Homework questions are welcome as long as they are asked honestly, explain the problem, and show sufficient effort. Please do not use this as the only tag for a question. For the answers on homework questions, helpful hints or instructions are preferred to a complete solution. Please do not add ...

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0
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0answers
53 views

Balls and Boxes [on hold]

How many ways are there to put 6 balls in 3 boxes if: a)the balls are not distinguishable and neither are the boxes? b)the balls are not distinguishable but the boxes are? c)the balls are ...
0
votes
1answer
263 views

Assignment: Find $a$ and $b$ such that a piecewise function is continuous

I'm having trouble solving a problem given in an assignment: If the following function $f(x)$ is continuous for all real numbers $x$, determine the values of $a$ and $b$. $$ ...
12
votes
1answer
139 views

$\int_0^{2\pi}e^{\cos x}\cos(\sin x)dx$ [duplicate]

$$\int_0^{2\pi}e^{\cos x}\cos(\sin x)dx$$ I tried Integration by parts but failed. Wolfram alpha gives answer in decimal points which are same as of $2\pi$. Any hints or suggestions will be helpful.
1
vote
1answer
48 views

property of Lebesgue measure

Suppose that $E \in \mathcal{M}$, show that for each $\epsilon > 0$ there is a closed set $F$ such that $F \subset E $ and $\lambda(E \setminus F) < \epsilon$, where $\mathcal{M}$ is the ...
3
votes
1answer
39 views

Show that $\lim_{s \to \infty}F_s(t) = F(t)$ uniformly for $t \in (0,+\infty)$

Given the following functions: $$ F(t)= \int_0^\infty e^{-tx}\dfrac{\sin{x}}{x}\,dx, \quad t>0$$ $$ F_s(t)= \int_0^s e^{-tx}\dfrac{\sin{x}}{x}\,dx, \quad t \geq 0, s>0$$ Show that $\lim_{s \to ...
0
votes
5answers
75 views

Some examples of applications of Game Theory

I'm approaching my junior year of HS now, and I'm looking for a good science fair project to do. I love mathematics, so I decided to a category of mathematics that can help base logical conclusions to ...
3
votes
5answers
192 views

Calculating the area

For the two graphs $ \frac{x^3+2x^2-8x+6}{x+4} $ and $ \frac{x^3+x^2-10x+9}{x+4} $, calculate the area which is confined by them; Attempt to solve: Limits of the integral are $1$ and $-3$, so I took ...
4
votes
0answers
90 views

Line integrals and path independence

Consider $\textbf{F}(x,y)=\frac{-y}{x^2+y^2}\textbf{i}+\frac{x}{x^2+y^2}\textbf{j}$. Let $C_1$ be the upper half of the unit circle oriented counterclockwise, and let $C_2$ be the lower half of the ...
0
votes
2answers
24 views

Trigonometry Identities questions

Given that $\sin\theta =\dfrac15$ and $0<\theta <\dfrac{\pi}2$, without evaluating the angle $\theta$, find the exact value of $$\sin\left( \frac{\theta}2-\theta \right)\tag1$$ I know that ...
0
votes
1answer
21 views

Confidence Interval for Regression Coefficient ,$\beta$

In the book 'Applied regression Analysis' by Draper/Smith, it is written that : Obtain individual $100(1-\alpha)\%$ confidence interval for the various parameters separately from the formula ...
0
votes
2answers
48 views

Differentiability at x=0 [on hold]

Discuss the differentiability of the following function in $x$ = $0$: $ f:\mathbb{R} \to \mathbb{R}: x\mapsto \begin{equation} f(x)= \begin{cases} \sqrt{x} & \text{if } x \geq 0 \\-\sqrt{-x} & ...
3
votes
1answer
265 views

sin(x) infinite product formula: how did Euler prove it?

I know that $\sin(x)$ can be expressed as an infinite product, and I've seen proofs of it (e.g. Infinite product of sine function). I found How was Euler able to create an infinite product for sinc by ...
0
votes
1answer
20 views

How to find normal of an intersecting plane

The equations of two different planes that intersect are the following: (1)---> x-3y+5z+8=0 (2)---> 5x+y-2z+7=0 It says that I must create a third plane that passes through the line of intersection ...
4
votes
6answers
387 views

Formula to estimate sum to nearly correct : $\sum_{n=1}^\infty\frac{(-1)^n}{n^3}$

Estimate the sum correct to three decimal places : $$\sum_{n=1}^\infty\frac{(-1)^n}{n^3}$$ This problem is in my homework. I find that n = 22 when use Maple to solve this. (with some programming) ...
0
votes
0answers
20 views

Convergence of Beta Distribution to Bernoulli Distribution [on hold]

How will I show that the $$\beta\left(\frac{a}{n} , \frac{b}{n} \right)$$ distribution converges to the $$\operatorname{Bernoulli}\left( \frac{a}{a+b} \right)$$ distribution?
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vote
3answers
299 views

pyramid with a trapezoid as a base.

