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

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10
votes
4answers
270 views

Calculate $f^{(25)}(0)$ for $f(x)=x^2 \sin(x)$

Calculate $f^{(25)}(0)$ for $f(x)=x^2 \sin(x)$. The answer is too short for me to understand, and the answer is $- 25 \cdot 24 \cdot 8^{23}$
1
vote
0answers
68 views

Why does it exist?

I can show that $\lim_{a \rightarrow \infty} \int_{-\infty}^{a} (\cos(tu)-e^{-\frac{t^2}{2}})e^{-\frac{u^2}{2}}du$ converges to 0 but I am not sure why this implies the convergence of $$\lim_{b ...
7
votes
3answers
274 views

How prove this $F'(x_{0})=f(x_{0})$ if $F(x)=\int_{a}^{x}f(t)dt,x\in[a,b]$

Question: let $f$ be Riemann integrable on $[a,b]$,Assmue that $x_{0}\in [a,b]$,and let $f(x)$ is continuous on point $x=x_{0}$,and define $$F(x)=\int_{a}^{x}f(t)dt,x\in[a,b]$$ show that ...
1
vote
1answer
44 views

Can we solve this equation?

Can we solve this equation or just prove that the solution exists: $$2^n = 100\log_2(n)$$
3
votes
1answer
150 views

Functions with the property: $f(\infty)=0$ and $f''(\infty)=\infty$

Please help me with an interesting question. There are functions $f: D \subset \mathbb{R} \to \mathbb{R}$ such that for some $a \in \mathbb{R}$ $f(a) = 0$ and $f''(a) = +\infty$; e.g., ...
0
votes
1answer
37 views

How can I define a “gradient discontinuous function”?

I am writing a report and need to know how I can define the "kink" in |x|. The function technically adheres to the definition of continuity, and the left and right limits appear to agree here... I ...
11
votes
4answers
399 views

When is the moment of inertia of a smooth plane curve is maximum?

Given a smooth plane curve $(x(s),y(s))$, parameterized in arc length $s$, of fixed finite length $L$, its moment of inertia about its center of mass (axis perpendicular to the plane) is given as $$MI ...
2
votes
2answers
94 views

Which functions satisfy $ g(x) d^n f(x)/dx^n = x^n/n!$ for all positive $n$?

Are there any $f(x)$ and $g(x)$ real functions which satisfy $$ g(x) \frac{d^n}{dx^n} f(x)= \frac{x^n}{n!}$$ for all positive $n$? EDIT: More generally, are there functions which satisfy $$ ...
3
votes
0answers
38 views

A simple question about Bohl functions, functions that are linear combinations of $t^ke^{\lambda t}$ where $\lambda \in \Bbb C, k\in\Bbb N_{> 0}$

A Bohl function is a linear combination of terms of the form $t^ke^{\lambda t}$ where $k$ is a non negative integer and $\lambda \in \Bbb C$. We denote the set of exponents of a Bohl function $p$ as ...
20
votes
1answer
363 views

$n^{th}$ derivative of a tetration function

I stumbled upon this very peculiar function last summer, namely: $f(x)=x^{x^{x^{...^{x}}}}$, where there is a number $n$ of $x$'s in the exponent, I tried to find the derivative for the function and I ...
20
votes
3answers
751 views

Who came up with the $\varepsilon$-$\delta$ definitions and the axioms in Real Analysis?

I've seen a lot of definitions of things like boundary points, accumulation points, continuity, etc, and axioms of the real numbers. But I have a hard time accepting these as "true" definitions or ...
0
votes
2answers
755 views

How to learn calculus for beginners? [duplicate]

As a precalculus student interested in teaching myself calculus, where should I start and how should I go about learning? This question is different than past questions as I am not solely interested ...
0
votes
2answers
22 views

One more statement regarding differentiation (one variable)

I will be glad to receive your thoughts about the following: If a function $f$ is differentiable in a neighberhood of $x=a$ then it must also be differentiable at $x=a$ itself .I guess it is true, ...
1
vote
2answers
137 views

