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

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1answer
19 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 ...
1
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0answers
36 views

It's raining at a 45 degree angle. Should I walk or run?

A variation on the classic problem, in which you are walking home and you have to calculate whether you get wetter from vertically failing raindrops by walking or running an equal distance. In the ...
0
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0answers
17 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
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1answer
12 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 ...
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0answers
8 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
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1answer
41 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 ...
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1answer
53 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. ...
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0answers
74 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 ...
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0answers
12 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 ...
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0answers
10 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?
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1answer
13 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
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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 ...
0
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2answers
48 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?
1
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1answer
28 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 ...
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0answers
15 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 ...
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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} = ...
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0answers
21 views

Arc length using bisection error - root finding methods.

Apologies if this is poorly formatted - it is my first time posting a question. I'm trying to find evenly spaced arc lengths along a function (in this case a sinusoid) using the bisection method. ...
0
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1answer
26 views

Radius of Convergence of $\sum_{n = 0}^{\infty} \frac{(-1)^nn!x^n}{n^n}$

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 ...
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0answers
16 views

How to determine when the Green's function do not exist?

I've been solving some problem which asks us to find the Green's functions for some problems when it exists. Now, there's a theorem which allows us to guarantee that it exists. The theorem is as ...
0
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1answer
16 views

Prove point-wise convergence for the sequence $\{f_k\}$ of functions

Consider the sequence $\{ f_k\}$ of functions in $C[0,1]$ defined by $$ f_k(x) = \begin{cases} 0, & 0 \leq x \leq \frac{1}{k} \\ 2(k^{3/2} x - k^{1/2}), & \frac{1}{k} \leq x \leq ...
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1answer
21 views

How would I show $|x| \le 1$ given the equation for $x$ the expression in the equation?

The expression is $x = \sin(\theta /2)$. I am asking how would I show that $\sin(\theta/2)\le1$ based on the expression? I already know that the biggest $\sin$ will ever get is $[-1, 1]$ which is the ...
0
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1answer
18 views

Regarding smoothness of a function defined by integral

While reading a differential geometry book, I came across a theorem which makes the following claim: Let $f$ be a $\mathcal{C}^\infty$ function on a convex open subset $U\in \mathbb{R}^n$, and let ...
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0answers
17 views

Alternating Euler sums with even index

We are all aware of the generating function of $\frac{x \arctan x}{x^2+1}$ which is: $$\frac{x \arctan x}{x^2+1} = \sum_{m=1}^{\infty} (-1)^m \left ( \mathcal{H}_{2m} - \frac{1}{2} \mathcal{H}_m ...
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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.
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0answers
15 views

Can a curve be represented by a differential equations? [on hold]

How from the variational principle curvature shape of a free hanging chain held at two fixed ends.. ?
1
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1answer
30 views

Functions of the form $f(x) = k^x - x^k$

Let $f: \mathbb{R} \rightarrow \mathbb{R},\ f(x) = k^x - x^k$ where $k \in \mathbb{R}$ is a given constant. Currently I am thinking of positive $k$ and positive $x$ because there would be complex ...
0
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1answer
19 views

Volume 4-dimensional sphere

I'm studying Fubini's Theorem and Change of Variables Theore in class, and one of the exercises from last year exam was calculate the volumen of the 4D sphere. I searched on Internet how can I do that ...
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5answers
78 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 ...
1
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0answers
21 views

Let $\alpha$ be a real number. Find the value of $\alpha$ for which the given function is continuous and differentiable.

Let $\alpha$ be a real number. Consider the function $$g(x)=(\alpha+|x|)^2e^{(5-|x|)^2}, \ \ \ -\infty<x<\infty $$ $(i)$ Determine the values of $\alpha$ for which $g$ is continuous at all $x$. ...
0
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1answer
14 views

Proving the interval in which a solution is valid

Question: Verify that both $y_1(x) = 1-x$ and $y_2(x)= \frac{-x^2}{4}$ are solutions of the initial value problem $$\frac{dy}{dx}=\frac{-x+(x^2+4y)^\frac{1}{2}}{2}, \ \ \ y(2)=-1$$ and determine ...
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2answers
27 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 ...
1
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2answers
50 views

Find the Laurent Series for $\frac{1}{e^z-1}$ for $0<|z|<2\pi$

Since $\frac{1}{e^z}<1$ I figured I could rewrite it into a geometric series: $$\sum_{n=1}^{\infty} \frac{1}{(e^z)^n}$$ But this seems to be way off the mark. I think I am confused about when ...
1
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1answer
17 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: ...
0
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1answer
20 views

Find the radius of convergence for $\sum_{n=1}^{\infty}\frac{2^n}{3^n+4^n}z^n$

$$\sum_{n=1}^{\infty}\frac{2^n}{3^n+4^n}z^n$$ What I've done is try to evaluate the expression sans $z^n$ with the root test. $$\sqrt[n]{\frac{2^n}{3^n+4^n}}=\frac{2}{\sqrt[n]{3^n+4^n}}$$ But I'm ...
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2answers
61 views

Prove that $a \sqrt{b + c} + b \sqrt{c + a} + c \sqrt{a + b} \le \sqrt{2(a+b+c)(bc + ac + ab)}$ for $a, b, c > 0$

Prove for $a, b, c > 0$ that $$a \sqrt{b + c} + b \sqrt{c + a} + c \sqrt{a + b} \le \sqrt{2(a+b+c)(bc + ac + ab)}$$ Could you give me some hints on this? I thought that Jensen's inequality might ...
2
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2answers
93 views

How evaluating this integral: $\int_0^\infty \frac{e^{-x}+x-1}{x(e^{2x}-e^{-2x})}dx$?

