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

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2
votes
2answers
41 views

Help with derivative of integral function?

How do you differentiate: $\displaystyle f(x)=\int_{a}^{b}e^{x^{2}+t^{2}}dt$ I tried writing $f(x)$ as the difference of the antiderivative of the function $\displaystyle e^{x^{2}+t^{2}}$ and I get $\...
0
votes
1answer
22 views

Differentiation product of functions in multidimensional Analysis

Define $k: \mathbb{R}^d \to \mathbb{R}^{m\times m}$ such that $ k(x)=g(x)f(x)^T$, where $f: \mathbb{R}^d \to \mathbb{R}^m, g: \mathbb{R}^d \to \mathbb{R}^m$ are differentiable functions. Prove that $k$...
1
vote
1answer
26 views

Difficulty setting up a solid of revolution integral

I am trying to find the volume of the region between the parabola $y^2=8x$ and the line $x=2$ revolved around the $y$ axis. My intuition is that taking an infinitesimal horizontal slice of the ...
3
votes
3answers
116 views

Does $\int_0^{1/2} \frac{1}{x\ln x}dx$ converge?

I tried this: $$ \begin{align*} \ln x &= t \\ \frac{1}{x} dx &= dt \\ \lim_{x \to 0^+} \ln x &= -\infty \end{align*} $$ So now we have $$ \int_{-\infty}^{\ln(1/2)} \frac1t dt $$ which ...
-1
votes
1answer
52 views

Chemistry using integration [on hold]

I was given this homework problem and I cant really figure out how to start or finish it. In Purdue's Chemistry department, the chemists have found that in a water based solution containing $12$ ...
2
votes
2answers
101 views

Integrating $\cos (x+\sin (x))$

I tried to solve $$\int\cos(x+\sin(x))\,dx$$ but it seems to be way out of my league (tried u-substitution with $u=x+\sin(x)$ and couldn't find an answer). Also, no one on the Internet seems to have ...
4
votes
3answers
96 views

How to solve the differential equation: $x \frac{d^2y}{dx^2}+2(3x+1)\frac{dy}{dx}+3y(3x+2)=18x$

How to solve the differential equation $$x \frac{d^2y}{dx^2}+2(3x+1)\frac{dy}{dx}+3y(3x+2)=18x$$ I think I could let $u=xy$ but I don't know how to proceed it.
5
votes
2answers
227 views

Explanation needed for a statement about power series convergence

I got a task in front of me but I don't really understand it. If someone could explain, I think I would be able to solve it myself. $P(x) = \sum_{k=0}^{\infty}a_{k}x^{k}$ is a power series. There ...
1
vote
3answers
78 views

How to get the correct angle of the ellipse after approximation

I need to get the correct angle of rotation of the ellipses. These ellipses are examples. I have a canonical coefficients of the equation of the five points. $$Ax ^ 2 + Bxy + Cy ^ 2 + Dx + Ey + F = 0$...
6
votes
2answers
109 views

Why is $\int_{-1}^{1} \frac{1}x \mathrm{d}x$ divergent?

Isn't $$\int_{-1}^{1} \frac{1}x \mathrm{d}x=\lim_{\epsilon\to 0^{+}} \int_{-1}^{-\epsilon} \frac{1}x \mathrm{d}x+\int_{-\epsilon}^{\epsilon} \frac{1}x \mathrm{d}x+\int_{\epsilon}^{1} \frac{1}x \mathrm{...
-4
votes
2answers
46 views

How to show that $\lim_{(x,y)\rightarrow (0,0)} 2^{\frac{xy}{x^2+3y^2}}$ does not exist? [on hold]

How to show that $\displaystyle\lim_{(x,y)\rightarrow (0,0)} 2^{\frac{xy}{x^2+3y^2}}$ does not exist? In other words, how can I solve this: $\lim_{(x,y)\rightarrow (0,0)}\frac{xy}{x^2+3y^2}$?
2
votes
3answers
44 views

Proving $-1$ is the infimum of $E=\{\frac{2}{n}+(-1)^n\mid n\in \mathbb{N}\}$

Prove: $\inf\{E\}=-1$ for $E=\{\frac{2}{n}+(-1)^n\mid n\in \mathbb{N}\}$ Let assume the contrary , there is $x\in E$ such that $\frac{2}{n}+(-1)^n\leq-1$. Because we are looking at negative ...
1
vote
2answers
41 views

