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

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3
votes
4answers
350 views

How to solve certain types of integrals

I'm asking for a walk through of integrals in the form: $$\int \frac{a(x)}{b(x)}\,dx$$ where both $a(x)$ and $b(x)$ are polynomials in their lowest terms. For instance $$\int ...
1
vote
1answer
23 views

About Fourier transform and complex conjugate

why this passage is correct ? \begin{equation*} \mathscr{F}[h(-\tau)] = H^*(f), \end{equation*} when $h(\tau)$ is a real function of real variable $\tau$, and $H^*(f)$ is the complex conjugate of ...
0
votes
1answer
29 views

Check if the following are perpendicular.

I have these expressions : $$2x+2y-5=0 \\ x=3-t,y=2+t,z=1-3t$$ I need to check if they are perpendicular. This is what I did : The following vectors represent the expressions $\langle ...
2
votes
1answer
40 views

Evaluating a triple integral by inspection

I would like to evaluate the triple integral: $$\iiint\limits_D {2 + 3{x^2} + 3{y^2}dV}$$ where $D$ is a conic domain with vertex $(0,0,b)$ and axis along the $z$-axis with a base (disk) with radius ...
5
votes
2answers
123 views

Infinite integrals$\int_0^{ + \infty } {\frac{1}{{\left( {x + 1} \right)\left( {{x^n} + 1} \right)}}dx} .$

How to calculate $$\int_0^{ + \infty } {\frac{1}{{\left( {x + 1} \right)\left( {{x^n} + 1} \right)}}dx} .$$
0
votes
0answers
15 views

How to express $[u,v,w]$ as a function of $\phi,\theta$ and the norms of the vectors?

$u,v$ are linearly independent and $w$ is a non-zero vector. Let $Angle(u,v)=\phi$ and $Angle(u \times v,w)=\theta$. Express $[u,v,w]$ as a function of $\phi,\theta$ and the norms of the vectors. ...
1
vote
1answer
24 views

For what $\alpha$ does the integral absolutely and for what conditionally converge?

For what $\alpha$ does the integral absolutely and for what conditionally converge ? $$\int_{0}^{1}\frac{\ln^{\alpha} (1+x^4)}{x^4}\cos{1 \over x}dx$$ Not sure which criteria to use to prove the ...
2
votes
1answer
52 views

Let $a_n>0$ for $n \geq 1$ and let series: $\sum_{n=1}^{\infty}a_n$ diverge. Let $S_n=a_1+a_2+…+a_n > 1$ for $n \geq 1$

Prove that the series: $$\sum_{n=1}^{\infty}\frac{a_{n+1}}{S_n \ln S_n}$$ diverges and the series : $$\sum_{n=1}^{\infty}\frac{a_{n}}{S_n \ln^2 S_n}$$ converges. (Using the famous criteria I ...
-2
votes
1answer
40 views

Integration by parts prove integral of cos^n x dx [on hold]

I'm having a problem with one of my questions. How can I prove that $\begin{align}\int\cos^n x dx&=\sin x\cdot\cos^{n-1}x+(n-1)\int\sin^2x\cos^{n-2}x dx\end{align}$ ?
2
votes
1answer
37 views

Are the extrema of this function global or local?

Last question about this function, I promise. The function $f: \mathbb R \rightarrow \mathbb R$ is given by $$f(x) = \begin{cases} \frac{x^2+5x+7}{x+3} & \mathrm{for} \; x < -3 \\ 0 & ...
0
votes
1answer
31 views

Fourier Transform of sin function

Hi there I'm trying to find the fourier transform of the following: \begin{equation*} x(t) = \sin(\pi t + \pi). \end{equation*} From what I know, I would integrate this using: $FT =\int x(t)e^{-iwt} ...
-3
votes
0answers
18 views

find the inverse Laplace transform of complex function3 [on hold]

It would be appreciate if someone help me to obtain the inverse Laplace transformation of the complex function F(s) is $$ 1\over\ \sqrt{s^{2}+1} $$ Thanks.
4
votes
0answers
36 views

