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

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2
votes
0answers
166 views

About differentiation under the integral sign

I would like to ask something related to the application of the differentiation under the integral sign (Leibniz rule) given by ...
0
votes
2answers
35 views

Indefinite Multiple Integration

In multivariable calculus, definite integrals with multiple variables seem routine. However, I have not seen any example of an indefinite multiple integral. In fact, it seems as if limits of ...
1
vote
1answer
32 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 ...
1
vote
1answer
69 views

Curve sketching for $\ln(x^3 - x)$

A few weeks ago I failed my math exam and today I revisited some problems. We needed to analyse $\ln(x^3 - x)$ and I have problems finding the X and Y intercept. The way I approached it was Y ...
2
votes
1answer
60 views

Power series - Calculate radius of convergence

Let $$\sum {n\over{n+1}} \cdot \left({{2x+1} \over x}\right)^n$$ I was asked to calculate the radius of convergence. We can write the series as: $$\sum {n\over {n+1}}\cdot \left(2+{1\over ...
1
vote
1answer
71 views

Calculate $ \frac{\partial f}{\partial x} (x,y) $ [closed]

Calculate $ \frac{\partial f}{\partial x} (x,y) $ of $$ f(x,y) = \int_{x^2}^{y^2} e^{-t^2}\, dt$$
4
votes
1answer
66 views

Proving for all polynomials $p(x)=p_0+p_1x+…+p_dx^d$, $ \ \sum^\infty_{n=1}p(a_n)$ converges iff $p_0=0$

The series $\displaystyle\sum^\infty_{n=1}a_n$ converges absolutely. Prove that for every polynomial $p(x)=p_0+p_1x+...+p_dx^d$, $\ \displaystyle\sum^\infty_{n=1}p(a_n)$ converges iff ...
0
votes
1answer
45 views

Find $\dfrac{dy}{dx}$ and $\dfrac{d^2y}{dx^2}$ at the given point without eliminating the parameter

Find $\dfrac{dy}{dx}$ and $\dfrac{d^2y}{dx^2}$ at the given point without eliminating the parameter: $x = \dfrac{t^2}{2} + 1$, and $y = \dfrac{t^3}{3} - t$ at $t = 2$ I have a pic of it worked. ...
0
votes
2answers
145 views

True or False Stokes'/Gauss Theorem Problems

I'm having difficulties deciding the truths of the two following statements. The first one I believe is false, but I'm not entirely sure and the second one I don't know how to make heads or tales of. ...
1
vote
0answers
50 views

bilinear form, anti symmetric part

$\mathcal{H}$ : real Hilbert space with inner product $(\,,\,)$ and norm $||\,||:=(\,,\,)^{1/2}$ Let $D$ be a linear subspace of $\,\mathcal{H}$ and $\mathcal{E}$ : $D\times D\to \mathbb{R}$ a ...
0
votes
7answers
159 views

Is $\sum_{n=1}^{\infty} \frac{(-1)^n}{n \sin^2(n)}$ converging?

Statement: let $a_n$ positive sequence such that $\lim_{n \to \infty} a_n=0$. Prove or disprove that $\sum_{n=1}^{\infty} (-1)^n\cdot{a_n}$ converges. It's obvious that the series converges if $a_n$ ...
1
vote
3answers
144 views

Why does $\lim\limits_{t\to-\infty}te^t=0$?

Intuitively, $\lim\limits_{t\to -\infty}te^t=0$, since $e^t\to 0$ much faster than $t\to -\infty$. Is there a way to more rigorously compute this? Writing it as $e^t/(1/t)$ seems resistant to ...
30
votes
4answers
2k views

Is it necessary that every function is a derivative of some function?

I thought about this a lot and consulted a lot of people but everyone had contradicting answers. I am a high school student. please help.
6
votes
7answers
613 views

Find the tenth derivative of $x^2e^x$

If $$f(x) = x^2e^x$$ find $$f^{({10})}x$$ I'm not familiar with doing the series expansion for this and the solution that was provided to me did not help me at all, as it wasn't explained. I think ...
0
votes
1answer
25 views

$g(f(x))=x$, find whether f,g are injective/surjective/both

let $f: \ A \to B \ , \ g: B \to A$ s.t $\forall x \in A$ $g(f(x))=x$. Is f is bijection? surjection? both? Same for g. I started by defining f to be constant function - $f(x)=C$, so we get ...
1
vote
2answers
56 views

Find general solution of first order DE using integrating factor

I have the equation $$R\frac{dq(t)}{dt}+\frac{q(t)}{C}-V_0=0$$ And am asked to find the general solution using the integrating factor. I am a bit confused as I have been shown two ways to do it. ...
0
votes
3answers
74 views

