theorem is applied in this case. Evaluate $\oint\limits_C {xydx + yzdy + zxdz} $ around the triangle with vertices $(1,0,0), (0,1,0), and (0,0,1)$, oriented ...
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0answers
47 views

Problem 6 of calculus [on hold]

I am having a hard time on problem 6 in the calculus book. How do you arrive at this result?
1
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1answer
27 views

Finding Recursive Function

Let $f(x)=e^\frac{-1}{x}$ Prove in induction that the general form of the n-th derive is: $$f^{(n)}(x)=P_n(\frac{1}{x})\cdot e^\frac{-1}{x}$$ For $n=0$: $P_0(x)=1$ Assume for n: ...
7
votes
4answers
174 views

solution to differential equation from deriving power series

Find the solution of the differential equation $$y'= 2xy$$ statisfying $y(0)=1$, by assuming that it can be written as a power series of the form $$ y(x)=\sum_{n=0}^\infty a_nx^n.$$ Im advised to ...
2
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0answers
32 views

Show that the set of all points $x \in \mathbb R$ where $f$ is differentiable is definable in $\mathcal M=(\mathbb R; +,-(), \cdot, \lt, 0,1,f)$

For the structure $\mathcal M=(\mathbb R; +,-(), \cdot, \lt, 0,1,f), n_f=1 $ show that the set of all points $x \in \mathbb R$ where $f$ is differentiable is a definable set. My issue here is how to ...
-1
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1answer
15 views

Concavity and quasiconcavity…

How do you explain the difference between concavity and quasi concavity? or convexity and quasi convexity?
2
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1answer
32 views

Understanding the Definition of a derivative as slope of a tangent line

I'm trying to understand the derivative and am wondering why the derivative is described as the slope of the tangent line and not the slope of a function itself. Say $f(x) = 2x+5$ where ...
3
votes
3answers
68 views

Solving with integration by parts: $\int \frac 1 {x\ln^2x}dx$

Solving: $$\int \frac 1 {x\ln^2x}dx$$ with parts. $$\int \frac 1 {x\ln^2x}dx= \int \frac {(\ln x)'} {\ln^2x}dx \overset{parts} = \frac {1} {\ln x}-\int \frac {(\ln x)} {(\ln^2x)'}dx$$ $$\int ...
1
vote
1answer
22 views

Proving Two Taylor Polynomials Are Equal

I am trying to prove the Following: Let there two polynomials: $p(x),q(x)$ at a degree on $n$ at most, and $$f(x)=p(x)+o(x-x_0)^n=q(x)+o(x-x_0)^n$$ therefore $p(x)=q(x)$ I have come to the ...
1
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4answers
33 views

Limit of a rational function with radicals [on hold]

How do I solve this limit: $$\lim_{x\to0}\frac{\sqrt{x^2+p^2}-p}{\sqrt{x^2+q^2}-q}$$
1
vote
1answer
16 views

How we can find $A_{(\Gamma_f)}$?

We have $f,g:[-4,4]\rightarrow\mathbb{R}$, $f(x)=x^2+2$ and $g(x)=x+4$. We need to find the crowd area between the graphs f and g. I know that $A_{(\Gamma_f)}=\int_a^b|f(x)|dx$ but in this case how ...
0
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0answers
51 views

Evaluate $\large \int_0^1\left(\frac{1}{\ln x} + \frac{1}{1-x}\right)^2 \mathrm dx $ using elementary, high school techniques [duplicate]

Evaluate $\large \int_0^1\left(\frac{1}{\ln x} + \frac{1}{1-x}\right)^2 \mathrm dx $ $$$$ I was given this integral by a friend who saw this here on MSE. He asked me if I could solve it using the very ...
1
vote
1answer
26 views

Proving a statement about probability theory

Let X be arandom variable. Consider any constant $c\gt 0$ how to prove the following satement $$\sum P(|X|\ge cn) \lt \infty \iff E(|X|)\lt \infty $$ My answer trail: $E[|X|]=\sum_X|X|P_x(X)\lt ...
2
votes
2answers
31 views

nonnegative Riemann-integrable function, infimum

$f$ is a nonnegative Riemann-integrable function on $(0,1)$ and $f(x)\ge\sqrt{\int_0^xf(t)dt}$ for $x\in(0,1)$. Find $\inf\frac{f(x)}{x}$ I have no idea how to work out the assumption, for equality ...
1
vote
2answers
33 views

Convergence of series 4

Determine if the following series is convergent or not: $$\frac{1}{\sqrt{n} \log n}$$ I tried: $a_k = \frac{1}{\sqrt{n} \log n}$ $b_k= \frac{1}{\sqrt{n}}$ then did: $\frac{a_k}{b_k}$ and got ...
0
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0answers
10 views

proof with value functions

I have a following setting ; $$V_{1}\left(S\right)=V_{1}^{-}\left(S\right)+V_{1}^{+}\left(S\right)$$ where $V_{1}^{-}\left(S\right)$ is the value function before time $T_{1}^{*}$ and ...
1
vote
0answers
16 views

does this yield convergence to smooth limit function?

let $u: M \times [0,T) \mapsto \mathbb{R}$ with $M$ compact Riemannian manifold (but would also be helpful first to just consider compact $K \subset \mathbb{R}^n$), and $u$ smooth as a function of $M$ ...
-1
votes
3answers
44 views

Maximum & minimum area of rectangle outside another.

Find the maximum & minimum area of an outer rotated rectangle when the inner rectangle has the side lengths $a$ and $b$. Here's an image: I have already tried to relate the side of ...
2
votes
1answer
47 views

If a polynomial maps a region onto a neighborhood of zero, does it follow that it has a zero in some “robust” sense?

Let $B^n\subseteq\Bbb R^n$ be a unit ball, $P: B^n\to\Bbb R^m$ is polynomial in each component, and assume that the image of $P$ contains $0$ in its interior. Does it follow that for some $\epsilon$, ...
0
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0answers
18 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
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4answers
38 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
80 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
4
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1answer
26 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, an 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
48 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
58 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
42 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
votes
2answers
46 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
20 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 ...
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0answers
30 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
21 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
225 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
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0answers
38 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
56 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 ...
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votes
2answers
39 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
49 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
14 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
39 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
votes
2answers
75 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
13 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 ...
11
votes
4answers
164 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
21 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
vote
0answers
40 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 $(f_n)$ ?