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

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2
votes
3answers
84 views

Compute $\frac{d}{dx}\left(\int_0^{x^2}\sin(s^2)ds\right) $

Is it correct to do the following? $$\frac{d}{dx}\left(\int_0^{x^2}\sin(s^2)ds\right) = F(b)-F(a)$$ $$=\frac{d}{dx}(\cos(x^4)2x^2-\cos(0)(0))$$ $$=\frac{d}{dx}(\cos(x^4)2x^2)$$ ...
0
votes
1answer
63 views

Why can a unit normal be simplified like this?

In my maths textbook, when trying to find a unit normal $n$, $$n = (2xi + 2yj)/\sqrt{4x^2 +4y^2} = xi + yj$$ How is this possible?
17
votes
1answer
320 views

Calculus over $\mathbb{Q}$

The mismatch between the sensitivity of 'mathematical calculus' and the flexibility of 'real world calculus' has been bothering me a bit recently. What I mean is this: in the real world, I can trust ...
2
votes
1answer
130 views

Verify solution: Is this gradient correct?

So I want to calculate minus the gradient of $$\Phi_1=\sum_{l=0}^{\infty}f(l)r^{l}P_l(\cos(\theta))$$ where $P_n$ is the $n$-th Legendre polynomial then we have $$-\nabla ...
1
vote
4answers
247 views

Proof of the greatest integer theorem

To define the function $f(x)=|[x]|$ where $|[x]|$ is the greatest integer that is less or equal to $x$, we need to prove that indeed such an integer exists. In other words $\forall x\in ...
1
vote
0answers
38 views

Approximative solution to PDE with additional term.

I am currently struggeling with the following problem: If I have a solution to the partial differential equation $ \Delta \Psi(r,\theta) = \rho(r,\theta)$ on $\mathbb{R}^3\backslash B(0,R)$(so the ...
10
votes
1answer
467 views

Interpretation of $\epsilon$-$\delta$ limit definition

The epsilon-delta definition for limits states that (from Wikipedia) for all real $\epsilon > 0$ there exists a real $\delta > 0$ such that for all $x$ with $ 0 < |x − c | < \delta$, we ...
2
votes
1answer
90 views

Need a hint on what's wrong - polar coordinates

I'm asked to solve the following $$ \int^2_0 \int^\sqrt{4-y²}_0 \sqrt{4-x^2-y^2} dxdy $$ I thought about using polar coordinates: (1) $0 \le x \le \sqrt{4-y^2}$ is the upper half of a circumference ...
1
vote
1answer
70 views

Functional equations very like the Taylor Series

Let $g(x,y)=0$ be a closed curve, that means, any point inside that curve satisfies $g(x,y)<0$ and any point outside that curve satisfies $g(x,y)>0$. Given a point $(a,b)$ outside the curve ...
0
votes
1answer
64 views

How to understand and to construct $m-1\leq n x<m$

$x, y\in R$, and $y>x$, prove: there exist $p\in Q$, such that $x<p<y$ Proof: Since $y>x$ is equvalent with $y-x>0$, by Archimede's property, there exists positive integer $n$, such ...
0
votes
2answers
114 views

denseness of rational numbers in $Q$

Firstly, Let's consider the denseness of rational numbers in real numbers. Between two arbitrary real numbers, there is at least one rational number. And we say ...
2
votes
1answer
159 views

Uniqueness of approximations like the Taylor polynomial

Given a function $f: \mathbb {R} ^n \to \mathbb {R} $, I am curious about the uniqueness of a $k$th-order approximation at $c \in \mathbb {R}^n $, i.e. a function $\phi(x)$ such that $$ \frac {f(c ...
1
vote
4answers
263 views

Ratio test: Finding $\lim \frac{2^n}{n^{100}}$

$$\lim \frac{2^n}{n^{100}}$$ as n goes to infinity of course. I know that the form os $\frac{a_{n+1}}{a_n}$ $$\frac {\frac{2^{n+1}}{(n+1)^{100}}}{\frac{2^n}{n^{100}}}$$ ...
-2
votes
3answers
213 views

Prove that the sequence $y_n$ converges [closed]

$y_n$ =$\sqrt{n} (\sqrt{n+1}− \sqrt{n})$ , n ≥ 1. Please help me..
5
votes
1answer
441 views

Finding the derivative of $\sin \sqrt {x^2+1}$ from the definition?

