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

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3
votes
1answer
107 views

Convergence of Integral of Matrices

I have proved the following convergence, but I'm not convinced with my answer. Suppose we have the following $$\lim_{t \rightarrow \infty} \text{tr}\int_{0}^{t}e^{sQ^T}Pe^{sQ} ds$$ where tr is trace. ...
0
votes
0answers
21 views

Variant of CMVT [duplicate]

If $\Phi(x)$ and $\Psi (x)$ are continuous for $a \leq x \leq b$ , differentiable for $a \lt x \lt b, $ and $\Psi'(x)$ never vanishes , then for some $\xi$ in $(a,b)$ ...
5
votes
0answers
87 views

An alternative proof of Cauchy's Mean Value Theorem

Let's focus on the following version of Cauchy's Mean Value Theorem: Cauchy's Mean Value Theorem: Let $f, g$ be functions defined on closed interval $[a, b]$ such that 1) Both $f, g$ are ...
0
votes
1answer
75 views

Possible values of a $f(x)=(ab-b^2-2)x+\int_{0}^{x} x^2(\cos^{4}t+\sin^{4}t)\mathrm{d}t$

Suppose $f(x)=(ab-b^2-2)x+\int_{0}^{x} x^2(\cos^{4}t+\sin^{4}t)\, \mathrm{d}t$ is a decreasing function of $x$, $x$ is a real number. What are the possible values of $a$? $b$ is independent of $x$. I ...
0
votes
1answer
16 views

find the equation of a tangent surface which is parallel to a plane surface

How to get the equation of the tangent surface of $F=x^2 + 2y^2 +3 z^2=21$, which is also parallel to the plane surface $ x+ 4y + 6z=0 $? Here is what I've tried: $n = \{1, 4, 6\}$ from $ x+ 4y + ...
0
votes
0answers
48 views

Is $i^i$a real number or not? [duplicate]

How might we go about proving $i^i$ is not a real number? I don't know in general how to exponentiate to complex powers, I found this question in an introductory calculus course, so maybe there is an ...
2
votes
1answer
50 views

Show that a closed $1$-form on ${\bf R}^2 - 0$ has the form $\omega=\lambda \,d\theta+dg$

This is Problem 4-30 from Spivak's Calculus on Manifolds: If $\omega$ is a $1$-form on ${\bf R}^2 - 0$ such that $d\omega = 0$, prove that $$\omega = \lambda \,d\theta + dg$$ for some $\lambda ...
3
votes
2answers
528 views

A curious proof of L'Hospital's rule

I quote P. Nahin When Least is Best (2004), pp. 173-174 "Since $g(x)=R(x)h(x)$, then differentiation of both sides gives $$g'(x)=R(x)h'(x)+R'(x)h(x).$$ Since $\lim_{x \rightarrow 0} h(x)=0$, and we ...
0
votes
1answer
30 views

What condition do I have set to have $x_j$ for $j=1,…,n$ to be non-negative?

I have $\sum_{j=1}^n q_{jj}(x_j-y_j)^2\le1$ What condition do I have set to have $x_j$ for $j=1,...,n$ to be non-negative? The book I am reading says $\sqrt{q_{jj}}\ge1/y_j$ but why? edit: ...
1
vote
1answer
87 views

Why prove that area is unique?

in the book Apostol's Calculus Volume 1, in the proof of the area of under of the parabola $x^2$ from $x=0$ to $x=b$ it is shown that the area $A$ must satisfy ...
0
votes
1answer
344 views

finding unconditional distribution by integrating conditional distribution

Given $$ f_Y (y)= \begin{cases}\frac{1}{120} e^{-\frac{1}{120}y} &, y\ge 0 \\ 0, &, y< 0 \end{cases}$$ and $$f_{X|Y} (x|y) = \begin{cases}\frac{1}{y} &, x\in [0, y] \\0 &, ...
4
votes
1answer
57 views

Why can I multiply both sides by $dx$? [duplicate]

When we start learning about differential equations sometimes we "multiply" both sides of the equation by a differential and then integrate. Example: $\frac{dy}{dx}=x$ then $dy=x*dx$ and so on. I ...
2
votes
0answers
24 views

Zero locus of 2-variate real polynomial are smooth curves

This seems like it should be an easy question, and probably already has already had answer in advanced mathematics, but I only know some basic calculus, so I would like to know how do I go about doing ...
0
votes
0answers
25 views

Is this upper bound ok to use when bounding the error between the Riemann sum and its integral?

