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

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1
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3answers
59 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
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2answers
43 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
23 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 ...
4
votes
1answer
38 views

A Question about Nested Maximizations

I am working on labor demand models where firms have to choose the optimal level of employment by maximizing profits. In particular, I have faced the following problem: Maximize with respect to $l$ ...
1
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1answer
163 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 ...
3
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1answer
53 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 ...
-1
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1answer
81 views

Lemme itô and Martingale [closed]

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
22
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6answers
2k views

Is the theory of dual numbers strong enough to develop real analysis, and does it resemble Newton's historical method for doing calculus?

I've been interested in non-standard analysis recently. I was reading up on it and noticed the following interesting comment on the Wikipedia page about hyperreal numbers, right after giving an ...
1
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1answer
70 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
2answers
71 views

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

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
37 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 ...
0
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1answer
75 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
67 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 ...
3
votes
0answers
160 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
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1answer
28 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
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0answers
30 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 ...
4
votes
2answers
61 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 ...
3
votes
2answers
82 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 ...
3
votes
1answer
33 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 ...
0
votes
2answers
41 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 = ...
0
votes
3answers
11k views

How does the difference quotient with a square root in the numerator end up with square roots in the denominator?

I don't understand when I apply the difference quotient to: $f(x) = \sqrt{x} $ , to get: $$\frac{\sqrt{x+h} - \sqrt{x}}{h}$$ To simplify it.. How does it end up like this?: $$\frac{x + h - x}{h ...
1
vote
0answers
128 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 ...
8
votes
7answers
769 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
1answer
47 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: ...
1
vote
1answer
47 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
1answer
44 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 & ...
3
votes
5answers
111 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? $$\lim_{n \to \infty} \ln \frac{n!^{\frac{1}{n}}}{n}$$
1
vote
1answer
39 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
35 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 ...
3
votes
1answer
45 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? $$\sum_{n=1}^{\infty} \frac{1}{n^x}\arctan\Biggl(\biggl(\frac{x-4}{x-1}\biggr)^n\Biggr)$$
5
votes
2answers
159 views

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

How to calculate $$\int_0^{ + \infty } {\frac{1}{{\left( {x + 1} \right)\left( {{x^n} + 1} \right)}}dx} .$$
1
vote
1answer
112 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 ...
1
vote
1answer
40 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 ...
0
votes
0answers
25 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. ...
0
votes
1answer
78 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} ...
2
votes
2answers
102 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 ...
3
votes
0answers
112 views

Integral formulas involving continued fractions

Ramanujan posed the following formulas as questions in the Journal of Indian Mathematical Society: $$\int_{0}^{\infty}\dfrac{\sin nx\,\,dx}{{\displaystyle x + \dfrac{1}{x +}\dfrac{2}{x +}\dfrac{3}{x ...
3
votes
3answers
122 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
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1answer
26 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 ...
5
votes
3answers
94 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.
1
vote
1answer
61 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
124 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 ...
2
votes
1answer
54 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 ...
0
votes
2answers
83 views

Name of a limit theorem [closed]

What is the name of the theorem below? I have tried googling it with no luck.
2
votes
3answers
48 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 ...
1
vote
1answer
28 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 ...
0
votes
2answers
49 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 ...
1
vote
3answers
77 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 ...
0
votes
1answer
113 views

Geometric series for this problem..?

I am trying to work this out with geometric series: ...
1
vote
2answers
43 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 ...