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

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3
votes
0answers
180 views

Prove an image of function $f:[a,b]\to\mathbb R^n:t\mapsto(f^1(t),f^2(t),\ldots,f^n(t))$ where $f^i\in C^\infty$ doesn't contain open ball

$\mathbb R^n\supset[a,b]$ is domain of definition $f:[a,b]\to\mathbb R^n:t\mapsto(f^1(t),f^2(t),\ldots,f^n(t))$ where $f^i\in C^\infty$ I need to prove that image of $f$, that means $f[a,b]$, doesn't ...
3
votes
0answers
297 views

Fastest convergence Series which approximates function

The question is the following: Is there any proof that shows that the Taylor series of an analytical function is the series with the fastest convergence to that function? The motivation to this ...
3
votes
0answers
137 views

Quantized calculus

I have heard about a generalization of the calculus named 'quantized calculus'. In this calculus the derivative is defined as $$ df= [F,f]=Ff-fF $$ Here $ F(g(x))= ...
3
votes
0answers
768 views

Spherical coordinates grad and div.

Struggling with the following: Prove the identity $$ \nabla = e_{r}(e_{r} \cdot \nabla) + e_{\theta}(e_{\theta} \cdot \nabla) + e_{\phi}(e_{\phi} \cdot \nabla).$$ Given the vector fields ...
3
votes
0answers
1k views

Find the shortest distance from the point (1, 1, 1) to the surface z = xy

This was a question on my math exam. I received zero points, so I obviously don't know how to do it. How would I go about this: Find the shortest distance from the point $(1, 1, 1)$ to the surface ...
3
votes
0answers
86 views

Notation for a certain kind of discrete measure

Suppose $\phi:\mathbb{R}^n \rightarrow \mathbb{R}$ is smooth, $Z=\{x: \phi(x)=0\}$ and $D\phi\neq0$ on $Z$. Is anyone familiar with use of the notation $dZ$ for the measure $$\sum_{x \in Z} ...
3
votes
0answers
188 views

Maximum size of a rotated-then-cropped rectangle

With regard to topic/question New size of a rotated-then-cropped rectangle: The answer by Isaac, the maximum area is $b^2\csc\alpha\sec\alpha$ when $x=0.5b\csc\alpha = 0.5b/\sin\alpha$ seems to ...
3
votes
0answers
110 views

Measurability of the derivative

A local analysis textbook I have used has the following exercise: Let $X$ be a finite-dimensional, $Y$ a separable Banach-space, $f\colon X\rightarrowtail Y$ any function. Show that $f'$ is ...
2
votes
0answers
23 views

Proof involving the Cauchy mean value theorem, Taylor series and induction.

I've been working on this problem for a long time and I'm stuck on it (nothing that I come up with is working). If someone could give me a hint or do the first part of the proof, that would be nice. ...
2
votes
0answers
16 views

On the uniform convergence of generalized integral

Is the integral $$ \int_{1}^{\infty} e^{-yx^2}\sin{y}dx.$$ uniformly convergent in $y \in [0,\infty]$? Why or why not?
2
votes
0answers
37 views

Green's Theorem

Hey guys I am having difficulties in problem 5. I thought I understood it, but I suppose I was mistaken. I will now explain what I planned to do to solve this problem and where I got stuck. So I ...
2
votes
0answers
37 views

Differentiation under integral sign

There is this integral that I used a lot in my research: $$\int_{-\infty}^{\infty}\exp\left(-b^{2}\left(x-c\right)^{2}\right)\mathrm{erf}\left(a\left(x-d\right)\right)\,\mathrm{d}x = ...
2
votes
0answers
51 views

$\sum_{n=-\infty}^\infty e^{-\alpha n^2+\beta n}$

Hi I am trying to calculate the sum given by $$ \sum_{n=-\infty}^\infty e^{-\alpha n^2+\beta n}=\ = \sqrt{\frac{\pi}{\alpha}} e^{\beta^2/(4\alpha)} ...
2
votes
0answers
38 views

Estimate $\displaystyle\left|\int_{\frac{\pi}{k}}^{\frac{\pi}{2}} (\sin \theta)^{-1+\frac{i(n+1)y}{2}} d\theta \right|$

I have to estimate the following integral $$\left|\int_{\frac{\pi}{k}}^{\frac{\pi}{2}} (\sin \theta)^{-1+\frac{i(n+1)y}{2}} d\theta \right|,\quad \forall k,n\geq 2 $$ According to Sogge (Oscillatory ...
2
votes
0answers
15 views

Quick way to determine the number of horizontal asymptotes

I understand how to calculate horizontal and vertical asymptotes, both by using the trick of comparing the degrees of the numerator/denominator and by using calculus. What I would like to know is ...
2
votes
0answers
27 views

Implicit Differentiation Proper Answer?

