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

learn more… | top users | synonyms

1
vote
0answers
7 views

How do ideas in differential geometry expand upon ideas from introductory calculus

I just went through first year in mathematics and used Stewart's book for calculus. I am trying to self study differential manifold and I find many concepts such as chart, atlas very similar to that ...
0
votes
0answers
14 views

Solution for a differential equation

I am stuck in getting the solution for the following non-linear differential equation: \begin{equation*} x^2 + B\frac{dx}{dt} = A\sin(wt) \end{equation*} Is there any method to solve this kind of ...
1
vote
0answers
24 views

The Coin-Exchange Problem (Application of the Residue Theorem)

These day, I have met a problem about application of the Residue Theorem, see section 10.4 of enter link description here.Could anybody help me solve it? (The Coin-Exchange Problem) Suppose $a$ and ...
0
votes
2answers
24 views

First order differential equation with initial conditions

I solved the differential equation $$\frac{dy}{dx} = \frac{x}{x^2+1}$$ to get the general solution $$y = \frac{ln|x+1| +c}{2}$$ Im given the initial condition $$yy' − 2e^x = 0, y(0) = 3$$ but ...
0
votes
0answers
18 views

Integrating $\operatorname{Log}(z+2)$ along the unit circle [duplicate]

For the function $f(z) = \operatorname{Log}(z + 2)$, where we choose the principal branch of logarithm (namely, $−\pi < \operatorname{Arg}(z) < \pi$), and the contour $C := \{z \in ...
-2
votes
2answers
32 views

Being a fresher i wanted to understand this [on hold]

So this is Calculus two and i have been having diffculties with math and our teacher asked us to do this, and i just want to give up in math but i think it is important to try. What is given.. Compute ...
1
vote
1answer
22 views

Question about conditions for conservative field

Question about conditions for conservative field In common textbooks' discussions about conservative vector field. There is always two assumptions about the region concerned, namely the region is ...
2
votes
1answer
14 views

How to calculate the continuum limit of a discrete system?

The question is based on the following excerpt from the book "Symmetries and Integrability of Difference Equations" Link: Book Excerpt Consider the discrete equation ...
-4
votes
1answer
18 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 $0$.
1
vote
1answer
42 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
29 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
43 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
39 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?
0
votes
1answer
25 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. ...
-7
votes
2answers
46 views

Is $(-1)^{n!}$ convergent?

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
24 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
54 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
43 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
26 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
25 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
43 views

Proof of sum in an inequality

I was having hard time solving this one, any help will be greatly appreciated. prove that: $$ {39\over e^2}\le\sum_{n=1}^\infty {4n^2-1\over e^n}-{3\over e}\le{54\over e^2} $$
-1
votes
2answers
26 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
51 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
23 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
26 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
26 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
51 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
34 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 ...
2
votes
4answers
63 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
73 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
24 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 = ...
2
votes
1answer
28 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
83 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
18 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
34 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 ...
2
votes
4answers
345 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 ...
1
vote
3answers
60 views

Prove that $f : [a,b] \rightarrow \mathbb{R}$ is a bijection from $[a, b]$ to $[f(a), f(b)]$

I'm a 1st year mathematics student, and in my analysis class I'm having trouble with proving the following: Let $a < b \in \mathbb{R}$, and let $f : [a,b] \rightarrow \mathbb{R}$ be a continuous ...
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
119 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
14 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
51 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} ...