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

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0
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3answers
19 views

Power series solution (Why the constant of the recurrence relation can be chosen arbitrarily?)

Please help me understand this: Solve $y''-xy=0$ First, since there are no singular points, it can be guaranteed that we can always find two power series independent solution, centered at $0$, and ...
1
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3answers
16 views

How to know differentiation of the function at zero

Suppose we have function $f(x)=\frac{x^2}{2+|x|}$. Can anyone tell me that this function is differential at zero or not? Thanks
85
votes
18answers
8k views

Is there any integral for the Golden Ratio?

This is a curiosity. I was wondering about math important/famous constants, like $e$, $\pi$, $\gamma$ and obviously $\phi$. The first three ones are really well known, and there are lots of integrals ...
0
votes
1answer
20 views

AP Calculus BC - Area integration question

I'm taking the AP Calculus BC Exam next week and ran into this problem with no idea how to solve it. Unfortunately, the answer key didn't provide explanations, and I'd really, really appreciate it if ...
0
votes
1answer
20 views

For what values of $k$ to both of the following series converge?

I'm taking the AP Calculus BC Exam next week and ran into this problem with no idea how to solve it. Unfortunately, the answer key didn't provide explanations, and I'd really, really appreciate it if ...
0
votes
3answers
30 views

Find area of shaded area in curve with range of values for $y$

The parabola in the diagram has equation $y = 32 - 2x^2$ The shaded area lies between the lines $y=14$ and $y=24$ Looking at the graph, I only need to find half the area and multiply by ...
1
vote
1answer
31 views

Area enclosed by the curve $\lfloor |x''| \rfloor +\lfloor |y''| \rfloor = 2$

The area enclosed by the curve $$\bigg\lfloor \frac{|x-1|}{|y-1|}\bigg\rfloor +\bigg\lfloor \frac{|y-1|}{|x-1|}\bigg\rfloor = 2\;,$$ Where $-2 \leq x,y\leq 0$ $\bf{My\; Try::}$ Let $x-1=x'$ and ...
18
votes
3answers
315 views

Why is this definite integral antisymmetric in $s\mapsto s^{-1}$?

I recently happened into the following integral identity, valid for positive $s>0$: $$\int_0^1 \log\left[x^s+(1-x)^{s}\right]\frac{dx}{x}=-\frac{\pi^2}{12}\left(s-\frac{1}{s}\right).$$ The ...
1
vote
3answers
97 views
+50

Find the latus rectum of the Parabola

Let $y=3x-8$ be the equation of tangent at the point $(7,13)$ lying on a parabola, whose focus is at $(-1,-1)$. Evaluate the length of the latus rectum of the parabola. I got this question in ...
2
votes
1answer
94 views

Does this expression have a closed form?

Does $\displaystyle \underbrace{x\left(\dfrac{d}{dx}\left(\cdots x \left(\dfrac{d}{dx} \left( \dfrac{x}{1-x}\right)\right)\cdots\right)\right)}_{\text{$x \frac{d}{dx}$ $m$ times}}$ have a closed ...
2
votes
2answers
23 views

Integral of bounded function with limit zero at $\pm \infty$

Very simple question here, I almost feel bad for asking it.. Lets say we have a function bounded between $0$ and $1$. This function is high dimensional: $0<f(X) \le1, ~~~ X \in \mathbb{R}^D$ Now, ...
0
votes
2answers
58 views

If the integral of $c/x$ is $c.log(x)+C$ what is the base?

This question is a follow up to an answer I gave here: How to integrate $1/x$? After the algebra I said that 'This step of course gives the argument of $ln()$ the value $e$ and note that so far we ...
4
votes
3answers
67 views

Please prove the following: Given $ƒ(x) = e^x$, verify that $\lim_{h\to 0}\frac{e^{x+h} – e^x}{h} = e^x$.

Given $ƒ(x) = e^x$, verify that $$\lim_{h\to 0}\frac{e^{x+h} – e^x}{h} = e^x$$ and explain how this illustrates that $f'(x) = \ln e \cdot f(x) = f(x)$.
-1
votes
0answers
13 views

Mean value of periodic function

$f(t) = \begin{cases} A \sin\Omega t, & {-{T\over 2}\le t \le 0} \\ 0, & {0 \lt t \lt {T\over 2}} \end{cases}$ where $A, \Omega, T$ are constants If I want to calculate the mean value of ...
-1
votes
3answers
72 views

How do you integrate $e^{-st}t\cos(t)$?

