1
vote
1answer
31 views

Optimization with both equality and inequality constraints

I need to minimize the following quantity: $$\min x_1^{-1/n}- \left(1-x_2 \right)^{-1/n}$$ subject to: $1-x_1-x_2=\gamma$ and $0<x_1+x_2<1$ $\gamma$ being a constant. Had it been two ...
1
vote
0answers
45 views

Most Suitable Book after Kline's Calculus?

I've been working through Morris Kline's Calculus: An Intuitive and Physical Approach and it's an absolutely excellent book for self-studying applied single-variable (and some multi-variable) calculus ...
1
vote
2answers
62 views

How wrong is it? - A “proof” of the FTC that I came up with in high school by hand-waving.

In high school calculus, I was first taught that the area under a curve $f(x)$ between $x=a$ and $x=b$ is given by: $$ A = \lim_{\delta x \rightarrow 0} \sum \limits_{a}^{b} f(x) \delta x $$ Then ...
6
votes
3answers
128 views

Beginning of Romance

I am a 17 guy from India. The fascination of maths has struck me recently, while I am in standard 12th. But all the resources I have, is some school textbooks. M.L Agrawal's of 11th and 12th. I don't ...
0
votes
1answer
24 views

$f$ is bounded $\iff$ $F/\log$ where $F(x)= \int_{[1,x]}f(t)/t \,dt$

Hi everyone I'm stuck with one exercise. This says the following: Let $F(x)= \int_{[1,x]}f(t)/t \,dt$ where $f$ is a non-decreasing function. Show that $f$ is bounded $\iff$ $F/\log$ is also ...
5
votes
4answers
109 views

Evaluate $\int_0^{\infty} x p^xe^{\Large-\frac{x}{a}}\ dx$

I need to evaluate the integral: $$\int_0^{\infty} x p^xe^{\Large-\frac{x}{a}}\ dx$$ for $0<p<1$. Unfortunately I do not know where to begin. I tried integration by parts but got nowhere ...
2
votes
1answer
89 views

Structured Self-Learning Program for Calculus I & II

I'm interested in a organised program which comprehensively covers the topics of Calculus I and Calculus II. I've recently finished taking my secondary school's university-level Calculus I course, ...
7
votes
3answers
288 views

Is formal logic necessary for pure/“higher” mathematics?

I'm asking this as an autodidact who wants to learn math rigorously for its own sake. And I was just wondering if understanding proofs could be achieved without a formal grounding in symbolic logic. I ...
2
votes
3answers
24 views

Find a vector orthogonal to other two given and ends at a plane

I am reviewing Calculus III using Mahavier, W. Ted's material and get stuck on one question in chapter 1. Here is the problem: Assume $\vec{u},\vec{v}\in \mathbb{R}^3$. Find a vector ...
1
vote
2answers
37 views

Problem of continuous function

Define the function $g(x) = x^2\cos\frac1x$ for $x\ne 0$. What should be the value of $g(0)$ if $g(x)$ is a continuous function? Explain your work and justify your answer. Frankly, I have no ...
0
votes
3answers
61 views

How do I find $\lim_{x\rightarrow \infty} x\sin \frac {c}{x}$?

How do I find the following limit for some real $ c $? $$\lim_{x\rightarrow \infty} x\sin \frac {c}{x} $$
1
vote
1answer
45 views

Washer method and shell method

(1) Sketch the region enclosed between the curve $y=sin^2x$ and the straight line $y =2x/π$ (2) Find the volume of the solid $S$ obtained by revolving the region in part (1) about the $y$-axis by ...
1
vote
3answers
48 views

Definite integrals: Calculating Volume

Suppose $D$ is the region in the $xy$-plane bounded by the parabola $y=x^2$ and the line $y=2x$. Find the volume of the solid generated by rotating $D$ about 1) $x$-axis 2) $y$-axis. Are the ...
1
vote
1answer
52 views

Calculus: Application of definite integrals

Suppose $a>0$ is a constant. Let $C$ be the curve $y=\cosh x$, for $-a \leq x\leq a$. Let $D$ be the region bounded by $C$, $|x| = a$ and the $x$-axis. 1) Find the length of $C$ 2) Find the area ...
2
votes
0answers
46 views

Continuity of the right-hand derivative of a Convex function (help with the proof)

Hi everyone I have some trouble with one point in the following proof. Let $f$ be a convex function (strict convex function) on a real interval. If $f'_-(a)=f'_+(a)$ where $f'_-$ and $f'_+$ are ...
1
vote
2answers
41 views

strict midpoint convex $\Rightarrow$ strict convex (help with a proof)

Hi everyone I have trouble with the following I think is something very simple, but I cannot figure out yet the correct approach for the strict inequality If $f$ is continuous and $f$ is strict ...
2
votes
3answers
165 views

Any good introductory book/tutorial on Fourier Transform (up to FFT) with plenty of exercises and solutions?