Limit with L'Hospital

Prove that $$\lim_{x\to0^+}\frac{e^x+e^{-x}}{e^x-e^{-x}}-\frac{1}{x}=0$$ Taking a look at asome excercises in my book, I found this one. Specifically, it is in the chapter where L'Hospital rule ...
2
votes
1answer
43 views

A question on Taylor expansion/approximation

Suppose we are given a continuos function $f(x)$ where $x \in [0,2]$, and the function $f(x)$ is $n$-th-order differentiable, for $n \in \mathbb{N}$ and $n>2$. Besides, we know that these ...
-1
votes
1answer
49 views

The phase plane and potential energy

I think I have spotted a mistake in my notes, however I need help verifying my assumption: I am given the equation: $d^2s/dt^2=-s$ m=1 for simplicity I have recast it at a first order system: ...
2
votes
3answers
104 views

How do we arrive at the definite integral to find area approximated by a sum of rectangles?

The area enclosed by a one variable function from a to b can be approximated by $n$ rectangles$$A \approx \sum_{i=1}^{n} f(x_i)(x_i-x_{i-1})$$ and if we let $n \rightarrow \infty$ we get $$A = ...
1
vote
0answers
29 views

How to find new region of integration after changing variable?

I'm not quite sure how to go about finding the new region of integration. The evaluation of an integral over a region D can be done by changing variables and integrating over a new region and ...
2
votes
2answers
36 views

Proving Identities

Can you help me prove this identity: $$\frac{\sin x}{2\csc x}\left(\tan^2 x + \sin^2 x + \frac{\sin^2x}{\tan^2x}\right)= \frac{1}{2} \tan^2x$$ I have tried working on the left side and changing ...
0
votes
1answer
51 views

Little “o” Notation

As h approaches $0$, show $\sin(x+h)=\sin(x)+h\cos(x)+o(h)$. What I've done is basically substituted $h=0$ and therefore $LHS=RHS$, but I realise I'm supposed to use limits and I somehow can't get rid ...
2
votes
0answers
113 views

Proof of integral inequality

How does one prove (without making use of any approximations whatsoever) the following inequality: $$\int_1^2 \left(\ln(x)\right)^{2013}dx\leq\dfrac{1}{2^{2013}}.$$
0
votes
2answers
31 views

Identity proving

Can you help me prove this identity: $\tan x\sin x = \sec x-\csc x$? I began working on the left side, by first changing $\tan x$ to $\sin x/\cos x$ and then multiplying by $\sin x$, I can get it down ...
0
votes
2answers
26 views

Separable Differential Equation Help

I just can't seem to get this one. I know the process at least what we learned is to: Get all the y's on one side and the x's on another Integrate each side Solve for some $C$ given a $x$ and $y$ ...
0
votes
3answers
63 views

Prove $\tan x\sec x= \sin^3x \sec^2x +\sin x$

Can you help me prove the identity $$\tan x\sec x= \sin^3x \sec^2x +\sin x$$ I began working on the left side and changed everything to sine and cosine terms.
0
votes
1answer
41 views

What Law prohibits me from doing this?

I am working on a differentiation question and my answer to simplify is $2x\ln \left(2x-1\right)+\frac{2}{2x-1}x^2$ Why cant I cancel out a x when multiplying to get $\frac{2x}{2-1}$= $2x$ ? So ...
0
votes
3answers
85 views

How to find $a+b$?

If $$ f(x)= \left\{ \begin{array}{lr} e^{ax} +3,& x<0\\ ax^2-3x+b & x≥0 \end{array}\right. $$ is differentiable at $x=0$, then $a+b$ is How do I go about solving this?
0
votes
0answers
19 views

Trouble calculating surface area

I am trying to calculate the surface area of a solid of revolution of f(x) about the x axis for the interval [1,6] $f(x) = \begin{cases} 1 & 1 \leq x< 2\\ 1/2 & 2 \leq x< 3\\ . ...
1
vote
2answers
94 views

Series: Let $S=\sum\limits_{n=1}^\infty a_n$ be an infinite series such that $S_N=4-\frac{2}{N^2}$.

Let $S=\sum\limits_{n=1}^\infty a_n$ be an infinite series such that $S_N=4-\frac{2}{N^2}$. (a) Find a general formula for $a_n$. (b) Find the sum $\sum\limits_{n=1}^\infty a_n$. Can you explain ...
0
votes
1answer
31 views

Optimization Intuition

Take this question for example: What is the smallest possible sum of the squares of two numbers, $a$ and $b$, if $ab = -16$ So you get $b = \frac{-16}{a}$ and substitute. Once you find your ...
0
votes
4answers
69 views

test for convergence $\sum_{n=1}^{\infty} \frac{1}{n^n}$

Test for convergence $$\sum_{n=1}^{\infty} \frac{1}{n^n}$$ I'm at a loss on what to do, is this a geometric series $\frac{1^n}{n^n}$?
0
votes
1answer
36 views

Finding surface area of revolution

Can anyone help me with finding the surface area of a solid of revolution of f(x) about the x axis for the interval [1,6]. It's supposed to be able to be done without needing calculus but I am having ...
0
votes
2answers
38 views

Proving divergence for piecewise defined sums

Show that the series $\sum(-1)^{n-1}b_n$, where $b_n=1/n$ if $n$ is odd and $b_n=1/n^2 $ if $n$ is even, divergent. I'm completely stuck on how to start the problem. I was told that I should use ...
1
vote
2answers
43 views

What test can you use to prove convergence/divergence?