I found this on some class notes, which gives several different estimates of the error term, when going from the Riemann sum to its corresponding Riemann integral: $$\frac{b-a}{n}[f(b)-f(a)]$$ Does ...
4
votes
3answers
107 views

Show that this difference goes to zero,

$$\frac{1+\sqrt{2} + ... + \sqrt{N}}{N} - \frac{2}{3}\sqrt{N} \to 0.$$ The hint given in the question is this: choose appropriate Riemann sums and estimate the approximation error. My current work: ...
17
votes
4answers
691 views

Computing $\int_{0}^{\pi}\ln\left(1-2a\cos x+a^2\right) \, dx$

For $a\ge 0$ let's define $$I(a)=\int_{0}^{\pi}\ln\left(1-2a\cos x+a^2\right)dx.$$ Find explicit formula for $I(a)$. My attempt: Let $$\begin{align*} f_n(x) &= \frac{\ln\left(1-2 ...
4
votes
4answers
317 views

What is the limit of $x/(x+\sin x)$ as $x$ approaches infinity?

I am trying to determine $$\lim_{x \to \infty} \frac{x}{x+ \sin x} $$ I can't use here the remarkable limit (I don't know if I translated that correctly) $ \lim (\sin x)/x=1$ because $x$ approaches ...
2
votes
3answers
80 views

Evaluating $\int\sec x \,\mathrm dx$ [duplicate]

$$\int\sec x \,\mathrm dx = \ln\left|\sec{x} + \tan{x}\right|+ C = \ln{\left|\tan\left(\frac{x}{2} + \frac{\pi}{4}\right)\right|} + C$$ My question is how? How are these derived?
1
vote
2answers
80 views

Is Spivak wrong here, or am I just missing something?

Chapter 1 Problem 18 has the reader doing various proofs with second-degree polynomial functions of the form $x^2 + bx + c$. My issue lies with problem 18d, but it uses knowledge from 18b and 18c, so ...
30
votes
3answers
655 views

A closed form for $\int_0^1{_2F_1}\left(-\frac{1}{4},\frac{5}{4};\,1;\,\frac{x}{2}\right)^2dx$

Is it possible to evaluate in a closed form integrals containing a squared hypergeometric function, like in this example? ...
2
votes
2answers
89 views

Is $\lim S_{n,m}=\sum_{k=1}^n({-1})^k{n\choose k}k^{-m}<\infty $ for $ n \to \infty$ and $m$ large?

Let $m$ be a fixed positive integer ($m>1)$ and let $$S_{n,m}=\sum_{k=1}^n({-1})^k{n\choose k}k^{-m}$$ be a partial sum of real series. My question here is : Is $\lim S_{n,m} <\infty $ as $ n ...
0
votes
1answer
125 views

Largest value of the function $f(x) = \sqrt{x^4-3x^2-6x+13} - \sqrt{x^4-x^2+1}$

Largest value of the function $f(x) = \sqrt{x^4-3x^2-6x+13} - \sqrt{x^4-x^2+1}\;\;,$ where $x\in \mathbb{R}$ My Try:: Let $y=f(x)=\sqrt{x^4-3x^2-6x+13} - \sqrt{x^4-x^2+1}$ $\displaystyle \Rightarrow ...
3
votes
1answer
37 views

Convergence of fixed-point iteration for $p$ times continuously differentiable function

I am stuck at this problem: Let $\alpha\in\Bbb{R}$ be some number that satisfies $g(\alpha)=\alpha$ for some function $g$ that is $p$ times continuously differentiable on some neighborhood of ...
0
votes
1answer
48 views

Evaluations of a Definite Integral with cosine function

How do you evaluate this integral? Does it involve an elliptical integral? What technique do I use to evaluate this integral? $$\int _{ 0 }^{ 2\pi }{ \sqrt { 5-4\cos { \theta } } d\theta } $$
1
vote
5answers
69 views

Does the function $|x^2-4|/x$ have critical points?