I'm studying for my calc final tomorrow, and am going through some practice questions and I'm not sure if the solution is wrong or I'm just miss understanding. The question is about finding a tangent ...
2
votes
0answers
41 views

Is there a generalization of integration by parts?

here is what i concerned: there are $u(x)$ and $v(x)$ in the original integration by part formula, what if the integral involve with one more function $w(x)$. Second of all, i know that there are ...
2
votes
0answers
66 views

Calculus book advice

I'm reading Thomas Calculus now but I don't think it includes Mellin transform or Riemann-Stieltjes integration... Can you recommend an advanced calculus book which includes all of this stuff?
2
votes
0answers
35 views

Integral inequality related to derivation

While trying to understand a proof, i have stumbled upon the following statement: Let $f \in L^p(a,b)$ be a $p$-integrable function. Then the inequality $$\liminf_{s \rightarrow t} \frac{1}{t-s} ...
2
votes
0answers
33 views

Solve by separating variables

$$\frac{dy}{dt}=e^y +1$$ I've tried: $$dy/dt - e^y = 1 $$ $$\Leftrightarrow y' - e^y dt = 1 dt$$ But I'm not sure what to do next or if I'm even doing this right!
2
votes
0answers
59 views

Integral $\int_0^\infty \frac{x^n}{(x^2+\alpha^2)^2(e^x-1)^2}dx$

Hey I am trying to integrate $$ \int_0^\infty \frac{x^n}{(x^2+\alpha^2)^2(e^x-1)^2}dx,\quad \alpha,n \geq 1. $$ Thanks. This integral is old and has a 4th order pole. I am also looking for literature ...
2
votes
0answers
20 views

Direct Partial Integration - How do I solve this?

I have two questions that I need to work out and I have lost my notes, so I have no idea how to go about solving these: $\dfrac{\partial u}{\partial t}=2t\cos\theta$ with boundaries of $u=2t$ and ...
2
votes
0answers
39 views

Exponential integral equation

I want to find the function $q(x)$ such that the following integral equation is satisfied for all $s\ge 0$ $$ \frac{p}{as + 1}\int_0^\infty \exp \left( -\frac{xs}{as + 1} \right)q(x)dx = ...
2
votes
0answers
31 views

Improper integrals in curve length

I am supposed to find the length of curve of the following: $ y = \sqrt{2-x^2}$ ; $0\le x\le 1$ $y =\ln(\cos x) $; $0\le x\le \frac{\pi}{3}$ I followed the directions found from this question : ...
2
votes
0answers
42 views

Wrong answer within 'Calculus Solution Manual, Michael Spivak, 3rd ed'

I have a problem with the answer provided in the solution manual of Calculus, Michael Spivak, 3rd ed, The Problem: Consider a hyperbola, where the difference of the distance between the two foci is ...
2
votes
0answers
48 views

Second order nonlinear ordinary differential equation. Help please

Can someone help me with this differential equation $$ay''(t)y(t)+2y'(t)=\left(b+\frac{c}{t^2}\right)y(t)^2$$
2
votes
0answers
24 views

Decimal notation and serie convergente

Show that the series of the sum $\frac{1}{n}$ where $n$ runs through the set of integers not containing 3 in their decimal notation is convergent. An integer A with $n$ figures is written, in ...
2
votes
0answers
24 views

Implicit differentiation and rules

I'm supposed to write the rules used for some differentiable functions. I got all of them correct except for the last one which is $d(x^c)$. I put in $cx^{c-1}$ because I thought it was the power ...
2
votes
0answers
22 views

$C^\infty$ function which is constant on two intervals and implicitly defined inbetween

My textbook states without further reference that there exists a $C^\infty$ function $\alpha:\mathbb{R}\to\mathbb{R}$, such that $|\alpha'(t)| < K$ where $K>2$, and $$ \alpha(t) = \begin{cases} ...
2
votes
0answers
54 views

About derivatives

Let $f \in C^1(\mathbb{R})$ a monotonic function such that $$\lim_{x \to \infty} f(x) = m \in \mathbb{R}$$ Does this imply $\displaystyle\lim_{x \to \infty} f'(x) = 0$? If so, can the hypothesis be ...
2
votes
0answers
46 views

show that $(x^b+y^b)^{1/b} < (x^a+y^a)^{1/a}$ if $x>0$, $y>0$, and $0<a<b$

Show that $(x^b+y^b)^{1/b} < (x^a+y^a)^{1/a}$ if $x>0$, $y>0$,and $0<a<b$ by examining the sign of the derivative of an appropriate function. This is an exercise in middle part of ...
2
votes
0answers
54 views

Help me find a less tedious way to prove this theorem about inequalities

I'm currently working through Spivak's Calculus and am currently in Chapter 1. So far, the book has covered commutativity, associativity, existence of an inverse and existence of an identity for ...
2
votes
0answers
24 views

Limits of summations with different indices

In general, is it true that the limit as $n$ goes to infinity of $\sum_{i=1}^n x_{i}$ is the same as the limit as $n$ goes to infinity of $\sum_{i=1}^{n-1} x_{i}$ for some sequence $x_{i}$?
2
votes
0answers
29 views

Calculus: Reduction formula

For this question, I can find out $I3$, but I have no idea how to find the reduction formula. Please advise me.
2
votes
0answers
85 views

