3
votes
1answer
24 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 ...
1
vote
1answer
33 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
39 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
20 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
44 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
224 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
74 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
222 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
104 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
212 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
113 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
193 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
65 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
54 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
62 views

Generate a formula for x in my case

Kindly see the following, ...
1
vote
1answer
66 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
61 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
131 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
208 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
35 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
103 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
131 views

Need help on this problem about maximizing a function

"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 $k$, ...
9
votes
4answers
506 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
310 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= ...
0
votes
1answer
42 views

About continuous functions and aritmethic progression

I've try solve this question, but I haven't sucess... The problem is the following: A continuous functions $f:[a,b]\rightarrow \mathbb{R}$ assume positive and negative values in its domain, show ...
9
votes
3answers
289 views

Solve the functional equation $f(x)=f\left({x\over 3}\right)+f\left({2x\over 3}\right)$ with $f : [0,\infty) \to \mathbb R$ continuous

Solve the functional equation $$f(x)=f\left({x\over 3}\right)+f\left({2x\over 3}\right)\qquad \forall x\geq 0$$ with $f : [0,\infty) \to \mathbb R$ continuous. I can't manage to get this one ...
2
votes
0answers
158 views

Personal project help needed [closed]

This is the first time i have posted on this site. If the people here judge post as brutal as the programming forums. I apologize ahead of time my math background isn't deep enough to make a thesis ...
0
votes
2answers
46 views

How to find initial function of a function

I am stuck in this problem. How do I find the initial function of a given function? I am learning integral and there in the formula $S=F(b)-F(a)$ is $F$ the initial function of $f$. For example, ...
2
votes
4answers
86 views

rewrite $2ie^{i\pi}+i^3$

i am asked to rewrite $2ie^{i\pi}+i^3$ into $x+iy$ form. i just tried all what i know so far, but couldnot come to solution. i said: $2ie^{i\pi}+i^3=2ie^{i\pi}-i$ but further i am stuck really. i am ...
1
vote
2answers
399 views

Need help with an integration word problem. This appears to be unsolvable due to lack of information.

I'm not sure I understand what to do with what's given to me to solve this. I know it has to do with the relationship between velocity, acceleration and time. At a distance of 45m from a traffic ...
9
votes
1answer
1k views

Finding the general term of a sequence

I would like to find an expression for the sequence $\;\{a_n\},\;n=0,\,1,\,2,\,3,\,\dots,\;$ $$-\frac{1}{6},\,\frac{2}{7},\,\frac{5}{8},\,\frac{8}{9},\,\frac{11}{10},\,\ldots$$ So by trial and ...
4
votes
2answers
316 views

prove the divergence of cauchy product of convergent series $a_{n}:=b_{n}:=\dfrac{(-1)^n}{\sqrt{n+1}}$

i am given these series which converge. $a_{n}:=b_{n}:=\dfrac{(-1)^n}{\sqrt{n+1}}$ i solved this with quotient test and came to $-1$, which is obviously wrong. because it must be $0<\theta<1$ so ...
1
vote
2answers
198 views

Prove $\sqrt[n]{n}\leq 1+\frac{2}{\sqrt{n}}$

I am given this statement and I need to prove it. the statement is for all $n\ge 1$: $$\sqrt[n]{n}\leq 1+\frac{2}{\sqrt{n}}$$ I am trying to prove with induction. But I am stuck for step n=k+1, how ...
1
vote
3answers
261 views

Polar Curve and maximum width

Find the maximum width of the petal of the four-leaved rose $r = \cos2\theta$, which lies along the x-axis Here is the solution Can someone tell me how on earth did the solution come up with ...
0
votes
3answers
154 views

Displacement from variable forces

Displacement from a singular force over time is given by the equation $${1\over2}{F\over m}t^2 $$ Where F is force, m is mass, and t is time. But what if F is variable over time? My best guess is to ...
1
vote
1answer
64 views

check for Continuity $x \rightarrow 2x^4-2$

Given the function: $f:\Bbb{R}\rightarrow \Bbb{R},\quad x \rightarrow 2x^4-2$ i am asked to check for Continuity for all values of $x$. i am now overasked how to do this since $\Bbb{R}$ is not a ...
1
vote
1answer
577 views

Maximum area for fixed perimeter of a triangle

I'm trying to prove that the triangle of largest area for a given perimeter is equilateral, but I'm having some difficulties. I've done 2 different proofs for a similar problem but for rectangles - ...
9
votes
2answers
461 views

Evaluating $\int_{0}^{x} e^t \sqrt{2 + \sin(2t)} \, dt$

I was recently asked to evaluate the following integral: $$\int_0^x e^t \sqrt{2 + \sin(2t)} \, dt$$ It was beyond the ken of WolframAlpha, which I find quite discouraging. Does anyone have an idea ...
1
vote
1answer
145 views

Related Rates calculus problem

For some reason I keep getting this question wrong. Suppose a 6 feet tall man is walking away from a 15 foot tall lamp post at 5ft/s. What is the rate at which the man's shadow is moving when he is ...
1
vote
2answers
71 views

simple Inequality.

Prove: if $x \in (0,1)$ then: $$ \sqrt{\frac{1-x}{1+x}} < \frac{\ln{(1+x)}}{\sin^{-1}x} < 1$$ I have been thinking on using maximization to solve this problem, but the function is not nice! Is ...
1
vote
0answers
61 views

Monotonicity of Solution to $∫_0^1(x- \frac 1 g )x^{ \frac 1{gr-1} }\frac {\ln x}{(1+xgr)^2}dx = 0$

Given any parameter $g > 1$, I can show that the following equation has a unique solution $r^{*} \in (\frac 1 g, \infty) $, $$∫_0^1(x- \frac 1 g )x^{ \frac 1{gr-1} }\frac {\ln x}{(1+xgr)^2}dx = ...
3
votes
2answers
1k views

What is the algorithm for long division of polynomials with multiple variables?

I was helping a high-school student last night whose teacher had given as a homework problem the division $$\frac{15x^4-y^2}{x^2+y};$$ I tried a heuristic involving splitting off a difference of ...
0
votes
1answer
30 views

Trouble on writing the function for an account record pagination problem

I have this problem about account record pagination. I have to write the function that describes the total pages needed to contain those account records. This is the problem: ...