I'm doing differential equations and specifically studying Laplace Transformations, where of course the Kernel is: $K(s,t) = e^{-st}$ And the Laplace Transformation $\mathcal{L}$ of a function ...
2
votes
2answers
62 views

Let $f:[0,1] \rightarrow \mathbb{R}$ be a continuous function.Show that $\lim_{n \rightarrow \infty} n \int_{0}^{1}e^{-nx}f(x)dx=f(0)$

Let $f:[0,1] \rightarrow \mathbb{R}$ be a continuous function.Show that $\lim_{n \rightarrow \infty} n \int_{0}^{1}e^{-nx}f(x)dx=f(0)$ I am making a claim is it really true? $\lim_{n \rightarrow ...
1
vote
2answers
40 views

Prove that $n^2 < n \cdot (n - 1) \cdot (n -2) $ [on hold]

How to prove or disprove that: $$ n^2 < n \cdot (n - 1) \cdot (n -2) $$ for every $n > 0$
1
vote
1answer
33 views

AP Calculus BC - Antideriative of cos(1-x^2)/(x^2 + root(x))

I'm taking the AP Calculus BC Exam next week and ran into this problem with no idea how to solve it. Unfortunately, the answer key didn't provide explanations, and I'd really, really appreciate it if ...
4
votes
7answers
102 views

Calculate the limit $\lim_{n \to \infty}\frac{ \ln(n)^{(\ln n)}}{n!}$

I wonder what the limit $\lim_{n \to \infty}\frac{ \ln n^{\ln n}}{n!}$ would be equal to. It is well known that the factorial function grow faster than an exponential but slower than $n^n$. But how ...
2
votes
1answer
852 views

Why is integral of $(\tan x)^3$ not $((\sec x)^2)/2 - \ln(\sec x)$?

Messing around with u substitution, I tried to integrate $\tan^3x$ as follows: $$ \tan^2 x \tan x = (\sec^2 x - 1)\tan x \\ u = \sec x \\ du = \sec x \tan x dx \\ du = u \tan x dx \\ du/u = \tan x dx ...
0
votes
2answers
37 views

With these two equations, how do I show that either a,b,c,d must be negative, if v is not 0?

If I have the equations $$ad-bc = u^2 +v^2$$ $$a+d = 2u$$ and I want $a, b, c, d \ge 0$, then how I can show that this is impossible, if $v \ne 0$? I.e., if $v \ne 0$, then one of $a,b,c,d$ must ...
2
votes
2answers
49 views

Show that $\lim_{h\to 0}\frac{e^h-1}{h} = \ln e = 1$ using at least two numerical examples.

Show that $$\lim_{h\to 0}\frac{e^h-1}{h} = \ln e = 1$$ using at least two numerical examples. To solve this should I find numbers for $h$ that makes those equations equal to $1$? And how should I ...
1
vote
1answer
26 views

Find the first terms of the Laurent series for: $\frac{e^{\frac{1}{z}}}{z^2-1}$

$\frac{e^{\frac{1}{z}}}{z^2-1}$ for $|z|>1$ I factored out the denominator and rewrote it to a geometric series and got the following expression: ...
-2
votes
0answers
23 views

Why is it that when n ≥ 1 the series is $\le$ 1/4 [on hold]

So how is the series $\sum_{n=1}^\infty \frac{1^2 * 3^2 * 5^2 ... (2n-1)^2}{2^2 * 4^2 * 6^2 ... (2n)^2}$ < 1/4 for n $\ge$ 1 is it because the series is divergent outside of the interval of ...
0
votes
2answers
15k views

Use the shell method to find the volume of the solid by rotating the region bounded by the given curves about the x-axis.