List of Common or Useful Limits of Sequences and Series

There are many sequences or series which come up frequently, and it's good to have a directory of the most commonly used or useful ones. I'll start out with some. Proofs are not required. ...
2
votes
0answers
55 views

Calculate $\displaystyle \int_{-\infty}^0 \int_0^\infty e^{-(x-2y)^2} \ dx \ dy $

$\displaystyle \int_{-\infty}^0 \int_0^\infty e^{-(x-2y)^2} \ dx \ dy $ so we're given the substiution of $ x = 2u + v$ and $y = u - v$ I calculate the jacobian to be $-3$, then from the given ...
2
votes
0answers
22 views

Asymptoticity of a definite integral

friends! I read on a book that, for $\alpha>1$, "being $g$ continuous in 0 [really $g$ is continuous in $[0,1]$, if it were useful to know] and approaching the extremes of the integral 0 for $n\to ...
2
votes
0answers
38 views

Cesàro Sum of Tangent

Can you proof or disprove the following? $\lim_{n \to \infty} (\frac {\tan1+\tan 2+\cdots+\tan n}{n})=0$. Is there any ergodic type theorem that can come to help?
2
votes
0answers
41 views

Change of Variable issue

I got across the integral $$\int_{\omega} \nabla y(x) \mathrm{d}x$$ $$y\colon \mathbb{R}^{3} \to \mathbb{R}^{3}$$ It would be better to perform the integration over the domain $\Omega$, the two ...
2
votes
0answers
41 views

Convergence of composition of functions sequences

Let $X$ be a metric space, $f_n: X \to X$, $g_n: X \to X$, $f_n(x) \to f(x)$, $g_n(x) \to g(x)$ ($n \to \infty$). Is $f_n(g_n(x)) \to f(g(x))$ ? Here: 1) pointwise convergence; 2) uniform ...
2
votes
0answers
37 views

How to simplify path integral?

I am trying to integrate a function, $f(x,y)$, over the straight line path connecting $(0,k)$ to $(k,0)$ in the x-y plane, where $k>0$ (the diagonal part of the boundary of a simplex in ...
2
votes
0answers
49 views

integral along closed curves

Let $f:\mathbb{C}\longrightarrow \mathbb{C}$ be a continuous function. If $\int_\gamma f=0$ for any closed curve $\gamma$, can we conclude $f$ holomorphic? Note that $f$ is not assumed to be ...
2
votes
0answers
40 views

Proving differentiability

I'm trying to do Spivak's Calculus on Manifold excersise 2-4. It goes as follows: Let $g$ be a continuous real valued function on the unit circle $\{x\in\mathbb{R}^2:||x||=1\}$ such that ...
2
votes
0answers
32 views

Question about finding volume using integration?

The question is: Find the volume of the solid whose base is a circle $x^2 + y^2 = 81$ and the cross sections perpendicular to the $x-axis$ are triangles whose height and base are equal. Now what the ...
2
votes
0answers
28 views

If $\emptyset\subsetneq A\subseteq (0,1)$ and $a,b\in A, a<b\implies a/b\in A$ then $\sup A=1$or $\sup A\in A$

I am looking for a simple and short solution of this problem using only basic definitions of $\sup$ etc. Here is my somewhat complicated solution that uses sequences and the Axiom of Countable Choice. ...
2
votes
0answers
63 views

Evaluation of $A$ in $2K(\sqrt{x}) = -\log(1 - x) + A + o(1)$ when $x \to 1^{-}$

Let $$K(k) = \int_{0}^{\pi/2}\frac{dx}{\sqrt{1 - k^{2}\sin^{2} x}}$$ be the complete elliptic integral of first kind where $0 < k < 1$. Let $k' = \sqrt{1 - k^{2}}$ be the complementary modulus. ...
2
votes
0answers
60 views

Integrating With Respect To $x$

Suppose I have the first derivative of the function $y$, $\displaystyle \frac{dy}{dx} = g(x)$. Furthermore, suppose I want to obtain the function $y$ by integrating with respect to $x$: ...
2
votes
0answers
33 views

Differential calculus question.

Find a constant c so that at any point of intersection of the two spheres $(x-c)^2+y^2+z^2=3$ and $x^2+(y-1)^2+z^2=1$ , the corresponding tangent planes are perpendicular to each other.
2
votes
0answers
96 views

integral involving upper incomplete gamma function

I trying to evaluate the following integral $$\int_0^\infty \dfrac { x^{m-1} \Gamma(A,\mathcal B x^q)} {\left[1+(\eta x)^n\right]^p} \,\mathrm dx$$ where the integration is w.r.t. $x$, and the ...
2
votes
0answers
44 views

Prove composite is smooth given condition on derivative

The problem I am attempting to prove is: Let $f:\mathbb{R}\to\mathbb{R}$ be smooth. Define $\psi$ by $\psi(x)=x^3$ Show $f\circ\psi^{-1}:\mathbb{R}\to\mathbb{R}$ is smooth $\iff\ f^{(n)}(0)=0$ if ...