1
vote
2answers
44 views

Damped simple harmonic oscillator, phase space

I want to calculate and draw the phase space trajectory of this damped harmonic oscillator: $$\ddot{x}+\gamma\,\dot{x}+\omega^2x=0$$ for the two cases $\gamma=2\omega$ and $\gamma=\omega$. I'm ...
1
vote
1answer
24 views

Train distances leaving at certain times

A train leaves Boston to Fort Lauderdale traveling at $125$ mph. An hour later, another train leaves Fort Lauderdale traveling to Boston at a rate of $140$ mph. When the two trains meet each other, ...
2
votes
0answers
56 views

Least sum of power of distances

Let $n$ points in a $3$-dimensional space. Find the point $X$ that minimizes the sum of distances $\|A_1X\|^q+ \|A_2X\|^q + ... +\|A_nX\|^q $ (where $q \in \mathbb{Q}$). Are there any general ...
4
votes
2answers
169 views

Least sum of distances

Problem: Let $A, B, C, D$ be points in a $3$-dimensional space. Find the point $X$ that minimizes the sum of the distances $AX+ BX + CX + DX$. Context: During a course, I was assigned a ...
2
votes
1answer
49 views

Integral of $\frac1{\cos^n x}$

Hi guys I have already proven for an assignment that: $$\int\cos(x)^n dx=\frac{1}{n}\cos(x)^{n−1}\sin(x) + \frac{n-1}{n}\int\cos(x)^{n−2}dx$$ Now we have been asked to calculate ...
1
vote
2answers
40 views

Showing the summation of numbers

Using each of the digits 1 through 9 once, form numbers whose sum is 100. If you think it can't be done, then prove it. My attempt: I say it can't be done because the sum of all numbers $1-9$ is ...
0
votes
1answer
33 views

Integer solutions of an equation that is set to a number

How many integer solutions for $a$ and $b$ in $(ab)/(a+b)=3600$? My attempt: $(ab)/(a+b)=3600$ = $ab=3600(a+b)$ = $ab=3600a+3600b$ =$ab=3600a=3600b$ Dividing $3600b$ on both sides ...
-1
votes
2answers
48 views

Solving an algebraic equation for x

$(($ 3^$2\sqrt{3x})$/4$)$ $+3=$ 3^$\sqrt{3x}$ = $($ (3^${2}*{3x^{1/2}}$)/4$)$ $+3=$ 3^${3x^{1/2}}$ After simplifying: = ($3^{6x^2}$ $+ 3$)/4 $= 3^{3x}$ = $3^{6x} + 3 = 12^{3x}$ I tried ...
10
votes
8answers
252 views

Evaluate $ \int_{0}^{1} \ln(x)\ln(1-x)\,dx $

Evaluate the integral, $$ \int_{0}^{1} \ln(x)\ln(1-x)\,dx$$ I solved this problem, by writing power series and then calculating the series and found the answer to be $ 2 -\zeta(2) $, but I don't ...
1
vote
1answer
39 views

How do i solve this to find PMT?

I know this may seem like a stupid question but i've been up late working on this math assignment and this question just isn't working when i transpose it. So this is the formula to find Present ...
0
votes
3answers
69 views

Derivation of the “Combined Work Formula”

Before I get to my question, some background: Person $A$ can paint a fence at the rate $9 \frac{hour}{fence}$ (or equivalently $\frac{1}{9} \frac{fence}{hour}$) Person $B$ can paint a fence at the ...
0
votes
1answer
51 views

Finding distance using rates of change — best approach?