I wonder what could be a good book to start learning in depth all aspects of the Fourier transform up to the FFT algorithm, and beyond. I am going to dedicate quite some time on the subject, so I ...
3
votes
1answer
72 views

Application of Picard-Lindelöf to determine uniqueness of a solution to an IVP

I am still struggling quite a lot with the Picard-Lindelöf-Theorem (also known as the Cauchy-Lipschitz-Theorem). Problem: Consider the following IVP with $\alpha \neq 1$ $$\begin{cases} y'&= ...
1
vote
1answer
35 views

Show that for $f,g: [0, + \infty) \to \mathbb{R}$ convex functions of Class $C^2$ their product $fg$ is convex

Problem: Let $f,g: [0, + \infty[ \to \mathbb{R}$ be two convex functions of Class $C^2$. Assume that $$ f(0) \geq 0, g(0) \geq 0 \text{ and } f'(0) \geq 0, g'(0) \geq 0 \tag{!}$$ Show that $fg$ ...
0
votes
1answer
18 views

Differentiability of Norm $N: U \subset \mathbb{R}^n \to \mathbb{R}, \ x \mapsto \sum_{i=1}^n i|x_i|$

Problem: Let $U:= \lbrace x \in \mathbb{R}^n \mid x_i \neq 0 \text{ for } 1 \leq i \leq n \rbrace $ and show that the Norm given by $$ N: \begin{cases} U & \longrightarrow \mathbb{R} \\ x ...
1
vote
1answer
27 views

Question on closed sets using a convergent sequence

Intro: The following two questions are from my exam preparation sheet, it is not mandatory and will not be accredited (or improve marks and the like). There won't be a correction, merely an online ...
5
votes
2answers
127 views

Easy exercise (hint) Real Analysis

I've been stuck for a while with this problem. I suppose is something very easy, but I cannot figure out yet the correct approach. I'd really appreciated not a complete solution just some hints ...
3
votes
2answers
88 views

Show that $f(x,y)= \|x-y\|_2^2$ is differentiable

Problem: Show that $f: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ with $f(x,y)=\|x-y\|_2^2$ is differentiable and compute its differential at every point in the domain of $f$Note: $\| \cdot ...
2
votes
2answers
63 views

Evaluating integral limit in two ways gives different limits

Problem Show that $$\lim\limits_{h \rightarrow 0^{+}} \int_{-1}^{1} \frac{h}{h^{2}+x^{2}} \, dx = \pi.$$ I can do this by evaluating the integral directly and showing that it is equal to ...
0
votes
1answer
72 views

Sequence of learning mathematics from basic algebra to calculus.

What would be a step by step sequence of learning mathematics from basic algebra to basic calculus? I pose this question because I am in the process of self-learning mathematics as a preparation for a ...
1
vote
3answers
40 views

Definition of limit of function

I'm reading Calculus: Basic Concepts for High School Students and am trying to digest the definition of 'limit of function'. There are two details that I am struggling to fully accept: If you are ...
0
votes
0answers
77 views

Using Picard-Lindelöf Theorem to elegantly demonstrate uniqueness of an IVP

I am trying to keep this question clean and short, therefore I won't write down the entire theorem of Picard-Lindelöf here. Problem: $$y'=1+y^2 =:F(y), \ y(0)=0 $$ Find a solution on a ...
0
votes
0answers
33 views

Reference for power series

I would need some references for power series, Taylors series of elementary functions, derivation and integration of power series, convergence of sequences of functions and series of functions. The ...
0
votes
1answer
45 views

Calculating mean vector of a multivariate distribution

I have a question concerning calculating the mean vector (vector of expected values) of a general multivariate distribution. I try to obtain the mean vector by doing a vector integration and I ...
4
votes
1answer
162 views

Using the Parseval Identity to compute $ \sum_{n=1}^{+ \infty} \frac{1}{(4n^2-1)^2}$

Parseval's Identity: For continuous $f: [- \pi , \pi] \to \mathbb{R}$ $$ \sum_{n=- \infty}^{+ \infty} |c_n|^2 = \frac{1}{2 \pi} \int_{ - \pi}^{ \pi} |f(x)|^2dx, \text{ where } c_n = ...
1
vote
4answers
162 views

Which book among these would you recommend for first year calculus?

I'm struggling a bit with functions(limits, squeeze theorem, etc). I have done some research and found a list of books on calculus but I'm not sure which one would be better suited for me, so I would ...
0
votes
1answer
46 views

circular differentiation

Suppose one starts with a function $f: \mathbb R^2 \rightarrow \mathbb R$ using $\mu, \sigma^2$ as its input, i.e. $f=f(\mu, \sigma^2)$. (Note that here I omitted the specific form of $f$ since I ...
0
votes
3answers
127 views

Derivatives of sine and cosine at $x=0$ give all values of $\frac{d}{dx}\sin x$ and $\frac{d}{dx}\cos x$?