I don't even know where to start, please can someone help me on this integral. $$ 4\int_0^\infty \frac{e^{-x}+x-1}{x(e^{2x}-e^{-2x})}dx=\gamma+\ln\frac{16\pi^2}{\Gamma^4(\frac{1}{4})} $$
6
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1answer
93 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: ...
3
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3answers
123 views

How is the last “=” true?

How can the last equality be true? $$ G(u)=\frac{g}{(1+u)^g-1}-\frac1u=\frac{g}{gu + \cdots + u^g} - \frac{1}{u}=\sum_{k=0}^\infty c_ku^k $$
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0answers
21 views

solve the pde z=pq?

i tried tried it using charpit method $$f=z-pq$$ $$\frac{\partial f}{\partial x}=0,\frac{\partial f}{\partial y}=0,\frac{\partial f}{\partial p}=-q,\frac{\partial f}{\partial q}=-p,\frac{\partial ...
0
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2answers
51 views

For what values of $a$ will $y=ax$ be a tangent to $x^2+y^2+20x-10y+100=0$

For what values of $a$ will $y=ax$ be a tangent to $x^2+y^2+20x-10y+100=0$ I tried to solve this question by differentiating and making it equal to $0$ and solving for $x$ and i got $-10$ as an ...
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0answers
32 views

The area of trapezium is given by $A=(a^2-x^2)(x+a)$. Find x for the area to be a maximum and find A max.

For the diagram, the area of trapezium is given by $A=(a^2-x^2)(x+a)$. Find $x$ for the area to be a maximum and find $A$ max. Hi, I'm not sure how should i do this question. Can anyone help me with ...
3
votes
2answers
34 views

$\lim x_n = a$, $\lim \frac{x_n}{y_n}=b$ then $\lim y_n = \frac{a}{b}$

I must prove: $\lim x_n = a$, $\lim \frac{x_n}{y_n}=b$ then $\lim y_n = \frac{a}{b}$ Well, I know that $$\lim x_n = a \implies |x_n-a|<\epsilon$$ $$\lim \frac{x_n}{y_n} = b \implies ...
6
votes
2answers
66 views

Solution of functional equation $f(x+y)=f(x)+f(y)+y\sqrt{f(x)}$

If $x,y\in \mathbb{R}$ and $f(x+y)=f(x)+f(y)+y\sqrt{f(x)}$ and $f'(0)=0\;,$ Then $f(x)$ is $\bf{My\; Try::}$ Using $$f'(x) = \lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h} = \lim_{h\rightarrow ...
0
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2answers
19 views

Taking derivatives of the integration bounds

It has been a while since I took rudimentary calculus classes, so I might be slipping on the basics. I tripping on how to differentiate the lower and upper bounds of an integral. For example, lets ...
0
votes
1answer
45 views

If $\int_{a}^b f(x)g(x) dx = 0$ for all continuous functions $g$ satisfying $g(a) = g(b) = 0$, show $f = 0$

Suppose $f$ is a continuous function over $[a,b]$ and that $\int_{a}^b f(x)g(x) dx = 0$ for all continuous functions $g$ satisfying $g(a) = g(b) = 0$. Show $f = 0$. We use the mean value theorem ...
1
vote
1answer
22 views

$a_n, b_n$ bounded, $a_n+b_n=1$,$z_n\to a$ and $t_n\to a$, then $(a_nz_n+b_nt_n)\to a$

I must show that if $a_n, b_n$ are bounded such that $a_n+b_n=1$, and if $z_n\to a$ and $t_n\to a$, then $(a_nz_n+b_nt_n)\to a$ My idea was: $$(a_n+b_n)(z_n+t_n) = a_nz_n+a_nt_n+b_nz_n+b_nt_n$$ I ...
1
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2answers
23 views

most general antiderivative involving sec x

I'm stumped on how to get the most general antiderivative, $F(x)$, of $f(x)=e^x+3secx(tan x + sec x)$. First, I split the equation on addition, since $\int[f(x)+g(x)]dx=\int f(x)dx+\int g(x)dx$ ...
1
vote
1answer
23 views

Write the series using sigma notation: $f(x)= 1 - (x^2)/2! + (x^4)/4! - (x^6)/6! +\cdots$

$$f(x)= 1 - (x^2)/2! + (x^4)/4! - (x^6)/6! + \cdots$$ I don't know how to get the signs to work like negative, then positive. I have tried to make it like the following: $(-1)^{n-1} \frac{x^{2n-1}}{ ...
1
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1answer
15 views

Trouble with parametrizing function

hope you're all well! I just started learning about line integrals in class today, and I'm having a difficult time understanding how and why the solution manual came up with the parameterization for ...
0
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0answers
13 views

Matrix Derivative/Operation of Flat(A).dot(B)?

In calculating the gradient of a convolutional neural net by hand, I am running into a snag. In the middle of the net, going forward, there is a layer where I take an array $A$, flatten it into a ...