The question: A man drives from state $A$ to state $B$ going $60 \frac{miles}{hour}$. Then he returns from state $B$ to state $A$, driving $45 \frac{miles}{hour}$. His total driving time is $2.5 ...
1
vote
0answers
27 views

Slicing through a cuboid containing spheres, how many are exposed to the surface and what is their combined volume

So I place spheres of radius chosen at random from a normal distribution of known mean and standard deviation in a cub or cuboid at random (not overlapping) until a known density of the entire cube is ...
1
vote
1answer
138 views

Comparing $\pi^e$ and $e^\pi$

Comparing $\pi^{e}$ and $e^{\pi}$ I read the answer there but I didn't understand one thing. How I should know to put $\dfrac{π}e-1$ instead of $x$? If I had this question on a test, I had no idea ...
15
votes
4answers
426 views
+50

Calculate $\frac{1}{5^1}+\frac{3}{5^3}+\frac{5}{5^5}+\frac{7}{5^7}+\frac{9}{5^9}+\cdots$

I'm an eight-grader and I need help to answer this math problem. Problem: Calculate $$\frac{1}{5^1}+\frac{3}{5^3}+\frac{5}{5^5}+\frac{7}{5^7}+\frac{9}{5^9}+\cdots$$ This one is very hard for ...
6
votes
2answers
145 views

Advice for self-studying Inequalities and Calculus

I'm interested in self-studying the following books over the next year or so: Spivak's Calculus (I'm already in Ch. 5 and it is very slow going) The Cauchy-Schwarz Master Class by J. Michael Steele ...
0
votes
0answers
22 views

Difficult Integral in functional basis

Let $$g(x)=\int f\prime(x)\left[\frac{4}{3}x^2+4x^3+(2x^2+4x^3)f(x)+6x^2f^2(x)+xf^3(x)\right]dx$$ express $g(x)$ in terms of $\{1,x,x^2,x^3,....\}$ and $\{f(x),f^2(x),f^3(x),...\}$. Is there a clever ...
2
votes
1answer
56 views

Motivation and Derivation of the Riccati Equation Transformation

Given a Riccati Equation which is differential equation of the form: $$ \frac{dy}{dx} = a_0 (x) + a_1 (x)y + a_2 (x)y^2 $$ It is well known that the transformation: $$ y = -\frac{1}{a_2(x)} ...
0
votes
1answer
36 views

How do I prove this statement?

I have to prove that if $$u=t^{\lambda}y(z)$$ and $$z=\frac{x}{\sqrt{t}} \,\,,$$ then $$\frac{\partial{u}}{\partial{t}}=\frac{\partial ^{2}{u}}{\partial{x}^{2}} \Rightarrow ...
0
votes
1answer
49 views

Using Poisson's integral formula

The problem asks to prove the following equality using Poisson's integral formula (or Poisson kernel, if I understood correctly from Wikipedia): $$\int_0^{2\pi} \frac{e^{\cos ...
9
votes
2answers
196 views

Computing $\int {\dfrac{\csc^{2014}x-2014}{\cos^{2014}x} dx}$

I don't know how to compute: $$\int {\dfrac{\csc^{2014}x-2014}{\cos^{2014}x} dx}$$ I have tried substituting $t=\tan ^{2} x$ but got nothing out of it. I know there's some trick involved, but ...
3
votes
2answers
82 views

How do I evaluate this integral by hand?

TL;DR how do I evaluate $\int_0^{2 \pi } \frac{1}{\cos ^2(\theta )+1} \, d\theta$ by hand? I'm trying to solve this problem: Find the volume of the region defined by $x^2+xy+y^2+yz+z^2\le1$. ...
0
votes
1answer
39 views

Finding/approximating 2 unknowns using one equation

I’m doing experimental data in a chemistry lab and I have faced this mathematical problem at a point of my work. Hope you guys can help me with that. What would be the best way to find two constants m ...
3
votes
1answer
32 views

Maximum likelihood to throw exactly two 6s

One throws a dice $n$ times. For which value of $n$ is maximum the probability to obtain exactly two 6s? I get $$n=11 \text{ or } n=12.$$ My solution: the probability to obtain exactly two 6s in ...
2
votes
1answer
38 views