This means finding $\lim_{h \to 0} \large \large \frac{\sin \sqrt {(x+h)^2+1}-\sin \sqrt {x^2+1}}{h}$ . The only way I could think of to do this is to replace $h$ by some function $f(h)$ such that ...
3
votes
0answers
39 views

Attach term to solution of PDE(perturbation theory)

Currently I am struggeling with the following problem: Actually I have a found a solution to the PDE $\Delta \Phi(r,\theta)=f(r,\theta)$ and now I want to include a small extra term given by ...
0
votes
1answer
57 views

Finding an $n$ so the sequence $\left\{\frac{1}{n}\right\}_{n = 1}^\infty$ satisfies $|a_n| < 10^{-4}$

How to find $n$ so that $\left\{\frac{1}{n}\right\}_{n = 1}^\infty$ satisfies $$|a_n| < 10^{-4}$$ I can't find this formula in my book anywhere. It seems like it would be very time consuming to ...
1
vote
3answers
129 views

Sum of a geometric series $\sum_0^\infty \frac{1}{2^{1+2n}}$

$$\sum_0^\infty \frac{1}{2^{1+3n}}$$ So maybe I have written the sequence incorrectly, but how do I apply the $\frac{1}{1 - r}$ formula for summing a geometric sequence to this? When I do it I get ...
-2
votes
1answer
84 views

domain of convergence of power series

I need to find radius of convergences and domain of convergence for the: $\sum_{n=1}^{\infty}3^{n^{2}}x^{n^{2}}$. Can you help me please?
0
votes
3answers
44 views

$(\sum_{i=1}^n\omega_i)^t$: A Generalization on Term Cardinality of Multinomials

I'm asked to compute the number of terms in the expansion of the expression below: $$(w+x+y+z)^{10}$$ What is meant by "terms"? How can this be generalized so that we can find something like this: ...
9
votes
3answers
1k views

$\epsilon$-$\delta$ proof that $\lim_{x \to 1} \sqrt{x} = 1$

I'm trying to teach myself how to do $\epsilon$-$\delta$ proofs and would like to know if I solved this proof correctly. The answer given (Spivak, but in the solutions book) was very different. ...
5
votes
6answers
753 views

Integral of $\frac{2x}{\sqrt{1+x^2}}$

I'm trying to find the following indefinite integral: $$ \int \frac{2x dx}{\sqrt{1+x^2}} $$ I have already done it via u-substitution, however, I've been asked to solve it with two methods. I think ...
0
votes
2answers
171 views

Check calculation of mean value of a vector field over a sphere

Let $E=-\nabla(\Phi)$ be a vector field, where $\Phi:\mathbb{R}^3 \rightarrow \mathbb{R}$. Is it true that the mean value $$\bar E:=\frac{-1}{V_{\text{sphere}}}\int_{V{\text{sphere}}}\nabla \Phi = ...
1
vote
2answers
161 views

Tough integrals with Legendre polynomial

Does anybody here know how to integrate $\int_0^\pi P_n(\cos(x))\sin(x)\cos(x) dx$, $\int_0^\pi P_n(\cos(x))\sin^2(x) dx$, where $P_n$ is the n-th Legendre polynomial? They are actually extremely hard ...
1
vote
5answers
211 views

Differentiating $\tan\left(\frac{1}{ x^2 +1}\right)$

Differentiate: $\displaystyle \tan \left(\frac{1}{x^2 +1}\right)$ Do I use the quotient rule for this question? If so how do I start it of?
0
votes
1answer
108 views

How to solve this minimization (maximization)?