Does the function $|x^2-4|/x$ have critical points? I tried differentiating and putting the derivative equal to 0.But I'm still a bit confused (as I got no solution).
6
votes
1answer
155 views

Infinite Product Representation of $\sin x$

I've recently taken interest in infinite products, and I'm having trouble with a proof I found in this PDF file: "Infinite Products and Elementary Functions": An intermediate step in finding an ...
3
votes
0answers
62 views

Is my proof rigorous? (Archimedes area of parabola)

I am currently reading Apostol's Calculus volume 1 and was revising the part where the area of a parabolic segment is found. I decided to write my own proof similar to the one in the book, which I ...
1
vote
2answers
36 views

Linear combination of basis function in logarithm space. Is it possible?

I have a function $f(x)$. As theory said that it can represent by linear combination of basis functions such as $$f(x)=\sum_{i=1}^{N}\alpha_ig_i(x)$$ where $\alpha$ is coefficient and $g(.)$ is basis ...
-8
votes
0answers
41 views

I need help on problem 54 and 55 (the way to solve this kind of quiz.) [on hold]

I need help in problem $54$ and $55$. How do I solve these kinds of questions? I know how to find gradients and how to find the tangent plane equation and the normal line too.
30
votes
6answers
921 views

Closed-form of $\int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx $

I'm looking for a closed form of this integral. $$I = \int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx ,$$ where $\operatorname{Li}_2$ is the dilogarithm function. A numerical ...
0
votes
3answers
47 views

Add or subtract something to a number to reduce it to the range 0 to 24

I'm developing a C++ program and I need to find a formula that given a number to reduce and a limit number, get a value between 0 and this limit number. I don't know if it is allow to put C++ code ...
8
votes
3answers
171 views

Closed-form of $\int_0^\infty \frac{1}{\left(a+\cosh x\right)^{1/n}} \, dx$ for $a=0,1$

While I was working on this question by @Vladimir Reshetnikov, I've conjectured the following closed-forms. $$ I_0(n)=\int_0^\infty \frac{1}{\left(\cosh x\right)^{1/n}} \, dx \stackrel{?}{=} ...
1
vote
2answers
52 views

Maximum value of expression

Let the maximum value of the expression $y=\frac{x^4-x^2}{x^6+2x^3-1}$ for $x>1$ is $\frac{p}{q}$,where p and q are relatively prime positive integers.Find (p+q). My ...
5
votes
0answers
36 views

Differential equation with shifited term

I have a differential equation (Or integral equation) of the form: $$ f(x) = a e^{-x} + b \int_0^x f(cz+dx) e^{-z} dz$$ $a,b,c,d$ are constants. I am considering whether the above equation has a ...
5
votes
5answers
97 views

$\int\dfrac{dx}{x^2(x^4+1)^{3/4}}$ [duplicate]

Evaluate $$\large{\int\dfrac{dx}{x^2(x^4+1)^{3/4}}}$$ I thought of rewriting this as $$\large{\int\dfrac{dx}{x^5(1+\frac{1}{x^4})^{3/4}}}$$ and substituting ...
4
votes
1answer
94 views

Backwards Heat Equation $ u_{t} = -\lambda^2 u_{xx}$

Problem Consider the backwards heat equation of the form $$ \left\{ \begin{aligned} u_{t} & = \lambda^2 u_{xx}, & x\in[0,L], \quad t\in[0,T]\\ u(0,t) &= u(L,t) = 0 \\ u(x,T) &= ...
0
votes
0answers
34 views

To determine the points of $\Bbb R^2$ at which $(i) f_x$ exists, $(ii) f_y$ exists.

Let $f : \Bbb R^2 → \Bbb R$ be defined by $f(x, y) := x^2 + y^2$ if $x$ and $y$ are both rational, and $f(x, y) := 0$ otherwise. To determine the points of $\Bbb R^2$ at which $(i) f_x$ exists, $(ii) ...
3
votes
0answers
57 views

List of techniques to evaluate limits?