A tough limit problem involving $1/(\sin x - \sin a)$ and its generalization

Long back I had encountered the following problem in Hardy's Pure Mathematics (originally from the infamous Mathematical Tripos 1896): If $$f(x) = \frac{1}{\sin x - \sin a} - \frac{1}{(x - a)\cos ...
-5
votes
3answers
44 views

Mean value theorem Problem? [on hold]

Using the "Mean value theorem" prove that $\tan(x)>x$ for $0 < x < \frac{\pi}{2}$
-2
votes
1answer
31 views

Is integration an injective operation on the set of (integrable) functions? [on hold]

Is indefinite integration an injective operation on the set of (integrable) functions? Thanks Alex
7
votes
4answers
173 views

Difficult Coordinate Geometry and Calculus Question

I was given this question by a friend and after tirelessly working on it I have not come up with anything substantial. I was hoping someone in the community could provide a pointer or possibly a ...
1
vote
1answer
19 views

Volume inside region delimited by surfaces $z=5-x^2$, $z=y$ and $y=1$.

I need to find the volume inside the region $E$ delimited by surfaces $z=5-x^2$, $z=y$ and $y=1$. I've spent few hours on this and would really need a hint from a charitable soul. I see that the ...
2
votes
2answers
30 views

How to find the length between 2 points given a pivot

I am not great at math but I have done the previous steps to my problem. This is the last step where I need to find out the distance between C,D. I am writing a program that will output this ...
4
votes
3answers
75 views

How do I solve the following differential equation

$$\frac{d^2y}{dx^2}=x^2y$$ Solving it by writing out a characteristic equation is not helping me find the solution to the above equation. Any help would be appreciated thanks.
2
votes
1answer
95 views

Calculate $\lim_{n \to \infty}(\sin nx)^\frac{1}{n}$

I know that $$(n)^\frac{1}{n} \to 1$$ and $(a)^\frac{1}{n} \to 1$ (with $a \in \mathbb{R+}$). However, I was wondering what can be said about $$\lim_{n \to \infty }(\sin nx)^\frac{1}{n}$$ and, more ...
2
votes
1answer
25 views

Finding volume using washer method

I'm supposed to determine the volume of the region obtained by revolving the region lying below the graph of the given function and above the $x$-axis about the specified axis. The problem I'm given ...
1
vote
1answer
30 views

Parameterization of the curve of intersection of a surface

I need to parameterize the curve of intersection of the surface: $x^2+y+z=2$ and $xy+z=1$ What I've done so far is said: $z=2-x^{2}-y$ therefore $xy+(2-x^{2}-y)=1$ (using substitution). Then, we ...
2
votes
6answers
100 views

Calculate the sum of three series which may be telescoping

Let $$\sum_{n=1}^\infty \frac{n-2}{n!}$$ $$\sum_{n=1}^\infty \frac{n+1}{n!}$$ $$\sum_{n=1}^\infty \frac{\sqrt{n+1} -\sqrt n}{\sqrt{n+n^2}}$$ I have to calculate their sums. So I guess they are ...
-3
votes
0answers
16 views

Applications of the Define Integral - Volume by moving square [on hold]

The point of intersection of the diagonals of a square is in motion along the diameter of a circle of radius a; the plane in which the square lies remains perpendicular to the plane of the circle, ...
-2
votes
1answer
36 views

Does $\sum a_n$ converge if $a_1 = 1$ and $a_{n+1}=\frac{2+\cos n}{\sqrt n}a_n$ [on hold]

Let $$a_1 = 1$$ and $$a_{n+1}=\frac{2+\cos n}{\sqrt n}a_n.$$ Consider $$\sum a_n.$$ How do I calculate if the series converges? The definition by recurrence troubles me a lot.
0
votes
2answers
69 views

Name of a limit theorem [on hold]

What is the name of the theorem below? I have tried googling it with no luck.
3
votes
3answers
112 views

Determine if a series defined by cases is convergent and calculate the sum

Consider $\sum_{n=1}^\infty a_n$, where $a_n$ is $$3^{-n}$$ if $n$ is even and $$\ln \frac{(n+2)(n+1)}{n(n+3)}$$ if $n$ is odd. I have to say if it is convergent and calculate its sum, but the ...
1
vote
1answer
23 views

