2
votes
2answers
53 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
0answers
60 views

Recommended Textbook/Resources

I'm looking for a textbook or resources my younger brother could use. (He is in year 9, equivalent to US high school freshman) He is wanting to advance upon his math, he currently does exercises out ...
0
votes
1answer
45 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
34 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
42 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
30 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
40 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
108 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
121 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
36 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
98 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
35 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
64 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
81 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
65 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
25 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
118 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 ...
-4
votes
1answer
75 views

The “calculus for dummies” books. [closed]

Does anybody have the ebooks: "Pre-calculus for dummies" and "Calculus for dummies"? The last one I found at the web but is from another author, i would like the one by Mark Ryan. Please share it ...
0
votes
0answers
119 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
88 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
65 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
108 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
452 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
57 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
110 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
52 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.
8
votes
2answers
179 views

Advice in Bachelor Degree

First of all, I´m very sorry for my bad english, especially writing. Ok, for differents problems i´m studing a Bachelor degree in Mathematics. These degree is online. Now, the problem with my school ...
1
vote
3answers
42 views

Why $\frac{d}{dy} \int_{-\infty}^{\frac{y-b}{a}}f(x)dx=\frac{1}{a}f \left ( \frac{y-b}{a} \right)$?

Could somebody explain why: $$\frac{d}{dy} \int_{-\infty}^{\frac{y-b}{a}}f(x)dx=\frac{1}{a}f \left ( \frac{y-b}{a} \right)$$ I don't get where this $\frac{1}{a}$ comes from. I assume that in general ...
3
votes
2answers
402 views

Learning maths as a hobby - tips and techniques not to forget

I learn math in my free time. I do it mainly as a hobby, but it also helps me as a software engineer. My main resources are MathTutor site and Stroud books. The problem I have is I can't work on math ...
5
votes
2answers
63 views

Mathematics lessons online?

IN school, I did not really care about my future, until I started working with computers, and getting into programming. I'm doing computer science at the moment & its currently really easy, but ...
1
vote
1answer
222 views

Find the integral curves of the equation

Question: Find the integral curves of the equation: $$\frac{dx}{y^2x-2x^4}=\frac{dy}{2y^4-x^3y}=\frac{dz}{2z(x^3-y^3)}$$ I could not find any similar example to understand this type of questions ...
8
votes
5answers
523 views

What is the easy way to calculate the roots of $z^4+4z^3+6z^2+4z$?

What is the easy way to calculate the roots of $z^4+4z^3+6z^2+4z$? I know its answer: 0, -2, -1+i, -1-i. But I dont know how to find? Please show me this. I know this is so trivial, but important ...
2
votes
1answer
67 views

Verify that an ellipse has four vertices.

Verify that an ellipse has four vertices. The ellipse is given by $$ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ And I took $$x=a\cos t$$ and $$y=b \sin t$$ for $t\in [0,2\pi]$ Please can someone help ...
0
votes
0answers
30 views

Convergence of $xe^x - R$

Basing my question on one of the previous questions I have passed before Root of the function $f(x)=xe^x-R$, I was wondering why does $xe^x - R$ always converge? I was told that the function will ...
2
votes
2answers
44 views

how to calculate frenet serre eqautions

how to calculate frenet serre eqations of the helix $$\gamma : \Bbb R \to \ \Bbb R^3$$ $$\gamma (s) =(\cos (\frac{s}{\sqrt 2}), \sin (\frac{s}{\sqrt 2}), (\frac{s}{\sqrt 2}))$$ i know the ...
0
votes
1answer
51 views

calculation of vector product $\ddot{\gamma} (t) \times \dot{\gamma} (t)$

$$\ddot{\gamma} (t) \times \dot{\gamma} (t)=(-a\cos t, -a\sin t, 0)\times (-a\sin t, a\cos t, b)$$ The writer will get the following result $$(-ab\sin t, ab\cos t, -a^2)$$ but I don't know how to ...
7
votes
4answers
249 views

Advanced Mathematics

I am a high school student and would like to pursue a career in mathematics and I am hoping to find a serious explanatory book on math (geometry, algebra, calculus, functions and trigonometry) for ...
1
vote
2answers
404 views

Proof of the limit laws (Analysis)

Hi everyone I'd like to know if my arguments for the next proof are sound or needs some changes to be correct. I hope they are not a little flaws. Proposition (limit laws): Let $(a_n)_{n=m}^\infty$ ...
0
votes
1answer
48 views

A question about continuity at a point

I have a question about continuity of a function defined in terms of limits. For simplicity consider only a function $f: \mathbb{R} \rightarrow \mathbb{R}$. I have seen a sufficient condition for ...
1
vote
1answer
258 views

Equivalent Cauchy sequences.

Hi everyone I'm having a bad time with two questions in the Analysis book of Terry Tao. I finally finished one of the exercises and I'm wondering if the next reasoning is correct or maybe needs some ...
0
votes
1answer
78 views

Help to understand a proof of Tao's book (analysis)

Hi I have troubles to understand a proof that is in the the notes and in the book of Terry Tao of Analysis. I Proposition in question is: The problems that I have it's to understand some tricky ...
1
vote
1answer
157 views

Exercise in Tao Analysis Book

I'm currently studying in the book of Analysis of Terry Tao, amazing book by the way. In one exercise I'm not pretty sure about how can do it (I know that will be almost trivial but I'm stuck in it). ...
0
votes
1answer
76 views

Find the limit $a$ of the sequence $(a_n)_n$

I want to find the limit $a$ of the sequence $(a_n)_n$ and the smallest natural number N such that $|a_n - a| < \epsilon \forall n \geq N$ My $a_n = 5/3 - 3^n/4^n$ for all $n \in \mathbb{N},$ ...
1
vote
1answer
293 views

one-sided continuity and one-sided derivative?

A continuous function is continuous at an $x$ value (call the $x$ value that we're interested in $c$) if both of these conditions are met and are true: $f(c)= \text{some real number}$ ...
1
vote
1answer
197 views

Finding domain of a rational function

Find the domain and graph: $$f(t)=\frac{-t}{|t|}$$ My book says to define it piecewise. My questions: $\mathbf{1)}$ Do all rational functions have to be defined piecewise, or just this ...
1
vote
0answers
187 views

Calculus math references for self-study?

I'm trying to learn mathematics and to understand calculus for the first time, I'm using this book Higher Math For Beginners: Mostly Physicists And Engineers by Y. B. Zeldovich and After that I 'll ...
6
votes
4answers
381 views

Can a function with just one point in its domain be continuous?

For example if my function is $f:\{1\}\longrightarrow \mathbb{R}$ such that $f(1)=1$. I have the next context: 1) According to the definition given in Spivak's book and also in wikipedia, since ...