I'm facing this problem: $$ \large \min_{x \in \mathbb{R}_+^3} \max \left\{ { \sum_{i=1}^3 x_i^2-2 x_1 x_3 \over \left(\sum_{i=1}^3 x_i \right)^2} , { \sum_{i=1}^3 x_i^2 + 2 (x_1 x_3 - ...
3
votes
1answer
82 views

Loss of direction in Gauß' theorem?

I was wondering about the following: If I have a function $\phi:\mathbb{R}^3\rightarrow \mathbb{R}$ and I want to calculate the mean value of $E=-\nabla \phi$ over a sphere, then $E$ of course if a ...
4
votes
3answers
227 views

Differentiate $\sin \sqrt{x^2+1} $with respect to $x$?

Differentiate $$ \sin \sqrt{x^2+1} $$ with respect to $x$? Can someone please help me with question, im very lost.
5
votes
1answer
124 views

When is $F(x,t)=\int_0^tf(x,\eta)\,d\eta$ a continuous function of $x,t$?

I have read some results about integrals of the form $$\int_D f(x,t)\,dt$$ for instance, the dominated convergence theorem and MCT. Also I see results for $F(x) = \int_0^\infty f(x,t)\,dt$ being ...
3
votes
3answers
97 views

Tackle this series

I am looking for the exact value or a smart approximation(if you have a good idea) of the following series: $$\sum_{n=0}^\infty \frac{1}{2n+1} (P_{n+1}(0)-P_{n-1}(0))$$ where $P_n$ is the n-th ...
3
votes
2answers
112 views

Is this integral correct?

I used substitution and got that: $$\int_0^\pi \sin x \cdot P_n(\cos x ) \, dx=0$$ where $P_n$ is the $n$-th Legendre polynomial.
1
vote
4answers
805 views

Explain this chain rule for differentiating $y=xe^{-kx}$

I am asked to differentiate $$y=xe^{-kx}$$ The answer I am given is $$e^{-kx}(-kx+1)$$ I understand that when e is differentiated, it remains the same. I also see the product rule, but I'm not sure ...
2
votes
2answers
36 views

Simplify this expression that came from integration

I was doing a calculation and arrived at a term $\left[P_{l-1}(\cos(\theta)) -P_{l+1}(\cos(\theta))\right]_{0}^{\pi}$(So this is the result of an integration). Does anybody of you know how to simplify ...
0
votes
0answers
140 views

How to prove that homometric sets lead to same result in this problem? (any justifications?)

First let me define Difference multiset for a set of integers $$P=\{p_1,p_2, \dots,p_K\} ,\quad p_i \in\{1,2,\dots,N\},\quad p_i\ne p_j $$ as below: $$ D = \{p_i-p_j \mod N ,\quad i \ne j\} $$ I ...
2
votes
1answer
301 views

Sum of series using integration

In certain special series, we can use the sigma notation to obtain the sum of $n$ terms. For example; $$1^3 + 2^3 + 3^3 + 4^3 + 5^3 +\cdots+ n^3 = \frac {n^2(n+1)^2}{4}$$ The sum can also be ...
1
vote
1answer
59 views

Interesting related rates question

A circle C in the xy-plane is described as follows: A point P on the circumference of C traces out the graph of $f(x) = \sqrt{x}$; the center of C is the y-intercept of the tangent line of $f(x)$ at ...
1
vote
2answers
165 views

Every bounded function has an inflection point?

Hello from a first time user! I'm working through a problem set that's mostly about using the first and second derivatives to sketch curves, and a question occurred to me: Let $f(x)$ be a function ...
0
votes
1answer
49 views

some integral and series whose value is $1$.

Give me some integral and series whose value is $1$. Where can I find a large number of these kinds of examples. I have two examples here, but I cannot think up more... This is geometry series, ...
1
vote
1answer
142 views

Prove $\sqrt{k}$ is not a rational number. [duplicate]

Suppose $k>1$ is an integer, and k is not a square number, then $\sqrt{k}$ is not a rational number. Proof: Let $\sqrt{k}=\frac{p}{q}$, and $(p,q)=1$,So $q^2|p^2$, $p\neq 1$, $k$ is not an ...
3
votes
2answers
128 views

Generalization of Jensen's inequality for integrals?

