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

learn more… | top users | synonyms

1
vote
1answer
42 views

Let $f$ and $g$ differentiables such that $|f'(t)| \le g'(t)$, for all $t \in [0,1].$ Prove that $|f(1)-f(0)| \le g(1)-g(0)$

Let $f:[0,1] \rightarrow \mathbb{R}^m $ and $g:[0,1] \rightarrow \mathbb{R}$ differentiables such that $|f'(t)| \le g'(t)$, for all $t \in [0,1].$ Prove that $$|f(1)-f(0)| \le g(1)-g(0)$$ Comments ...
2
votes
1answer
22 views

Find a the value of a point on the tangent line

Suppose that the line tangent to the graph of $y = h(x)$ at $x = 3$ passes through the points $(-2, 3)$ and $(4, -1)$ with a slope of $-2/3$. Find $h(3)$. Hey guys, here's a question from my ...
4
votes
2answers
69 views

An easier way to prove this

Suppose $f\in C^1[a,b]$, and $f''$ exists on $(a,b)$. Show that, for any $c\in (a,b)$, there is $\xi\in(a,b)$ s.t. ...
1
vote
1answer
59 views

Is this equality true? Why? Why not?

Let $$ \lim_{a\to 0} \frac{1}{2} \left( \left( \sum_{n=-\infty}^\infty \frac{1}{(n+a)^2} - \frac{1}{a^2} \right) \right) = \sum_{n=1}^\infty \frac{1}{n^2}$$ I already know that ...
0
votes
3answers
22 views

How to solve this system of equations (Lagrange Multipliers)

I was doing a question on Lagrange multipliers and stucked when trying to evaluate the point. The system of equations that I can't solve is this: $$y^2-x^2+3x-3y=0$$ $$-y^2-yx+3y-xy=0$$ I just ...
0
votes
1answer
20 views

Finding hypervolume lying between Gaussian function and x-y-z plane over $\mathbb{R}^3$

Define the 3-variable Gaussian function by $G(x,y,z) = e^{-(\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2})}$. Find the hypervolume lying between this surface and the x-y-z hyperplane, over the ...
1
vote
1answer
26 views

Simplification of integration region. (Shuffle product?)

Let us define the usual $n$-dimensional simplex: $$\Delta_{a,b}^n = \{x_1,\dots, x_n\in [0,1]^n: a<x_1<\cdots <x_n<b\}.$$ Imagine we have an integral like: $$I:=\int_{\Delta_{a,b}^n} ...
1
vote
5answers
65 views

Finding $\lim_{n \rightarrow \infty} \frac{\log n}{\sqrt{n}}$

Compute $$\lim_{n \rightarrow \infty} \frac{\log n}{\sqrt{n}}$$ It seems pretty obvious, but I have tried Stolz-Cesaro and other tricks and I still can't get a solution.
-1
votes
1answer
33 views

if $\lim_{x \rightarrow \infty } |f(x)| = \infty$ then $\lim_{x \rightarrow \infty } f(x) = \infty$? [on hold]

Let $f: \Re \mapsto \Re$ be continuous function. How to prove that if $\lim_{x \rightarrow \infty } |f(x)| = \infty$ then $\lim_{x \rightarrow \infty } f(x) = \infty$ or $\lim_{x ...
0
votes
0answers
11 views

Vanishing Jacobian and solutions of an equation $X=\Psi(X)$

An equation is given: $$ X=\Psi(X) $$ where $X \in \mathbb{R}^n$ and $\Psi(0) = 0$. No other information on the equation is given. A material I'm reading (a physics material) says, without any ...
2
votes
1answer
19 views

Prove the series converges uniformly at $[x_0, \infty)$

Let $\sum_{n=0}^\infty a_ne^{-\lambda_n x}$, where $0 < \lambda_n < \lambda_{n+1}$. It is given that the series converges at $x_0$. Prove that the series converges uniformly at $[x_0,\infty)$. ...
7
votes
2answers
116 views
+50

How to find the inverse arc in the configuration space

The following Figure shows the function from configuration space (Torus) to operational space (Annulus). There is a naturally defined continuous function from configuration space $(\theta_A, ...
2
votes
1answer
39 views

Inverse of $3$ by $3$ matrix with non-constant entries.