Find the Maclaurin series of f(x)=(arctan(x)-x)/x^3

What I think I need to to do is find a general series expansion of the function and then derive term by term to get the Macaurin series...but I'm not quite sure how to expand this function. Any help ...
-1
votes
1answer
17 views

Given $g(x,y)$ with continous and differnatiable derivations for all $(x,y)$ points

Given $g(x,y)$ with continous and differnatiable derivations for all $(x,y)$ points(I mean by that: $f'_x$ has a value, continous and differantiable, same for $f'_y$), and given $g'_x(0,0) = 1 , g'_y(...
1
vote
1answer
776 views

Centroids and volume

I'm not quite sure what i'm doing wrong here.. Find the centroid (bar-x,bar-y) of the plane region defined by Then use Pappu's theorem to find the volume V of the solid obtained by rotating the ...
1
vote
0answers
12 views

Variant of local inversion theorem in special case

Let $F:\mathbb{R}^2\to\mathbb{R}^2$ be defined by $F(x,y)=(x+2y+x^2\ ,\ y-x^3+y^2)$. Then show that for $p_0=(4,1)$ and $p_1=(1,1)$ there exists $\delta>0$ such that for every $\vec y\in B(p_0,\...
4
votes
0answers
94 views

$\displaystyle\int_1^2\sqrt\frac{x^6+4x^4-2x^3+1}{x^4}\ \mathrm dx$ [on hold]

Find the value of: $\displaystyle\int_1^2\sqrt\frac{x^6+4x^4-2x^3+1}{x^4}\ \mathrm dx$ I do not really know where to start, so please forigve me for not showing my attempt. Wolfram alpha gives $2....
0
votes
2answers
20 views

Given $h(x,y) = f(x^2y + y^3)$, and $f(t)$ is differentiable $\forall t$ and $f'(t) = \frac{1}{2}$, then

Given $h(x,y) = f(x^2y + y^3)$, and $f(t)$ is differentiable $\forall t$ and $f'(t) = \frac{1}{2}$, then compute $h'_x(1,1) + h'_y(1,1)$. Well, I'm having difficulties using the chain rule to solve ...
1
vote
3answers
70 views

About the convergence of $\sum_{n=1}^\infty(-1)^n\tan(\frac{\pi}{n+2})\sin(\frac{n\pi}{3} )$ [on hold]

Does the series $$\sum_{n=1}^\infty(-1)^n\tan\left(\frac{\pi}{n+2}\right)\sin\left(\frac{n\pi}{3}\right )$$ converge and why? I think that Leibniz' test may be helpful, but I wasn't able to find a ...
-7
votes
1answer
55 views

Help Me Solve A Few Problems [on hold]

Here is it $$2y^2y'-x^2=0,\quad y=\text{?}$$ For example $$y^2y'-5x=0$$ $$y^2 \,dy -5x \,dx = 0$$ $$y^2\,dy=5x\, dx$$ $$\int y^2\, dy = \int 5x\,dx$$
2
votes
1answer
27 views

Finding arc length of the curve $6xy=x^4+3$ from $x=1$ to $x=2$

Looking at this as a graph of a function of $y$ is more convenient $$ y=\frac{x^4+3}{6x}\Rightarrow \frac{dy}{dx}=\frac{x^3-1}{2x^2}\Rightarrow \left( \frac{dy}{dx} \right)^2=\frac{x^6-2x^3+1}{4x^4} ...
2
votes
2answers
45 views

Prove: Let $a,b\in R$ such that $a\lt b$, and $f:[a,b]\rightarrow R$ be monotonic, then $\frac{1}{f}$ is also monotonic

Prove or disprove: Let $f:[a,b]\rightarrow \mathbb{R}$ be monotone. If $f(x)\ne 0$ for all [a,b], then $1/f$ is also monotone on [a,b]. I've been sitting on this for quite a while trying to find a ...
3
votes
3answers
102 views

Are there many different power series representation for a given function?