Jensen's inequality for sums says that for $f$ convex, $$f\left(\sum_1^n \alpha_i x_i\right)\leq \sum_1^n \alpha_i f(x_i), \,\,\,\,\text{for } \sum_1^n \alpha_i = 1.$$ I have read that a ...
3
votes
2answers
70 views

Prove: For all $n\geq 1$, $a_{n+1}-a_n<8^ka_n^{\left(1-\frac{1}{k}\right)^3}$.

Set $S=\left\{\left.x^k+y^k+z^k\right|x,y,z\in Z^+\cup \{0\}\right\}$, k is a positive integer, sort elements of $S$ increasingly, that $a_1<a_2<a_3<\text{...}<a_n<\text{...}$. Prove: ...
27
votes
2answers
995 views

Limit of recursive sequence $a_{n+1} = \frac{a_n}{1- \{a_n\}}$

Consider the following sequence: let $a_0>0$ be rational. Define $$a_{n+1}= \frac{a_n}{1-\{a_n\}},$$ where $\{a_n\}$ is the fractional part of $a_n$ (i.e. $\{a_n\} = a_n - \lfloor a_n\rfloor$). ...
2
votes
2answers
76 views

$\frac{\sin x}{x^5} - \frac{1}{x^4} \underset{x\to 0}{\approx} \frac{-1}{6} \cdot \frac{1}{x^2}$, right?

I was reading an set of notes about Taylor series, and I came across a part I think is a typo. I want to make sure, because I want to understand this stuff correctly. Here is the relevant page of the ...
2
votes
2answers
450 views

Induction Proof for a series expansion of a function

I have done induction proofs of many different types, but trying to prove by induction that a derivative from the Taylor series expansion of a function has me stumped in terms of how to get the final ...
6
votes
2answers
126 views

Double integrals over general region -how to approach?

I'm in doubt on how to approach a problem of double integrals over a specific region. I have to calculate $\int\int\limits_R e^x dA$, R being the region between $y=\frac{x}{2}$, $y=x$, ...
6
votes
1answer
184 views

Spivak's Calculus exercise, possible error in text. Chapter 10, problem 24(a).

Working through all the problems in Spivak's Calculus (3E) and hit a snag here. Given $c, d \in R$, and distinct $x_1, ..., x_n \in R$, show that for any given $1 \leq i \leq n$ there exists a ...
8
votes
3answers
171 views

Compute $\lim_{n\to\infty} nx_n$

Let $(x_n)_{n\ge2}$, $x_2>0$, that satisfies recurrence $x_{n+1}=\sqrt[n]{1+n x_n}-1, n\ge 2$. Compute $\lim_{n\to\infty} nx_n$. It's clear that $x_n\to 0$, and probably Stolz theorem would be ...
3
votes
5answers
220 views

Proof of the product rule. Trick. Add and subtract the same term.

While I was looking at the proof of the product rule, there was something that I don't quite understand. Product Rule: $F'(x) = f'(x)g(x) + f(x)g'(x)$ The proof goes like, ...
1
vote
0answers
107 views

How do we go from $f'(x) = \frac{dy}{dx}$ to $dy = f'(x)dx$? [duplicate]

As far as I know, the derivative of $y$ is defined as: $$f'(x) = \lim_{h \rightarrow 0} \frac{f(x + h) - f(x)}h = \frac{dy}{dx}$$ So $\frac{dy}{dx}$ is a limit, not a fraction of real numbers. I ...
0
votes
1answer
146 views

Expand the function $\ln(27+x^3)$ as a power series at x=0

Expand the function $\ln(27+x^3)$ as a power series at $x=0$ A) What is the radius of convergence of this series? B) What is the coefficient of $x^{12}$ in this series? Not sure what to do?