I tried to use the comparison test on this series however it oscillates above and below $0$ as $n$ tends to $\infty$, which violates the constraints of the comparison test. Any help would be much ...
1
vote
2answers
52 views

Convergence/Divergence of $\sum_{n=2}^{\infty} \frac{2^n}{3^n+4^n}$ Using Comparison Test

Use the comparison test to show if the series converges/diverges? $\sum_{n=2}^{\infty} \frac{2^n}{3^n+4^n}$
0
votes
3answers
441 views

Infinite derivative

I have just discovered the second derivative of $\frac{d^2}{dx^2}$. However now I have a curiosity for the infinite derivative. I am asking for a proof on if the infinite derivative is possible. I got ...
5
votes
2answers
160 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 ...
0
votes
1answer
44 views

Find all four-digit numbers with both a square and a cube root

How would one find all numbers that have at most 4 digits and have both a square and a cube root? $$x\in\mathbb{I};0<x<10^{3}\\ \sqrt[2]{x}\in\mathbb{I}\\ \sqrt[3]{x}\in\mathbb{I}$$ While this ...
0
votes
2answers
34 views

What is $\int e^{-\frac{(y-\mathbf{x}^T\mathbf{\theta})^2}{2}}d\mathbf{\theta}$?

$y \in \mathbb{R}, \mathbf{x} \in \mathbb{R}^2$ are constants, and $\mathbf{\theta} \in [-5,5]\times[-5,5] \subset \mathbb{R}^2$. I'm just not familar with vector integrals, any reference to read? ...
4
votes
1answer
584 views

A few improper integral

$$\displaystyle \begin{align*} & \int_{0}^{+\infty }{\frac{\text{d}x}{1+{{x}^{n}}}} \\ & \int_{-\infty }^{+\infty }{\frac{{{x}^{2m}}}{1+{{x}^{2n}}}\text{d}x} \\ & \int_{0}^{+\infty ...
2
votes
2answers
44 views

Solving bernoulli differential equation

How to solve $$t \frac{dy}{dt} + y = t^4 y^3$$ First I divided by $t$ to get $$\frac{dy}{dt} + \frac{y}{t} = t^3 y^3$$ Then I multiplied through by $y^{-3}$ to get $$y^{-3} \frac{dy}{dt} + ...
1
vote
1answer
49 views

Question on the norms

I got stuck with the following (simple) question since the result I got seems to be counterintuitive: I have a function defined in terms of its Chebyshev expansion, i.e. ...
0
votes
1answer
62 views

basic thing that I always get confuse about

LEt $\epsilon > 0$ be given. if $a - b < \epsilon $ Why does it follow that $a = b $ ??? This bothers me a lot. Why does it follow? shouldn't be $a < b $ ?
1
vote
1answer
88 views

How do I show that $x=1$ is the only solution to $\textrm{e}^{x-1}=x$?

How can I solve this equation: $e^{x-1}=x$ ? I know that $x=1$ is a solution but how can I prove that it's the only one? I can sketch each graph but I don't think it's adequate for a proof. Thanks. ...
2
votes
1answer
57 views

telescoping series and its sum

Write $S=\sum\limits_{n=8}^\infty \frac{1}{n(n-1)}$ as a telescoping series and find its sum. $S_N = $ $S = $ I found for the first one to be $\frac{1}{N}-\frac{1}{N-1}$ but it says it's ...
1
vote
1answer
145 views

Functions with infinitely many critical points

Does there exist an $f\in C^2(\mathbb{R})$, $f:\mathbb{R}\rightarrow \mathbb{R}$ with infinitely many critical points but finitely many inflection points? (Not a homework question.) Such $f$ would ...
1
vote
1answer
25 views

Finding the slope using the implicit function theorem

If $F(x,y)=x^2+xy^3$ how can one find the slope of $F(x,y)=2$ at the point where y=1 and x>0 using the implicit function theorem?
1
vote
1answer
79 views

convergence of $\frac{x^n}{1+x^n}$

How do I check convergence/ uniform convergence of $\sum\frac{x^n}{1+x^n}$. Also for series $\sum \sin \left(\frac{x}{n^2}\right)$, can I use that $\sin x \leq x$?