I'd like to make a complete list of techniques to solve a limit. Definition of the limit Continuous functions Algebra of limits Addition, multiplication, division Composition Inverse function ...
0
votes
2answers
24 views

Convergence when the derivative is uniformly continuous

Let $f: \Bbb R \to \Bbb R$ be a derivable function. $f'$ is uniformly continuous in $\Bbb R$ Prove that $[n(f(x+1/n)-f(x))]$ converges uniformly to $f'(x)$ I'm having a hard time seeing why does ...
12
votes
1answer
467 views

Integration of $\sqrt{x+\sqrt{x^2+3x}}$

I faced the following indefinite integration problem: $$\int \sqrt{x+\sqrt{x^2+3x}}dx$$ This result by WolframAlpha suggests that there is an elementary way to compute this integration. But I don't ...
2
votes
3answers
3k views

Heaviside step function fourier transform and principal values

I found the following answer on SE: Fourier transform of unit step? However, it is still not clear to me and maybe somebody could explain it clearer. Problem I have the following in my notes of ...
0
votes
4answers
57 views

What does it actually mean if a cost function is differentiable?

I am just learning about optimization, and having trouble understanding the idea behind differentiating cost functions. I have read that for standard optimization problems, the cost function needs to ...
1
vote
0answers
80 views

Optimal allocation in network

We want to analyse specialization matters in a given network (N,g). Nodes represent individuals that can produce goods and services (just like in our usual economy) and that can be consumers too. ...
1
vote
1answer
34 views

Does $f_n(x)=\cos^n(x)(1-\cos^n(x))$ converge uniformly for $x$ in $[π/4 , π/2]$?

Does $f_n(x)=\cos^n(x)(1-\cos^n(x))$ converge uniformly for $x$ in $[π/4 , π/2]$? Its clear to see that the point-wise convergence is to $0$. By finding the derivative I obtained that the maximum of ...
5
votes
3answers
1k views

How to prove $f(x)=ax$ if $f(x+y)=f(x)+f(y)$ and $f$ is locally integrable

Suppose $f(x)$ is integrable in any bounded interval on $\mathbb R$, and it satisfies the equation $f(x+y)=f(x)+f(y)$ on $\mathbb R$. How to prove $f(x)=ax$?
4
votes
1answer
66 views

Calculation of integral using two different methods? [on hold]

Find $$\int \dfrac{x^3}{(x^2+1)^3}dx$$ in two different ways, first using the substitution $u=x^2+1$ and then using the substitution $x=\tan \theta$. I managed to do both of these but the answer is ...
3
votes
3answers
193 views

Indefinite integration: $\int x^{x^2+1}(2\ln x+1)dx$

Find the value of the integral: $$\int x^{x^2+1}(2\ln x+1)dx.$$ My attempt: I tried by using integration by parts, but not working since $x^{x^2+1}$ keeps coming again and again. Then I tried putting ...
5
votes
0answers
170 views

Can $\int_{0}^{1}\frac{x^{p}\ln^{q}(x+a)}{(x+a)^{b}}dx$ be expressed in a simple form?

I was browsing the book Irresistible Integrals and found this gem, at page 97, $$ \int_{0}^{1}x^{n}\ln^{k}(x)dx=\frac{(-1)^{k}k!}{(n+1)^{k+1}} $$ that resembles a previous question of mine here. So, ...
0
votes
1answer
23 views

How do I take the derivative of this vector valued function?

Problem: Find the velocity at time $t$ of the particle whose position is $\hat{r}(t)$: \begin{align*} \hat{r} = e^{-t} \cos(e^t) \hat{i} + e^{-t} \sin(e^t) \hat{j} - e^t \hat{k} \end{align*} This is ...
1
vote
6answers
75 views

For which values of $x$ does this series converge?

For which values of $x$ does the series presented below converge? $$\sum_{n=1}^{+\infty}\frac{x^n(1-x^n)}{n}$$ Neither the root test nor the ratio test is of much help - I've tried for ...