Find real numbers that makes two different equations the same curve

I'm taking a vector calculus course, and I'm having trouble with this question. I know that I can make $B$ into $A$ (for the $x$ component) by multiplying $X$ by $t^3$ , which will be $t + t^3$, but ...
1
vote
0answers
35 views

Skellam CDF Increasing vs Decreasing in a parameter

I'm working with the following Poisson difference distribution: $$\text{Prob}\{X_1-X_2 \geq 0\} $$ where $X_1 \sim$ Poisson $(\mu_1)$ is independent from $X_2 \sim$ Poisson $(\mu_2)$. I need to ...
0
votes
2answers
41 views

I need help with the integration order please

the integral is as follows: find the volume between these regions bounded by : $z = x^2 + 3y^2$ and $z = 9 - x^2$ I discovered that this would be the space bounded by the elliptic paraboloid and the ...
2
votes
3answers
40 views

Integral Question using the Rule of Subsitution

I'm confused as to why $ \int e^{kx}dx$ = $\frac{e^{kx}}{k} + C$. I'm using the rule of substitution and came to the conclusion that it should be $e^{kx}k$ because the derivative of $kx$ is $k$. What ...
0
votes
0answers
7 views

convergence improvement

I try working in chevyshef series of zeta[2] but althougth the convergence it is not to bad it is hard to compute it would posible to split the sum? $$\zeta (2)=\sum _{n=1}^{\infty } \frac{(-1)^{2 n} ...
1
vote
3answers
51 views

Derivative of a Rational function $f(x)=\sqrt{2x-5\over3x+1}$

I'm trying to find the derivative of, $$f(x)=\sqrt{2x-5\over3x+1}$$ I think I can change this into $$f(x)= \left({2x-5 \over 3x+1}\right)^{1\over2} \\ =[(2x-5)(3x+1)^{-1}]^{1 \over 2}$$ Am I not ...
1
vote
2answers
29 views

Is this an open set in $\mathbb R^2$?

Is $\{(x,y)\mid y = \sin \frac {1}{x}, x>0\}$ an open set? (It is living in $\mathbb R^2$.) I think it should be open because $(0,0)$ seems to be a limit point of this set while it is not an ...
4
votes
5answers
95 views

Calculate $\lim_{n \to \infty} \ln \frac{n!^{\frac{1}{n}}}{n}$

How can I calculate the following limit? I was thinking of applying Cesaro's theorem, but I'm getting nowhere. What should I do? $$\lim_{n \to \infty} \ln \frac{n!^{\frac{1}{n}}}{n}$$
2
votes
1answer
41 views

For which values of $x$ is the following series convergent: $\sum_0^\infty \frac{1}{n^x}\arctan\Bigl(\bigl(\frac{x-4}{x-1}\bigr)^n\Bigr)$

For which values of $x$ is the following series convergent? $$\sum_{n=1}^{\infty} \frac{1}{n^x}\arctan\Biggl(\biggl(\frac{x-4}{x-1}\biggr)^n\Biggr)$$
1
vote
1answer
26 views

Deciding whether $(ax+b)/(cx+d)$ is increasing

In order of studying usual function such us : 1) $f(x)=ax^2$ 2) $g(x)=\frac{a}{x}$ 3) $h(x)=\frac{ax+b}{cx+d}$ 4) $p(x)=ax^2+bx+c$ To know if a function is increasing or decreasing we can ...
1
vote
1answer
79 views

Feynman Integration Problem

$$ I = \frac{\pi^2}{8} - \int_0^1 \frac{\arctan(x)}{\sqrt{1-x^2}} \,dx $$ Evaluate $I$ $$ I = \frac{\pi^2}{8} - \int_0^1 \frac{\arctan(x)}{\sqrt{1-x^2}} \,dx$$ $$f(a) = \int_0^1 ...
2
votes
1answer
33 views

Find all asymptotes of the function

Follow-up on Is the function continuous and differentiable at $x=-2$? The function $f: \mathbb R \rightarrow \mathbb R$ is given by $$f(x) = \begin{cases} \frac{x^2+5x+7}{x+3} & \mathrm{for} \; ...
1
vote
1answer
27 views

Can someone please help me in proving this?