I'm solving a question in nonhomogenous ordinary differential equation system $x'=Px+q$, and to solve my question I need to compute the inverse of the matrix $A=\begin{pmatrix}e^{-2t} & e^{-t} ...
0
votes
1answer
19 views

Bounding quantities that appear after using the residue theorem

for an exercise using the residue theorem I need to prove that this term $$\left|\dfrac{e^{R+it}-e^{-R-it}}{\left(e^{R+it}+e^{-R-it}\right)^2}\right|$$ tends to zero as $R\to\infty$. It's clear that ...
2
votes
2answers
57 views

Convergence of $\sum_{n=1}^{\infty} \int_n^{n+1} e^{- \sqrt x} dx$

Test the convergence of $$\sum_{n=1}^{\infty} \int_n^{n+1} e^{- \sqrt x} dx$$ Attempt: For sufficiently large $x$, we have $e^{-\sqrt x} > e ^{- x}$. I also tried solving the integral by By Parts ...
0
votes
3answers
59 views

Solve this limit $\lim_{x\to0} (1+x)^{\frac{1}{x}}$ [on hold]

How would I go about solving this limit ? $$\lim_{x\to0} \,(1+x)^{\frac{1}{x}}$$ I tried several times to solve it but I can't...
8
votes
2answers
85 views

Compute the limit $\lim_{n\to\infty}\frac{1}{\log n}\sum_{k=1}^n\left(1-\frac{1}{n}\right)^k\frac{1}{k}$

Similar to this problem, how can one compute the following limit: $$\lim_{n\to\infty}\frac{1}{\log n}\sum_{k=1}^n\left(1-\frac{1}{n}\right)^k\frac{1}{k}\quad ?$$ Note that $$\log x = ...
0
votes
1answer
43 views

Integration by parts problem

If $\textbf{x}\in \Omega \subset\mathbb{R}^n,$ where $\Omega$ is a bounded open set, $u:\Omega\rightarrow\mathbb{R}, \;\eta:\Omega\rightarrow\mathbb{R},\;u'=\nabla u = ...
0
votes
0answers
10 views

Specific utility (error) function for machine learning

I need a differentiable analog of following piecewise-defined function for machine learning application: $E=E(x,y)$ when $y=1$, $E=1/(x+1)$ when $y=-1$, $E=-1/(x-1)$ $y\in \{-1,1\}$ (two values, ...
1
vote
1answer
32 views

convergence of $\sum_{n=1}^\infty \int_0^{\frac {1}{n}} \dfrac {\sqrt x}{1+x^2} dx$

Test convergence of $\sum_{n=1}^\infty \int_0^{\frac {1}{n}} \dfrac {\sqrt x}{1+x^2} dx$ Attempt: Since, $\lim_{n \rightarrow \infty} \dfrac {1}{n} \in (0,1) \implies \sum_{n=1}^\infty \int_0^{\frac ...
2
votes
1answer
34 views

To find value of n using taylor series expansion [on hold]

Let $$ f(x)=\begin{cases} 0& -1\le x\le0\\ x^4& 0\lt x\le 1\end{cases} $$ IF$$ f(x)=\sum_0^n\frac{f^{(n)}(0)}{n!}(x)^n + \frac{f^{(n+1)}(c)}{n+1!}(x)^{(n+1)}$$ is the Taylor's formula for ...
2
votes
2answers
35 views

How do i optimize a function with an integral in it?

I want to find the values of $t_1 t_2...t_n$ which would give the maximum value for the function below: $\int_a^b \{f_1(x-t_1)+f_1(x-t_2)+f_1(x-t_n)\} dx$ functions $f_1f_2....f_n$ are expected to ...
3
votes
2answers
86 views

Does this infinum tend to infinity?

Let $f:\mathbb{R}^2\to\mathbb{R}$ be a continuous function satisfying $$\lim _{x\to +\infty}f(x,y)=+\infty\quad\text{for each fixed }y\in\mathbb{R}.$$ Further, let $\mathcal{I}\subset\mathbb{R}$ be a ...
0
votes
1answer
26 views

'Meaning' of a triple integral with $f(x,y,z)\neq 1$

I'm studying for my Calculus II exam, and this question came to my mind while I was practising integals with spherical coordinates. Probably this question doesn't have sense at all, but there's a ...
0
votes
0answers
15 views