So I have to find the power series representation for $f(x) = \ln (3-x)$. I attempted the following: $$\ln(3-x) = \int {- \frac{1}{3-x} dx}$$ $$ = - \int { \frac{1}{1-(x-2)} dx}$$ $$ = - \int {\...
3
votes
2answers
47 views

Improper integral - checking convergence of $\int_{1}^{\infty} x^2 \sin(x^4) dx$

Does the following improper integral converges ? $$\int_{1}^{\infty} x^2 \sin(x^4) dx$$ Tried to find some known improper integral to compare this one to, but didn't find one. Thanks for helping!
0
votes
1answer
57 views

More complicated uSubstitution

I have absolutely no idea what to do here other than use uSubstitution. $$\int{{1}\over{4x^2 + 9}}\mathrm dx$$ I also tried looking at the output of an integral calculator but to no avail. I noticed ...
0
votes
0answers
20 views

Maxima and minima nth derivative reasoning

I found a statement somewhere in my notes that if we have a higher order function and lets say we take the nth derivative of it. If n is odd and the result turns out to be any number except zero then ...
0
votes
1answer
21 views

Stokes Theorem Parametrisation Question

Verify Stokes’ Theorem for the vector field $\boldsymbol{F}(x, y, z) =\langle 3y, 4z, −6x \rangle$, if the surface $S$ is the part of the paraboloid $z = 9 − x^2 − y^2$ that lies above the $xy$-plane, ...
1
vote
1answer
221 views

Cauchy repeated integration formula - different lower limits/ change of variable

Cauchy's repeated integration formula is as follows: Let $f^{(-n)}$ be a continuous function on the real line. Then the $n^{th}$ repeated integral of $f$ based at $a$, $$f^{(-n)}(x) = \int_a^x \...
1
vote
2answers
76 views

Continuous functions $f$ satisfying $\int_{0}^{x^2} f(x) = 1-e^{2x^2}$

Find all continuous functions $f$ satisfying $\displaystyle \int_{0}^{x^2} f(t)dt = 1-e^{2x^2}$. I differentiated both sides to get $2xf(x^2) = -4e^{2x^2}x$, so I concluded that $$f(x) =\left\{ \...
6
votes
4answers
3k views

Calculate $\pi$ precisely using integrals?

This is probably a very stupid question, but I just learned about integrals so I was wondering what happens if we calculate the integral of $\sqrt{1 - x^2}$ from $-1$ to $1$. We would get the surface ...
0
votes
1answer
31 views

Prove that $\lim_{x\rightarrow 1} \frac{\int_0^xg(t)dt-\int_0^1g(t)dt-\int_0^1f(t)dt(x-1)}{(x-1)^2=\frac{f(1}{2}}$

Prove that $$\lim_{x\rightarrow 1} \frac{\int_0^xg(t)dt-\int_0^1g(t)dt-\int_0^1f(t)dt(x-1)}{(x-1)^2}=\frac{f(1}{2}$$ Now I now this is a limit of the form $\frac{"0"}{"0"}$ which means I can use L'...
0
votes
1answer
34 views

How can i proof that every high order derivative of $\frac{1}{1+x}$ is equal to$ (-1)^kk!$ at point $0$.

In order to calculate the Taylor-Maclaurin polynomial $\frac{1}{1+x}$ of order $n$ at point $0$, i used the identity: $$\sum_{i=0}^n x^i + \frac{x^{n+1}}{1+x} = \frac{1}{1+x}\qquad (I)$$ and i ...
0
votes
0answers
23 views

Given $G = \{(x,y,z) | x^2 + y^2 + z^2 = 6\}$, $M_0(1,-2,1)$ find $n$ (normal vector)

Given $G = \{(x,y,z) | x^2 + y^2 + z^2 = 6\}$, $M_0(1,-2,1)$ find $n$ (normal vector) In the solution it is written that $n = (F'_x(M_0),F'_y(M_0),F'_z(M_0)$ Which means $n = (2,-4,2)$. Can anyone ...
1
vote
1answer
33 views

Proof using function continuity

Given that f(x) is continuous at 0 find a if: $$\lim_{x\to 0+} f(x) = 5 - a$$ And $$f(0) = a +3$$ Thanks for any help!
2
votes
1answer
87 views

Value of the integral $\int_{0}^{1}x^{n}(1-x)^{n}dx$

If $A=\int_{0}^{1}x^{n}(1-x)^{n}dx$ then which of the following is/are true? $1.$ $A$ is not a rational number. $2. 0<A\leq 4^{-n}.$ $3.$ A is a natural number. $4.$ $A^{-1}$ is a natural ...
0
votes
1answer
49 views

How to compute the definite integral $\int _0^{\infty }\:\frac{\left(2e^x+1\right)}{e^{2x}+2e^x+2}dx $

Good evening to everyone. I have an integral that I don't know how to compute: $$ \int _0^{\infty }\:\frac{\left(2e^x+1\right)}{e^{2x}+2e^x+2}dx $$.I've never computed integrals with $ \infty$ before ...
4
votes
1answer
67 views