Interesting question about functions

I saw the following question and I would like to share. I don't know the answer. Suppose that the function $f:\Bbb{N}\to\Bbb{N}$ has the property $f(f(n))<f(n+1)$ for any $n\in\Bbb{N}$. Prove that ...
1
vote
1answer
31 views

Random variable of a store

The weekly profit in thousands of dollars of Miller's Office Supply Store is random variable X whose cdf is given as follows: $F(x)=0$ for $x<0$; $F(x)=(3/32)(2x^2-x^3/3)$ for $0 \leq x \leq 4$; ...
0
votes
1answer
51 views

Solving $\int\sqrt{1+(-2ax+b)^2}\;dx$

List item What solution $$\int\sqrt{1+(-2ax+b)^2}\;dx$$Unable to develop anything ...$~$:'( I tried completing squares, but can not move much.
0
votes
0answers
22 views

The radius of a circle grows at a rate of $ 30$ cm/s, that rate increases the area of ​​the circle with respect to time?

I would like to address the question The radius of a circle grows at a rate of $ 30$ cm/s, that rate increases the area of ​​the circle with respect to time? I know I have to derive, but where I ...
0
votes
1answer
48 views

Solve equation of form $(d_B - 0.32)^{0.8} (d_B + 1.45)^{1.1} = exp(0.8)$ for the term $d_B$

I have the following equation: $$ \left(\frac{\sqrt{d_B}-\sqrt{d_{Beq}}}{\sqrt{d_{Bmin}}-\sqrt{d_{Beq}}} \right)^{1-\frac{c1}{c2}}\left(\frac{\sqrt{d_B}+\sqrt{c3}}{\sqrt{d_{Bmin}}+\sqrt{c3}} ...
4
votes
8answers
283 views

Evaluating $\int \frac{1}{\sqrt{x^2 + a^2}}\, dx$ without resorting to trigonometric $u$-substitution

I am looking for a quick and intuitive way to evaluate this indefinite integral without resorting to any trigonometric functions. I'm not sure if it is at all possible to do so, but I was just ...
2
votes
1answer
44 views

Simplify $y^\top x -\log(\sum_i e^{x_i})$

Simplify $\sup_x y^\top x -\log(\sum_i e^{x_i})$ The first order conditions yield $y_i=\frac{e^{x_i}}{\sum_i e^{x_i}}$. How do I eliminate $x_i$ from the equation? I know the answer to be $\sum ...
5
votes
1answer
78 views

An integration question.

An help in the following problem: Let $f:[-1,1] \longrightarrow \mathbb{R}$ a $C^1$ function, i.e., continuously differentiable. Suppose that we have $$\int_{-1}^{1} f(x)\;dx = \pi ...
0
votes
1answer
272 views

Finding dy/dx as a function of x for a dog-walker dragged by a dog travelling in a straight line

Hello. I was wondering if anyone could provide some insight into how to solve the following Calculus word problem: Max is walking his dog Beau in the Cartesian plane, with the leash between them at ...
0
votes
2answers
117 views

Solving a system of equation

Solve the following system of equations: $\left\{ \begin{align} & 2{{x}^{2}}-5xy-5{{y}^{2}}+x+10y-35=0 \\ & {{x}^{3}}+5x{{y}^{2}}+42=0 \\ \end{align} \right.$ By using a computer, I have ...
6
votes
5answers
234 views

How can I express the sum of $\sin a+\sin2a+\sin3a+\cdots+\sin(n-1)a$?