I'm having problems solving this indefinite integral

$$\int \frac{1}{x+1} \left(\frac{x+1}{x}\right)^{2/3}dx$$ I have tried a $u$-substitution on the whole cube root thingy with $t$ but it did not work. I get $-3\int\frac{tdt}{(t^3-1)^2}$ and I don't ...
7
votes
0answers
175 views

Definite integral $\int_0^{2\pi}\frac{ab}{\sqrt{b^2\cos^2(\theta)+a^2\sin^2(\theta))}}\cos^2(\theta) d\theta$

Could you help me finding the following definite integral, with $a$ and $b$ constants? Thank you! $$\int_0^{2\pi}\frac{ab}{\sqrt{b^2\cos^2(\theta)+a^2\sin^2(\theta))}}\cos^2(\theta) d\theta$$
-1
votes
2answers
76 views

Differential Calculus, Slope at a Given Point [closed]

If $$3x^2 + 2xy + y^2 = 2$$ then what is the value of $dy/dx$ when $x = 1$?
0
votes
2answers
529 views

Finding the point on the curve at which the normal is parallel to a given line. [closed]

What is the point on the curve $y = \sqrt{2x+1}$ at which the normal is parallel to the line $y = -3x + 6$ ? This is based on tangents and normals. In differential calculus. I have choices here. ...
2
votes
4answers
115 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
4answers
169 views

Show that $\int_{-\infty}^{\infty} \frac{1}{x^4 + 1} \,dx = \frac{\pi}{\sqrt{2}}$ [duplicate]

Show that the following integral: $$ \int_{-\infty}^{\infty} \frac{1}{x^4 + 1} \,dx = \frac{\pi}{\sqrt{2}}. $$ I know this is $\pi/\sqrt{2}$ which I was found with Mathematica. I think I could utilise ...
0
votes
1answer
472 views

Find the normal plane and osculating plane

Find the equations of the normal plane and osculating plane of the helix $r(t) = cost\mathbf{i} + sint\mathbf{i} + t\boldsymbol{k}$ at the point $P(0,1,2)$
4
votes
2answers
158 views

Maclaurin expansion of arctan: convergence?

In my textbook, the Maclaurin series expansion of $\arctan{x}$ is found by integrating a geometric series, that is, by noting that $\frac{d}{dx}(\arctan(x)) = \frac{1}{x^2+1}$ then rewriting the ...
1
vote
3answers
93 views

Problem solving Integration problem2

How can we solve $\displaystyle\int\frac{1}{(x^2+1)^2}dx$ for Riemanian integration and Lebesque integration mode?
3
votes
4answers
334 views

Does the series $\sum^\infty_{n=1}\frac{n!}{\sqrt{(2n)!}}$ converge/diverge?

Does the series $\displaystyle\sum^\infty_{n=1}\frac{n!}{\sqrt{(2n)!}}$ converge/diverge? I used the ratio test but I'm not sure: $\begin{align} ...
1
vote
2answers
42 views

Proving with epsilon for a continuous function [duplicate]

Let f be a continous function in the section $[0,1]$. $f(x)>x \ \ \forall x\in[0,1]$. Prove that $\exists\epsilon>0 \ s.t \ f(x)>x+\epsilon :\ \forall x\in[0,1]$ From ...
10
votes
8answers
546 views

Proving convergence of a sequence whose terms are integrals

How to prove the following sequence converges to $0.5$ ? $$a_n=\int_0^1{nx^{n-1}\over 1+x}dx$$ What I have tried: I calculated the integral $$a_n=1-n\left(-1\right)^n\left[\ln2-\sum_{i=1}^n ...
1
vote
2answers
54 views

Prove $0 \le e^{-\theta x^2} \le 1$ for $0 \le \theta \le 1$

Why is it that $$0 \le e^{-\theta x^2} \le 1$$ for $0 \le \theta \le 1$? My textbook told me this in the context of langrange remainder for taylor series, and I can't figure it out. (Also, I don't ...
1
vote
1answer
24 views

Finding the continuity of a function with parameters

For which values of the parameters $\alpha,\beta$ is the function continuous: $$f(x)= \begin{cases}\begin{align} &(1+\alpha x)^{\alpha /x} & x>0 \\ &\beta &x=0 \\ & ...
1
vote
2answers
211 views

How to calculate volume of a cylinder using triple integration in “spherical” co-ordinate system?

