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

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-3
votes
1answer
19 views

two ways to find these vectors ortho? [on hold]

What are two ways I can show $r'(t)$ is orthogonal to $r'(t)$? With $$r(t)=(f(t),g(t),h(t))$$, that is on a curve on a sphere I tried showing that the dot of the two were equal to $0$.
2
votes
2answers
76 views

Integrating a square's perimeter to get its area

I am trying to wrap my head around some integration applications. I went through the exercise of integrating the circumference of a circle, $2*\pi*r$, to get the area of a circle. I simply used the ...
0
votes
1answer
26 views

Help understanding the result of a formula

I need some help understand the middle section of this formula. $$OA^2 = (100-40)^2 + 50^2=10^2(61)\to OA = r = 10\sqrt{61} $$ and $$\sin(\angle OCB ) = \frac{30}{r} = \frac{3}{\sqrt{61}}, ...
1
vote
2answers
46 views

If $\frac{\partial F^i}{\partial x^j}=0$ on a connected open set, is $F$ constant?

Let $U$ be open in $\mathbb{R}^n$ and let $$F:U\to \mathbb{R}^m$$ be a smooth map, i.e. $F\in C^\infty(U)$. It is easy to prove that if $U$ is convex and $$\frac{\partial F^i}{\partial x^j}=0\tag{1}$$ ...
-8
votes
1answer
53 views

Brilliant formulaes [on hold]

Hey Brilliant mathematician, i am very honored for having your time. I need general Formulas on breaking down a number to a different and being able to derive that number back, my requirements is to ...
1
vote
1answer
51 views

How many real numbers satisfy the following

How many real numbers satisfy: $$\sin x=\frac{x}{100}$$ I don't know where to start it how to do this at all. Can someone please help me?
1
vote
2answers
30 views

Find the length of $2$ points based on intersection of a circle

Would anyone know the formula for finding the length of $BC$ giving the below data. $AB = 20$ $r = unknown$ $BC = ?$ The other dimensions that can be used if needed are written on the diagram. ...
-6
votes
2answers
59 views

Is $(-1)^{n!}$ convergent? [on hold]

I don't think I can use the alternating series test because of the factorial sign, but I don't know how else to solve this. can you please give any hints ?
2
votes
1answer
26 views

Non-linear differential equation I

What is the solution to the non-linear differential equation $$ \frac{d^2 y}{dx^{2}} = \left( \frac{2 y -1}{y^2 + 1} \right) \, \left( \frac{dy}{dx} \right)^2\ \text{ ?} $$ I would suspect it has a ...
2
votes
0answers
63 views

Geometric proof for Sophomore's dream

Is there a "visual proof" for sophomore's dream? $$\int_0^1 x^{-x}\,dx = \sum_{n=1}^\infty n^{-n}.$$ In the wikipedia article there are two algebraic proofs, but the integral and the summation has ...
0
votes
1answer
48 views

How can I determine the value of $a_1 + \displaystyle\sum_{i = 1}^{2012}\frac{a_{i + 1}^3}{a_i^2 + a_ia_{i + 1} + a_{i + 1}^2}$

For reals $x \ge 3$, let $f(x)$ denote the function $f(x) = \frac{-x + x\sqrt{4x - 3}}{2}$. Now suppose that $a_1, a_2, \ldots, a_{2013}$ is a sequence of real numbers such that $a_1 > 3, a_{2013} ...
1
vote
0answers
29 views

Show that a sequence is between a range

I got this question in class which I'm having trouble proving I tried investigate the sequence a little bit but it doesn't seem like I'm doing the right think, some help? $ \frac{39}{e^2} \le ...
0
votes
1answer
27 views

Show that the following sequence converges for $ 0 < a < e $ and diverges for $ a \ge e$

I have this question which I'm having trouble solving, can I use some help? :) Show that the following sequence converges for $ 0 < a < e $ and diverges for $ a \ge e$: $ \sum_{n=1}^{\infty} ...
1
vote
3answers
55 views

Proof of sum in an inequality

I was having hard time solving this one, any help will be greatly appreciated. prove that: $$ {39\over e^2}\le\sum_{n=1}^\infty {4n^2-1\over e^n}-{3\over e}\le{54\over e^2} $$
0
votes
2answers
43 views

A probability theory question [on hold]

let X be a rondom variable and coonsider a non-negative function $g: \Bbb R \to \Bbb R^+$ Please help me sshowing this following statement; for $r\in \Bbb R^+ $, $$P(g(X)\gt r) ...
4
votes
2answers
56 views

Is it true that: $|a_{n+1} - L| < |a_{n} - L| \forall n \in \mathbb{N} \implies \lim \limits_{n \to \infty} a_{n} = L ?$