Procentege of numbers extract more times. [on hold]

i can't find on your page one thing. Example : Lotton numbers extract every 5 minutes , and i need to know the procentege of the most numbers extracted in last 2 hours. So that my probability is ...
6
votes
1answer
53 views

How to prove $\lim_{n \to \infty}\frac{\pi}{2n+1}\sum_{k=1}^{n}(-1)^{k+1}\cot\frac{k\pi}{2n+1}=\ln2$

I am trying to prove the following: $$\lim_{n \to \infty}\frac{\pi}{2n+1}\sum_{k=1}^{n}(-1)^{k+1}\cot\frac{k\pi}{2n+1}=\ln2$$ I tried some values and it seems convincing. I wonder if this is a ...
-3
votes
0answers
40 views

A complex integral [on hold]

$$\Large\int_0^te^{-Ae^{\lambda u}-Be^{\lambda (t-u)}}du$$ There is an integral: A,B,λ are all constants, u is the variable of integration. Please help me to solve this problem.
1
vote
1answer
49 views

How do i find this limit as n goes to infinity $(1 - \frac {1}{3} ) (1 - \frac {1}{6} ) (1 - \frac {1}{10} )$ … $(1 - \frac {1}{n ( n+1) /2} )$

How do I find this limit as n goes to infinity $(1 - \frac {1}{3} ) (1 - \frac {1}{6} ) (1 - \frac {1}{10} )$ ... (1 - $\frac{1}{\frac12 n ( n+1)} )$
1
vote
1answer
62 views

Pullback of a differential form

My question is in regards to a proof in Lee's 'Introduction to Smooth Manifolds'. He proves a lemma about the pullback of a differential form on a manifold $N$, where $F:M\rightarrow N$ is a smooth ...
2
votes
5answers
68 views

the choice of 2 when proving the limit when $x\to\pm\infty$

Suppose that $f$ is a continuous function on $\mathbb{R}$ and $\lim_{x\to -\infty}f(x)$ and $\lim_{x\to -\infty}f'(x)$ exist. Show that $\lim_{x\to -\infty}f'(x)=0$ A common way to show this is ...
1
vote
3answers
53 views

Limit of logarithms exponential

$$ \lim_{x\to\infty}\biggl(\frac{\ln(x-2)}{\ln(x-1)}\biggr)^{x\ln x}. $$ L'Hopital seems like a very hardcore solutions given the situation.Are the any other options?
0
votes
2answers
20 views

domain of the function and it's simplified expression

I have 3 questions that I'm working on right now and it is these: give its domain and simply for each function and sketch the graph of function(I can sketch it by myself once I'm sure that it is all ...
1
vote
0answers
35 views

What is the intersection between $x + y - z = -2$ and $z^2 = x^2 + y^2$

I got the answer as $4x + 4y + 2xy + 4 = 0$ by substituting $z = x + y + 2$ into the second equation, but I feel as this is wrong since I am missing $z$ in the function. How do I approach this ...
3
votes
0answers
28 views

Verification of $\frac{d}{dt}(\hat{v}\cdot\hat{v})=0$

I would like to verify if the following is actually true: $$\frac{d}{dt}(\hat{v}\cdot\hat{v}) = \frac{d}{dt}\left|\left|\hat{v}\right|\right|^2 = 0$$ My thought on this is that since $\hat{v}$ is a ...
2
votes
0answers
26 views

Lagrange multiplier over two constraints

I'm having two constraints $g_{1}$=$x+y-z+2=0$ and $g_{2}$=$z^{2}-x^{2}-y^{2}=0$ and I want to determine the point on the intersection which is closest to the origin. The question asks us to use ...
0
votes
1answer
28 views

Draw $-dU(x)/dx$ for $U(x)$

It's been a little while since I've done any problems like this, but I just wanted to make sure I'm on the right track. Updated attempt:
0
votes
1answer
29 views

Pass the lower limit to $-\infty$ for an integral of positive function

Hello I have an very elementary calculus problem. Let $\phi(\eta)$ be a real value function satisfying \begin{equation} \phi(-\infty)=1,\quad \phi(+\infty)=0, \end{equation} Let $g$ be a positive ...
-2
votes
1answer
66 views

How to compute $\sum_{n\geq 0}\frac{\sin n}{n!}$?