Lets have a cylinder given by $x^2+y^2=1$ which is cut from the top by plane $z=2$ and bottom by $z=-2$.I am having problem regarding the limits of ρ for the equation ∭ ρ sin^2ϕ dρ dϕ dθ where ϕ is ...
0
votes
2answers
38 views

Proving that for a continuous function $\forall n: x_1…x_n\in(a,b):\exists x\in(a,b) \ s.t \ f(x)=\frac1n ( f(x_1)…f(x_n) ) $

We have a continuous function $f:(a,b)\to \mathbb R$ Prove that: $\forall n: x_1...x_n\in(a,b):\exists x\in(a,b)$ such that: $$f(x)=\frac1n ( f(x_1)+...+f(x_n) ) $$ Experience tells me ...
-1
votes
1answer
61 views

Prove some identites. [closed]

Prove the following identity. I do not know where to start. Any help would be most welcome. Thank you very much. $$ \int_0^1 \left[-\frac{1}{x} + \frac{1}{x\sqrt{1+x}}\right]dx=\sum_{l=1}^\infty ...
3
votes
3answers
57 views

Proving $\cot x =\alpha x$ has a solution $\forall \alpha>0$ in $(0,\frac\pi2)$

Prove $\cot x =\alpha x$ has a solution $\forall \alpha>0$ in the section $(0,\frac\pi2)$. Well let's define: $g(x)=\cot x -\alpha x$ I know that $\cot$ goes to infinity as x go to zero, and ...
3
votes
1answer
78 views

Proving that a polynomial $|P(x)|=e^x$ has a solution

Prove that $|P(x)|=e^x$ has a solution. $|P(x)|$ is a polynomial (that isn't $0$) and $x\in \mathbb R$. This is what I did: $g(x)=|P(x)|-e^x$ Now there are several cases: $g(0)=|P(0)|-1 ...
1
vote
2answers
51 views

Proving $(1-x)\cos x=\sin x$ has a solution in the section $(0,1)$

Prove that $(1-x)\cos x=\sin x$ has a solution in the section $(0,1)$ Well, this is what I did: $g(x)=(1-x)\cos x-\sin x \\ \displaystyle\lim_{0\neq x\to0}g(x)=1 , \ \lim_{1\neq x\to1}g(x)=-1 $ ...
3
votes
3answers
112 views

Proving that a continuous function has a solution $f:[0,1]\to \mathbb R$ [duplicate]

Let $f:[0,1]\to \mathbb R$ be a continuous function such that $f(0)=f(1)$. Prove that $f(x)=f\left(x+\frac12\right)$ has a solution for $x\in [0,\frac12]$. This question has to do with ...
0
votes
1answer
31 views

Prove a subset of $C^0$ is closed

I want to prove that the set $ A= \{ f \in C[-1,1]:( f(1)+f(-1)+ \int_{-1}^1 f(x)\,dx=0)\} $ is closed. I've already noticed that all the odd functions belong to it and I thought of considering a ...
1
vote
3answers
105 views

Understanding $n \left(\frac{2n \choose n}{4^n}\right)^2$ for large $n$

My answer over at cstheory.stackexchange.com involved the expression $$\lim_{n\to \infty} n \left(\frac{2n \choose n}{4^n}\right)^2$$ According to Wolfram Alpha, this expression is at most ...
1
vote
0answers
160 views

Random walk - expected distance not from origin

We have an assignment on random walk, but I can't figure out the expected value. The situation is as follows: In the origin there is a hunter that shoots at a duck, but misses. The duck starts at a ...
0
votes
1answer
42 views

A problem about a calculus equivalence of inflection point

The problem: Given $f: D \rightarrow \mathbb{R}$ a differentiable function on the interval $(a,b)$, and $g: D \rightarrow \mathbb{R}$ satisfying: $$g(x) = \begin{cases}\dfrac{f(x)-f(x_0)}{x-x_0} ...
0
votes
1answer
107 views

Cauchy's theorem, and Peano's theorem

We were asked to use Cauchy's mean value theorem In proving the following inequallities: $cosx < 1 - \frac{x^2}{2} + \frac{x^4}{24}$ for every $x \neq 0$ $sinx < x - \frac{x^3}{6} + ...
4
votes
2answers
111 views

Two notions of total derivative.