Rotate $y = x^3$ and $y = 8$, $x = 0$ Using the method of cylindrical shells, which is $2\pi rh \, dx$, or $2\pi xf(x) \, dx$ $$x^3 = 8$$ $$x = 2$$ $$2\pi \int_0^2 x(8-x^3) \, dx?$$ I have tried ...
0
votes
1answer
18 views

How to set the limit for interated integral of $f(x,y)$ over diagonally partitioned region

I would like to compute $$I = \int_{\mathcal{R}} f(x,y) d\mathcal{R}$$ $$ f(x,y) = \begin{cases} x^2, \quad 0 < x < y < \pi \\ y^2 , \quad 0 < y < x < \pi \end{cases}$$ ...
2
votes
2answers
38 views

Evaluate the following trignometric sum

I am interested in the following sum $$\sum_{\text{even } n=-\infty}^{\infty}\left(-\cos^2x\delta_{n,0}+\cos x\left(\frac{1-\cos x}{\sin x}\right)^{|n|}\right).$$ Wolfram alpha returns answer ...
0
votes
1answer
43 views

The volume of a Torus

A torus $\mathscr{T}$ with the equation $$z^2 + \left( \sqrt{x^2 + y^2} - 2 \right)^2 = 1.$$ (a) Give an equation with a close line in the plane $Oxz$ where $\mathscr{T}$ is a surface of ...
1
vote
1answer
30 views

Problem with multiple integrals of $\cos(x+y)$

I have a problem with this integral $\int_{0}^{\pi}\int_{0}^{\pi}\mid \cos\left(x+y\right)\mid dxdy$ I work with this problem, but the result of the book does not match with my result Note: The book ...
0
votes
0answers
20 views

how do I resolve equations that are both dependant on each other

I'm working on a project concerning the ideal power equation of aerodynamic bodies seen here: $$P = \frac{1}{2}C A D v^3 + \frac{W^2}{Db^2v}$$ where $P$ = power, $C$ = coefficient of drag, $A$ = ...
0
votes
1answer
57 views

Simple yet challenging integral, can it be solved analytically, and if so, the answer.

I'm trying to find solutions to the 3 following integrals. The first 2 are of the same form, only varying by a constant in the numerator within the cosine, and yes, x is a constant in the first one. ...
0
votes
1answer
14 views

Taking the derivative of $\epsilon \cdot(\ln X + \ln \beta) - \ln(1 + X^{\frac{\alpha}{\alpha - 1}})$ with respect to $\ln X$

So I am taking the derivative of $$\epsilon \cdot(\ln X + \ln \beta) - \ln(1 + X^{\frac{\alpha}{\alpha - 1}})$$ with respect to $\ln X$, where $X$ is a variable, $\epsilon, \beta, \alpha$ are ...
0
votes
0answers
13 views

O.D.E. in Homogeneity Lemma

Let $\psi: \mathbb{R}^{n} \to \mathbb{R}$ smooth such that $\psi(x) > 0$ for $x \in B(0,1)$ and $\psi(x) = 0$ for $x \notin B(0,1)$. Let $c \in S^{n-1}$ fix and arbitrary and consider the O.D.E. ...
1
vote
1answer
44 views

Radius of convergence: Why do we always use nth root test or ratio test?

Is this just definitional? I never did this in calculus but in complex analysis we defined the radius of convergence by the limit supremum of the nth root of the terms of a series. Why does this ...
0
votes
1answer
67 views

Calculate the upper sums Un and lower sums Ln,on a regular partition of the intervals, for the following integral:

Calculate the upper sums $U_n$ and lower sums $L_n$, on a regular partition of the intervals, for the following integral: $$\int_1^2 \lfloor x\rfloor dx$$ $$\Delta x=\frac{1}{n}$$ And then I'm ...
0
votes
0answers
16 views

Integrating product of powers of tangent and secant

In integrating trigonometric integrals of the form: $\int{\sec^m{x}\tan^n{x}}dx$ where $m$ and $n$ are positive integers, an exact definite method were known as written in any integral calculus ...
0
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0answers
12 views

Finding a relation between the integral bounds?

If $a,b,c,d,u_1,u_2$ are real numbers, what is $\frac{u_1}{u_2}$? I get that $\frac{u_1}{u_2} = \frac{b}{d} = \frac{a}{c}$ but apparently this is wrong and the answer is $\frac{b}{d}$. Why?
0
votes
1answer
15 views

Integrating equation with square on the bottom.