Let $k_{2}>k_{1}>0$, prove that for any $x>0$, $f(x)$ is a monotonically increasing function. $$ f(x)=\frac{1-e^{-k_{1} x}}{1-e^{-k_{2} x}}. $$ We can have ...
1
vote
1answer
36 views

Prove that $f$ in monotonic

In my assignment I have to prove the following: Let $f$ a continuous function in $\Bbb R$. Prove the following: if $|f|$ is monotonic increasing, in R then $f$ is monotonic in R. ...
8
votes
7answers
679 views

What mistake have I made when trying to evaluate the limit $\lim \limits _ {n \to \infty}n - \sqrt{n+a} \sqrt{n+b}$?

Suppose $a$ and $b$ are positive constants. $$\lim \limits _ {n \to \infty}n - \sqrt{n+a} \sqrt{n+b} = ?$$ What I did first: I rearranged $\sqrt{n+a} \sqrt{n+b} = n \sqrt{1+ \frac{a}{n}} \sqrt{1+ ...
0
votes
2answers
27 views

With $f(x)= 32 \cosh(x) \sinh(2x) $, determine the slope of its tangent at $( \ln 2 , \, 75)$

With $f(x)= 32 \cosh(x) \sinh(2x)$, determine the slope of its tangent at $( \ln 2 ,\, 75)$. My work $$\sinh x \cosh y = \frac{1}{2}(\sinh (x + y) + \sinh (x - y))$$ $$\cosh(x) \sinh(2x)= ...
0
votes
0answers
38 views

limit problem δ and ε

In this problem we consider the function $f(x)=x^2$. Suppose our friend Bob is attempting to use the definition to show that $\lim_{x \to 1} f(x) = 3$, which of course is not true. How small does ...
1
vote
2answers
35 views

Definite integral calculation

$$\int_0^{2\pi}a^2(1-\cos(x))^2\sqrt{((a-a\cos(x))^2+\sin^2(x))}\,dx$$ After some work I get $$2a^3\int_0^{2\pi}(1-2\cos(x)+\cos^2(x))\sin\left(\frac x2\right)\,dx$$ And stopped:( Any ideas?
2
votes
1answer
24 views

Calculating volume by shell integration

$y = \ln x$, region is delimited by $y = -1$, $y = 2$ and the $y$-axis, it rotates around the $y$-axis. It's quite simple to solve by using disk integration but I can't get it right with shell ...
0
votes
2answers
25 views

$y = 3\sin^{-1}(\sqrt{x})/x$ find $y'(1/4)$

$y = 3\sin^{-1}(\sqrt{x})/x$ find $y'(1/4)$ my work is that y'= (x*$ ('\sin^{-1}(\sqrt{x}))+\sin^{-1}(\sqrt{x}) *1$)/(x)^2 my problem how to Derivative $ \sin^{-1}(\sqrt{x})$ ...
2
votes
2answers
37 views

Symmetry argument and WLOG

Suppose that $f:[0,1]\rightarrow \mathbb{R}$ has a continuous derivative and that $\int_0^1 f(x)dx=0$. Prove that for every $\alpha \in (0,1)$, $$\left|\int_0^{\alpha}f(x)dx\right| \leq ...
0
votes
0answers
17 views

Find the normal and tangent given a level curve and a point

I'm having some trouble solving this question. Consider $f(x,y)=xy+e^{x+y}$. For the level curve $f(x,y)=1$ at the point $(0,0)$, compute the equations in the form of $y=mx+b$ for the normal and ...
2
votes
3answers
43 views

$f(x) = x \tan^{-1}(x\ln(x))$ find $f'(e)$

$f(x) = x \tan^{-1}(x\ln(x))$ find $f'(e)$ my work $f'(x)=\tan^{-1}(x\ln(x)) *1 + x$ ---> stack here I know $\tan^{-1}(x)'= \frac{1}{1+x^2}$ so $\tan^{-1}(x\ln(x)) = ???$ I need help to solve ...