Let $f:\mathbb R^n\rightarrow \mathbb R^m$ be a function. By definition, $f$ is differentiable at $a$ if there exists a linear map $D_af:\mathbb R^n\rightarrow\mathbb R^m$ such that ...
0
votes
3answers
624 views

Integral $\int_{0}^{3} \frac{e^{-x^2}}{\sqrt{1-x^2}} \,dx$

I recently got stuck on evaluating the following integral, $$ \int_{0}^{3} \frac{e^{-x^2}}{\sqrt{1-x^2}} \,dx. $$ Is it possible to evaluate this integral in a closed form? I am not sure if there is ...
6
votes
0answers
114 views

How to solve this integral equation?

Solve this integral equation: $$ {{\rm e}^{{\rm i}k\,\sqrt{\vphantom{\Large A}\,r^{2} + z^{2}\,}\,} \over \sqrt{\vphantom{\large A}r^{2} + z^{2}\,}} = \int_0^{\infty}{\rm K}_{0}\left(\lambda r\right) ...
1
vote
1answer
42 views

number of solutions of a system of linear equations

Consider a system $\Sigma (y)$ of $m$ linear equations in $n$ variables $x_1,\cdots,x_n$: $\sum_{j=1}^n a_{i,j}(y)\cdot x_j=b_i(y)$, $i=1,\cdots,m$, whose coefficients $a_{i,j}(y)$ and $b_i(y)$ are ...
5
votes
3answers
163 views

Fixed Point Proof

*I would only like a hint! Not a full proof. Prove: If a function $f(x)$ is differentiable on $\mathbb{R}$ and $f'(x) < 1$ for all $x\in\mathbb{R}$, then $f(x)$ has at most one fixed point. So ...
1
vote
3answers
215 views

showing $f_n(x):=\frac{x}{1+n^2x^2}$ uniformly convergent in $\mathbb R$ using $\epsilon-n_0$

I want to show $$f_n(x):=\frac{x}{1+n^2x^2}$$ is uniformly convergent in $\mathbb R$ using $\epsilon-n_0$ argument. So $|f_n-f|\leq\frac{|x|}{1+n^2|x|^2}$ I know $|f_n-f|\leq\frac1n$ since ...
1
vote
1answer
305 views

Proof of a claim on a continuous function in [0,1] [duplicate]

This question has given me a huge headache, and I can safely say I hate everything about it. I need to prove, that for a continuous function in [0,1],which has the property that for every x in the ...
10
votes
5answers
2k views

Calculus self taught? Books?

I recently graduated with a degree in bachelor of science with a focus interactive and multimedia design. I had to opportunity to take 1 C++ course and 1 HTML course. I was also only required to take ...
0
votes
0answers
24 views

A property of summable sequence [duplicate]

Let $(a_n)$ be a summable sequence of positive real numbers. I would like to know whether or not we can find a sequence $w_n\to +\infty$ such that the sequence $(a_nw_n)$ is still summable?
4
votes
3answers
161 views

Prove $\lim_{x\to0}\cos(\frac{1}{x})$ does not exist

Prove $\lim_{x\to0}\cos\left(\frac{1}{x}\right)$ does not exist using $\epsilon$-$\delta$ proof. I think my professor wants this to be done by letting $L$ be arbitrary. Then have two cases: $L > ...
1
vote
1answer
73 views

Trouble finding the volume using the shell method

A region is bounded by the line $y=3x+4$ and the parabola $y=x^2$ and is rotated about the line $x=4$. First I found the limits of integration by finding the points of intersection. They are $(-1,1)$ ...