Say you are working with acceleration as a function of displacement and you are using calculus. $a = \frac{1}{(s - 600)^2}$. If you wanted to obtain velocity you'd use $a = v\frac{dv}{ds}$ so $vdv = ...
2
votes
2answers
22 views

Langrange Multiplier, to find maximum volume of a cone

Question: A right-angled triangle is rotated about one of its sides that form the right angle to a cone. Given that the sum of the lengths of two sides of the triangle that form the right angle is ...
4
votes
2answers
93 views

Evaluation of $ \int_0^\infty\frac{x^{1/3}\log x}{x^2+1}\ dx $

The following is an exercisein complex analysis: Use contour integrals with $-\pi/2<\operatorname{arg} z<3\pi/2$ to compute $$ I:=\int_0^\infty\frac{x^{1/3}\log x}{x^2+1}\ dx. $$ I don't ...
1
vote
3answers
35 views

Prove $ 0 \leq \frac{k}{n} \sum_{i = 1}^n \alpha_i - \sum_{i = j}^k \alpha_j$

Given a set of numbers that are non-decreasingly ordered $$\alpha_1\leq\alpha_2\leq\cdots\leq\alpha_n$$ Prove that $$ 0 \leq \frac{k}{n} \sum_{i = 1}^n \alpha_i - \sum_{j = 1}^k \alpha_j$$ for ...
0
votes
2answers
52 views

find integral $\int_{1}^{-1} \sin\left(x^3\right) dx$

$$\int_{1}^{-1} \sin\left(x^3\right) dx$$ so I know the result is $0$ since above function is odd. But how to compute this integral?
0
votes
1answer
23 views

Proof of the Strenghtened Limit Comparison Test

I'm studying on my own using Bonar and Khoury's Real Infinite Series. I understand the proof of the "regular" Limit Comparison Test( a link to google books, page 23 ) but the book doesn't provide a ...
6
votes
1answer
97 views

Does rationality of $\cosh(nx)$ and $\cosh((n+1)x)$ imply rationality of $\cosh(x)$?

Suppose that $x\in\mathbb{R}^+$ and $n\in \mathbb{N}$. If $\cosh(nx)$ and $\cosh((n+1)x)$ are rational, can we show that $\cosh(x)$ is rational too? I guess the following equalities should be useful: ...
0
votes
0answers
16 views

$f:[-1,1]\to \mathbb{R}$ such that $f(x)=x$ if $x=1/n$ and $n$ is a nonzero integer, and $1-2x$ otherwise. Question: is $f$ differentiable at $1/3$?

My thought was that no, it wasn't. I was thinking by the Denseness of $\mathbb{Q}$, there would be irrationals $a,b$, such that $a<1/3<b$, which means $a,b$ would be on $1-2x$, and $1/3$ would ...
1
vote
1answer
30 views

Why is this piece-wise limit equal to 2?

$$f(x) = \begin{cases} 2x-2, & x < 3 \\ 2x-4, & x \ge 3 \end{cases} $$ Why is $$\lim\limits_{h \to 0^+} \frac{f(3+h)-f(3)}{h} = 2 ??$$ Note the (+) in the limit. If $h \to 0$ from ...
-1
votes
0answers
27 views

Calculate stationary points of $x^3 \textrm e ^{\frac {-x^2} {a^2}}$ [on hold]

Calculate all of the stationary points of $$x^3 \textrm e ^{\frac {-x^2} {a^2}}$$ where $a > 0$. Thanks in advance.
0
votes
1answer
24 views

$| \langle a , i \rangle| \leq \| a\|$ if $\|i\|=1$ this space is a normed vector space upon $\langle , \rangle$ . Why does this apply?

I tried over Cauchy Schwarz to conclude, but could not. Anyone see why this is ? The term: normed vector space upon $\langle , \rangle$ i hear for the first time, Im assuming it means that: $$\|a \| = ...
1
vote
1answer
52 views

Product of limits when one is zero and the other does not exist

We have two functions $f(x)$ and $g(x)$, such that $\lim_{ x\to 0}f(x)=0$ but $\lim_{x\to 0} g(x)$ does not exist. Would that mean that $\lim_{x\to 0}f(x)g(x)= 0$, assuming we don't divide by zero ...
0
votes
0answers
14 views

simple integral recurrence relaiton

I have $$I_n = 2\lambda\int_0^\infty x^nx\exp((-\lambda)x^2) \ dx$$ I have the relation: $I_n = \frac{n}{2\lambda} I_{n-2}$. I am trying to compute $I_{2n+1}$ I got that $I_{2n+1} = ...