I want to sum up the partials of a harmonic series, how do I do it? If I was using the 'Lagrange trigonometric identity to solve this problem', how would I plot it on Wolfram mathematica (using which ...
2
votes
1answer
28 views

Question of how to write certain response

What is the domain of the function $$f(x,y)=\frac{1}{x^2+y^2-1}$$ The answer is clear, right?! $$x^2+y^2-1\neq 0\Longrightarrow x^2+y^2\neq1$$What is the point contained in the circle of ...
1
vote
2answers
171 views

Problems with the existence of limits of several variables

I started some time studying calculus of two variables, and I'm having difficulty in knowing (and prove) that a limit does not exist, how could I resolve, for example, these, whose statement asks to ...
12
votes
1answer
228 views

Teaser or fun calc equation to surprise husband (physicist/EE) at work

I am a geneticist and unfortunately have not worked much with advanced calc since undergrad. In genetics, as you likely know, a male is denoted as XY and a female as XX. I plan to leave a riddle for ...
1
vote
2answers
68 views

Comparison between Bessel's coefficients

The spatial solution is written as $$\Phi_k(r) = r^{1-\frac{d}{2}} \left(c_1 J_{1-\frac{d}{2}}(k r) + c_2 Y_{-1+\frac{d}{2}}(kr)\right).$$ In the case $d=3$, the solutions can be written as ...
2
votes
5answers
55 views

Help with minimization problem

help me, if $x$ and $y$ are real such that $3x-4y = 12$, determine the minimum value of $z = x ^ 2 + y ^ 2$?$$$$I thought of $$3x-4y = 12\Longrightarrow x=4\frac{y+3}{3}\\z = x ^ 2 + y ^ ...
2
votes
1answer
95 views

Generate a formula for x in my case

Kindly see the following, ...
1
vote
1answer
70 views

Need help with boundary conditions of a differential equation.

QUESTION: A particle $A$ is moving along the $X$ axis at a constant horizontal velocity $u\hat{i}$. Another particle $B$ is moving such that its velocity vector always points towards the particle ...
1
vote
1answer
63 views

Optimisation Problem about convex curve

A smooth closed curve C is said to be convex if it lies wholly to one side of each tangent . Show that for the triangle of minimum area circumscribed about C that each side is tangent to C at its ...
2
votes
1answer
220 views

What is another, perhaps quicker and nicer way of solving this question?

In a calculus book, a question reads: A car is traveling at night along a highway shaped like a parabola with its vertex at the origin. The car starts at a point 100 m west and 100 m north of the ...
0
votes
1answer
291 views

Exact area using limits and Riemann sum

the i need to find the exact area under tha curve of the function $f(x)=4+3x-x^2$ on the interval $[-1,3]$ using limits and a Riemann Sum. I have nothing started, because I am confused on where to ...
0
votes
1answer
36 views

ball on the moon part 3

This the final parts to my previous question ( On the surface of the moon ) The original question reads "On the surface of the moon, acceleration due to gravity is approximately 5.3 feet per second ...
0
votes
2answers
124 views

On the surface of the moon

On the surface of the moon, acceleration due to gravity is approximately 5.3 feet per second squared. Suppose a baseball is thrown upward from a height of 6 feet with an initial velocity of 15 feet ...
-1
votes
2answers
153 views

Need help on this problem about maximizing a function [closed]

"As an epidemic spreads through a population, the number of infected people, $I$, is expressed as a function of the number of susceptible people, $S$, by: $$I(S)=k \ln(S/S_0)-S+S_0+I_0$$ For $k$, ...
10
votes
4answers
549 views

The value of $\sqrt{1-\sqrt{1+\sqrt{1-\sqrt{1+\cdots\sqrt{1-\sqrt{1+1}}}}}}$?

How to find value of $\sqrt{1-\sqrt{1+\sqrt{1-\sqrt{1+\cdots\sqrt{1-\sqrt{1+1}}}}}}$ ? I've calculated it by MATLAB for some finite terms and I've got : $0.3001 - 0.4201i$, but I don't know how to ...
3
votes
1answer
323 views

Evaluating the time average over energy

For more info see the article equations 37 Edit: The $\varepsilon ^3 $ has vanished due to time average. But how to get the 4th order? Let us define some function for scalar field $$\phi= ...