I want to calculate the sum of $$\sum_{n\geq 0}\frac{\sin n}{n!}.$$ I think I am supposed to use the Taylor polynomial of $\ e^x$ but I don't know how to solve it. Thanks for your help.
1
vote
1answer
27 views

Is the taylor polynomial of degree $2$ near $(0,0)$ of $𝑓(𝑥, 𝑦) = \frac{1}{ 2 - (𝑥 + 𝑦^2)}$ the following:

$ P(𝑥, 𝑦) = \frac{1}{2} + \frac{𝑥}{4} + \frac{𝑥^2}{4} + \frac{𝑦^2}{2}$ Is this right? I can't tell, as I can't seem to see the remainder going to $0$ when divided by $x^2 + y^2$ as $(x, y) → ...
0
votes
1answer
49 views

Solving a Variable Separable Differential Equation

The equation is $$y'=\frac{1}{18}x(81-y^2)$$ with $y(0)=81$, and I have to solve for an equation of the form $y(x)$ So I do $$\frac{dy}{(81-y^2)}=\frac{1}{18}x \ dx$$ I integrate both sides, and ...
1
vote
0answers
34 views

How does integration by parts work with multivariable functions

How does integration by parts work with multivariable function? Lets say I have the functions $f(\textbf{x})$ and $g(\textbf{x})$, where $\textbf{x}\in\mathbb{R}^n$. How would integration by parts be ...
0
votes
1answer
18 views

Particle Motion/Mean Value Theorem

Here's a two-part question: "Consider the function $f(x)=2x^3−9x^2−24x+1$ on the interval $[−6,8]$. Find the average or mean slope of the function on this interval." What I did in my initial attempt ...
1
vote
0answers
32 views

Solving a homogenous system of linear ODE with Pauli matrices

I was asked to solve find a general solution to $\overrightarrow{x'}=P\overrightarrow x$ where $P=\begin{pmatrix} -1 & 2 \\-1 & 1\end{pmatrix}$. Using the "regular" method of finding the ...
2
votes
2answers
30 views

Lagrange Multipliers Calculus II Question

It is given me that $f(x,y)=x^2+-x+2y^2$ subject to $g(x,y)=x^2+y^2=1$ and asks for maximum and/or minimum. What I did... Equalize the partial derivatives and add the function $g$ to the system: ...
0
votes
0answers
17 views

Modulus of Green's function

Consider the nonlinear differential equation $$y'' = f(x,y,y')$$ together with the boundary conditions: $y(\alpha) = A$ and $y(\beta) = B$. Now $y(x)$ is a solution of this problem if and only if ...
0
votes
1answer
22 views

Finding the volume using cylindrical shells about the x-axis

So I have spent about a hour on this problem and figured it was time to ask for some advice. The problem is to find the volume using cylindrical shells by rotating the region bounded by $$8y = ...
1
vote
1answer
85 views

Convergence of $\sum_{n=2}^\infty \frac {1}{(\log n )^3}$

Test Convergence of $\displaystyle\sum_{n=2}^\infty \dfrac {1}{(\log n )^3}$ Attempt: I haven't been able to find a suitable comparator for $\dfrac {1}{(\log n )^3}$ . The integration test also seems ...
2
votes
4answers
77 views

Evaluate $\lim_{n \rightarrow \infty} \frac {[(n+1)(n+2)\cdots(n+n)]^{1/n}}{n}$

Evaluate $$\lim_{n \rightarrow \infty~} \dfrac {[(n+1)(n+2)\cdots(n+n)]^{\dfrac {1}{n}}}{n}$$ Attempt: Let $$y=\lim_{n \rightarrow \infty} \dfrac {[(n+1)(n+2)\cdots(n+n)]^{\dfrac {1}{n}}}{n}$$ ...
0
votes
1answer
62 views

Find roots of $ω^x+(ω^x)^2+1=x$ [on hold]

We have to solve this equation at complex numbers group $ω^x+ω^{2x}+1=x$ I tried to find the roots, which led to $x = 0 , 3 $ But $0$ isn't right
0
votes
0answers
13 views

Differentiation of an inclination function

Let $f: \mathbb{R} \to \mathbb{R}$ be a twice differentiable function. Define $F \colon \mathbb{R}^2 \to \mathbb{R}$ by $$F(x,y) = \begin{cases} \frac{f(x)-f(y)}{x-y} & x\neq y \\ f'(x) & ...