If $a_{n}$ is a sequence and $|a_{n+1} - L| < |a_{n} - L|, \forall n \in \mathbb{N} $, then clearly the sequence $s_{n} = |a_{n} - L|$ converges (it's decreasing and bounded by $0$). Does it ...
-4
votes
1answer
24 views

How to use TI-Nspire CX CAS to solve Diophantine equation? [on hold]

Such as 58x+75y=1, please tell me the command(of ti-nspire). I don't know how to define a variable as an integer. I want the general formula of the solutions.
-1
votes
1answer
35 views

Lemme itô and Martingale [on hold]

I want to to find values of $a$, $b$ such that the process: $$e^{W_{t}^2+at+b\int_\limits{0}^{t}W_{s}^2\,ds}$$ be a martingale Could you please help me do that Thank you
1
vote
1answer
27 views

curves and integral

Find the area between these curves. $$y=\dfrac{3}{2x+1},\qquad y=3x-2;\qquad x=2\quad \text{et} \quad y=0 $$ indeed, I calculate the integral of the blue function between $1$ and $2$. Then, I ...
3
votes
2answers
55 views

Proving that a function is discontinuous

In my assignment I have to prove that the following function is discontinuous: $$f(x)=\begin{cases}2x-1&\text{if }x\notin\Bbb Q\\x^2&\text{if }x \in \Bbb Q\end{cases}$$ I have to prove that ...
1
vote
0answers
38 views

First fundamental theorem of calculus for line integrals [on hold]

Please, could someone look at this tricky question? Find the work done by force $F(x,y)=(3y^2+2) \hat i+16x \hat j$ in moving a particle from $(-1, 0)$ to $(1,0)$ along the upper half of the ellipse ...
3
votes
4answers
80 views

Solving $\int_0^{+\infty}\frac{e^{-\alpha x^2} - \cos{\beta x}}{x^2}dx$

I need to find solution of $$\int_0^{+\infty}\frac{e^{-\alpha x^2} - \cos{\beta x}}{x^2}dx$$ I know that Leibniz rule can help but I don't know how to use it. Could you help me please? Thank you.
2
votes
1answer
93 views

Study the following integral: $\int_0^\infty \frac{\mathrm{d} x}{x \cdot \ln x \cdot \ln^{(2)} x \cdot \ln^{(3)} x … (\ln^{(k)} x)^s }$

How do I calculate for which values of $s$ the following integral converges? $$\int\limits_{0}^{\infty} \frac{\mathrm{d} x}{x \cdot \ln x \cdot \ln^{(2)} x \cdot \ln^{(3)} x \cdots (\ln^{(k)} ...
0
votes
2answers
25 views

finding volume of solid

Suppose that a solid is formed in such a way that each cross section perpendicular to the x-axis, for $0 \le x \le 1$, is a disk, a diameter of which goes from the x-axis out to the curve $y = ...
3
votes
1answer
31 views

Single Variable calculus: trigonometric integral problem

Not a homework problem; I'm self-studying calculus from Serge Lang's book and am stuck on the following question. Question: $\int x^2\sqrt{1-x^2} dx$ My approach: Try to get rid of the square root ...
1
vote
0answers
35 views

Calculating total mass of a wire

I'm giving the following $$ \delta(x) = x + 7,\quad (0 \leq x \leq 4) $$ It says you are given the length-density function, $\delta(x)$, of an ininfinitesimally thin wire lying on the $x$-axis over ...
3
votes
5answers
90 views

How to prove $\sinh^{-1}x=\ln\left(x+\sqrt{x^2+1}\right)$ [on hold]

How to prove $$\sinh^{-1}x=\ln\left(x+\sqrt{x^2+1}\right)$$ I couldn't how to start to prove this. Any help ,thanks
0
votes
1answer
43 views

Confusion with Summations

I am having a little bit of confusion regarding summations. I know that $$\sum_{i=m}^n a_i = a_{m}+a_{m+1}+\cdots +a_{n-1}+a_n$$ Here is my confusion. How do we interpret/decompose the following: ...
2
votes
1answer
21 views

Fourier transform of $L^1$ function square summable?

It is known that for a $L^1$ function $f: \mathbb{R} \rightarrow \mathbb{C}$ the Fourier transform vanishes at infinity and is continuous. Does this even mean that $(\hat{f}(n))_{n \in \mathbb{Z}}$ is ...
1
vote
1answer
36 views

Solving second order nonhomogeneous linear equation

So i have the equation $$\frac{d^2y}{dt^2} + y = \sin(t)$$ I know the first step is to find the corresponding homogeneous equation, which i think would be: $$r^2+1=0$$ giving real roots and therefore ...
3
votes
4answers
364 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
25 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
51 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
132 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
16 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
26 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
53 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
41 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
38 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
34 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.
5
votes
1answer
64 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
8
votes
5answers
225 views

Difficult Coordinate Geometry and Calculus Question

I was given this question by a friend and after working tirelessly 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
21 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
32 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
76 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
98 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 ...