Proving that a function is integrable

Given that $f:[0, \infty] \to \mathbb{R}$ is decreasing with $\displaystyle\lim_{x \rightarrow \infty} f(x)=0$, prove that $$I=\int_{0}^{1}\frac{\cos(\frac{1}{x})f(\frac{1}{x})}{x^2}dx$$ converges. ...
4
votes
1answer
34 views

Find the radius of convergence of this power series: $\sum_{k=0}^{\infty } \binom{2k}{k}x^{k}$

Given: $\sum_{k=0}^{\infty } \binom{2k}{k}x^{k}$ I started by forming it: $\binom{2k}{k} = \frac{(2k)!}{k!*(2k-k)!} = \frac{(2k)!}{k!*k!}$ Now the problem is, I cannot write $2! * k!$ instead of $(...
1
vote
1answer
100 views

Inequality $ab\le \frac{a^p}{p}+\frac{b^q}{q}$ [duplicate]

If $\frac {1}{p}+\frac {1}{q}=1$ and $a,b \ge 0$ , then prove $ab\le \frac{a^p}{p}+\frac{b^q}{q}$ . I can't find a simple and short way to prove this. Any hint would work. Thanks in advance!
4
votes
4answers
237 views

Why do some say that $f(x)=\frac{1}{x}$ discontinuous?

A continuous function $f:X\to Y$ is one which satisfies the following property: for every open set $U\subset Y$, the set $f^{-1}(U)$ is open in $X$. I don't see why according to this definition $f(x)...
13
votes
3answers
660 views

Find a continuous function $f:[1,\infty)\to\Bbb R $ such that $f(x) >0 $, $\int_1^\infty f(x)\,dx $ converges and $\int_1^\infty f(x)^2\,dx$ diverges

I'm trying to find an example of a continuous function $f:[1,\infty) \to \Bbb R $ such that $$f(x) >0 $$ $$\int_{1}^{\infty} f(x) \ dx \ \text{converges and } \int_{1} ^{\infty} f(x)^2 dx \ \text{...
1
vote
1answer
45 views

A functional equation involving integration

What should I do here? I don't even know where to start from. Please help me by giving me a hint. Q: A function $f:\mathbb R\to\mathbb R$ satisfy the two conditions below: $\displaystyle\...
0
votes
0answers
31 views

Find an equation to the tangent line to the curve at the given point

\begin{align} x &= \cos t + \cos 2t, & y &= \sin t + \sin 2t, & \left(−1, 1\right) \end{align} Using the above information I found that $\;\frac{dy}{dx}\;$ is: \begin{align} \...
3
votes
0answers
17 views

Defining derivatives and integrals for hyperoperations > 2

Derivatives and Integrals are continuous generalizations of the Forward Difference and Summation additive operators respectively. We can do the same with multiplication and get multiplicative calculus ...
0
votes
0answers
25 views

Kline calculus intuitive chapter 3 problem 14

I am starting to get frustrated as I am not able to comprehend these questions because they simply do not make sense to me. Here is the problem and answer to it. https://scienceanswers.wordpress.com/...
26
votes
15answers
2k views

Intuitive proofs that $\lim\limits_{n\to\infty}\left(1+\frac xn\right)^n=e^x$

At this link someone asked how to prove rigorously that $$ \lim_{n\to\infty}\left(1+\frac xn\right)^n = e^x. $$ What good intuitive arguments exist for this statement? Later edit: . .&...
35
votes
8answers
42k views

Why is the area under a curve the integral?

I understand how derivatives work based on the definition, and the fact that my professor explained it step by step until the point where I can derive it myself. However when it comes to the area ...
0
votes
3answers
34 views

Slope of a Tangent without given x value

At what point on the graph of $y=-3x^3+2x-1$ is the tangent parallel to $y=2x+10$? Now do I solve this question algebraically or do I solve it graphically since there is no specific x value given to ...
0
votes
2answers
26 views

I have to maximize this function involving absolute values

f(x) = $\frac{1}{1+|x|}$ + $\frac{1}{1+|x-2|}$ needs to be maximized. Maximizing this function means minimizing the denominators simultaneously. So I have to find the minimum value of 1+ $|x|$ and 1+...
0
votes
1answer
30 views

How can this be minimized?

I have the following function of $x_1$ and $x_2$: $$e(x_1,x_2)= (x_1^2+x_2^2)(a+n)+2a(-x_1+x_1x_2-x_2)+a^2$$ where $a$ and $n$ are real numbers. I want to find the values of $x_1$ and $x_2$ that ...