In video 3 of the video lectures by MIT on Single Variable Calculus presented by David Jerison, the latter says: Remarks: $\dfrac{d}{dx}\cos x\left|\right._{x=0}=\lim\limits_{\Delta ...
0
votes
2answers
39 views

Calculus problem (Differential)

First off, no, this is not homework. This comes from self-study and has stymied me. Please explain your answer as thoroughly as you can! Find increment $\Delta y$ and differential $dy$ for the ...
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.
4
votes
0answers
74 views

Transitioning to Higher Level Mathematics

I am just finishing grade 12 pre-calculus at my school and have strong interest in math. The problem is, it seems some important elements of higher level math are not in my schools curriculum that are ...
2
votes
1answer
191 views

Calculus - Finding the minimum vertical distance between graphs

Question:Find the minimum vertical distance between the graphs of $2+\sin x$ and $\cos x$? In order to find out the required distance, what should I do? It seems that there is a problem if I ...
1
vote
2answers
67 views

Calculus: L′ Hopital's Rule

$\Large\displaystyle\lim_{x\to\frac{\pi}{2}}(\sin x)^{\tan x}$ $\Large\displaystyle\lim_{x\to0}x^2\ln x$ $\Large\displaystyle\lim_{x\to1^+}x^{\frac{1}{1-x}}$ Do I have to apply l'Hôpital's Rule to ...
0
votes
1answer
104 views

help and verification of 3 short exercises

I'm reading an old book and find the following three question. I'd like to know two things: if my attempts are correct and also it would be great if someone could give suggestions in more detail. ...
1
vote
1answer
27 views

Find the point of inflection

Will there be an inflection point if there is no solution for $x$ when $f ''(x) = 0$? For example, $$ f(x)=\frac{x^2-x+1}{x-1} $$ with domain $\mathbb{R}-\{1\}$ Also, is that when $x$ is smaller than ...
2
votes
1answer
161 views

Would you provide a study routine for Spivak's Calculus? [closed]

I've been working on Spivak's Calculus for the past few days and although I can manage to solve most problems, they take a lot of time. Some chapters have over 20 exercises and it can take several ...
0
votes
0answers
176 views

Problem with Spivak calculus.What should I do?

For past two weeks I have been working throught the Spivaks calculus book.Needless to say I am very pleased with his writing style,but I have a slight issue. The issue is namely that in those two ...
2
votes
3answers
100 views

Derivative of a determinant whose entries are functions

Happy New Year, everyone! I do not understand a remark in Adams' Calculus (page 628 $7^{th}$ edition). This remark is about the derivative of a determinant whose entries are functions as quoted below. ...
1
vote
1answer
66 views

Where am I wrong applying chain rule here?

Why is $$\dfrac{\partial}{\partial x_i}f(tx_1, \dots, tx_n)=\dfrac{\partial f}{\partial(tx_i)}(tx_1,\dots,tx_n)\cdot\dfrac{\partial(tx_i)}{\partial x_i}=\dfrac{\partial f}{\partial ...
3
votes
1answer
120 views

Prove that $\limsup_{n \to \infty} \left(a_n + b_n \right) \leq \limsup_{n \to \infty} a_n +\limsup_{n \to \infty} b_n$

I want to prove that for two sequences, say $a_k$ and $b_k:$ $$ \limsup_{n \to \infty} \left(a_n + b_n \right) \leq \limsup_{n \to \infty} a_n +\limsup_{n \to \infty} b_n$$ If we let $M_n =\sup \{ ...
0
votes
2answers
739 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 ...
1
vote
1answer
58 views

Testing Boundary Points Of $\sum_{n = 1}^{\infty} \frac{n!z^n}{n^n}$

I'm having some trouble testing the series indicated in the title at its boundary points. I'll sketch the preliminary work, then arrive at the problem. It is clear that the series converges absolutely ...
5
votes
0answers
143 views

Study group for working through Spivak/Wilson [closed]

I'm planning on working through Spivak's Calculus or Wilson's "Introduction to Graph Theory" and was wondering if anyone here might be interested in joining a study group for it. There's no ...
1
vote
1answer
53 views

Please explain. Really I dont understand and I need to learn. Pde: : example of finding particular integral

When we look at the solution part, there is a statement The PI of the given PDE is obtained as follows After the statement, I dont really understand all of the calculation. Espacially, After the ...
0
votes
1answer
70 views

Calculus: computation of $\sum \frac{2^i}{i!}$

$$\sum_{i=0}^\infty \frac{2^i}{i!}$$ Would anyone mind telling me what is the answer? I know this may be a silly question but I would like to know.