Prove that there exists a pyramid SABCD with a given trapezoid ABCD as a base (BC||AD; the trapezoid's lateral sides AB and CD are not parallel) such that the pyramid's lateral faces SAB and SCD are ...
0
votes
1answer
46 views

Estimate for the limit of the solution of an ODE system

I have this system: $$\begin{cases} \frac{d}{dt}x(t)=-axy\\ \frac{d}{dt}y(t)=axy-by\\ \frac{d}{dt}z(t)=by \end{cases} $$ Let be: $x+y+z=1$ for every $t$ $a>b$ and $a,b$ strictly positive ...
1
vote
1answer
15 views

Trigonometry Question, Finding the distance and angle of elevation.

So there is this question, and for some reason, whether it be the early time of day or my lack of skills, It seems I have no idea how to draw the required diagram. I have tried and tried but none of ...
1
vote
0answers
61 views

Ideals of $\Bbb Z/p^2q\Bbb Z$

Let $p,q$ be distinct primes. Then $\mathbb{Z}/p^2q\mathbb{Z}$ has 3 distinct ideals. $\mathbb{Z}/p^2q\mathbb{Z}$ has 3 distinct prime ideals. $\mathbb{Z}/p^2q\mathbb{Z}$ has 2 distinct prime ...
4
votes
5answers
380 views

Am I allowed to apply L'Hospital's Rule inside of the natural logarithm function?

I have the following limit: $$\lim_{x\rightarrow \infty} \ln\left(\frac{2x^2+1}{x^2+1}\right)$$ If I was finding the limit of only the terms inside the natural log function, I would have the ...
1
vote
1answer
213 views

Distribution of Brownian Bridge

PROBLEM $U_t = B_t - tB_1$, $B_t$ is a Brownian motion on $[0,1]$. What is a Brownian Bridge and give the twodimensional distributions of the vector $(U_s, U_t)$. I think that a Brownian ...
0
votes
1answer
17 views

Find the rate of change. $P=250(1+(2t/(49+t^2)))$

A population of bacteria is introduced into a culture. The number of bacteria $P$ can be modeled by $P=250(1+(2t/(49+t^2)))$ where $t$ is time (in hours). Find the rate of change of the population ...
3
votes
0answers
35 views

Line integral parametrization

We are given the field $\textbf{F}(x,y)=(x-y)\textbf{i}+xy\textbf{j}$ and C being $\frac{3}{4}$ of a circle of radius $2$ centered at the origin traversed from $(2,0)$ to $(0,-2)$. $$\textbf{F}(x,y) ...
1
vote
1answer
17 views

Normal line to a curve $C_1$

Find the interval for $a$ so that $(3-a)x+ay+(a^2-1)=0$ is normal to the curve $xy=4$ $(C_1)$. I approached it this way-- $C_1$ is $xy=4$. So, $\dfrac{dy}{dx}$ for $C_1$ is $\dfrac{-4}{x^2}$. ...
3
votes
3answers
59 views

Bound in Complex Analysis

Can someone direct me towards the right way to approach this problem? Show $$\displaystyle \left|\int_{|z|=R} \frac{Log{z}}{z^2} dz\right| \leq 2\sqrt{2}{\pi}\frac{\log{R}}{R},\; \text{ for } ...
0
votes
2answers
39 views

Need help finding a number x so that $\phi > 9x/10$?

I need help finding a number $x$ so that $φ(x) > 9x/10$? ($φ$ being Euler’s phi function.) I also need to find a number $x$ so that $φ(x) < x/3$?
1
vote
2answers
49 views

If $f(g(x))=\sqrt {x^2-2x+8}$ and $f(x)=\sqrt x,$ find $g(x)$.

If $f(g(x))=\sqrt {x^2-2x+8}$ and $f(x)=\sqrt x,$ find $g(x)$. There is no example like this in my math book.
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vote
3answers
51 views

Show a complex equation has one or two roots

Let $a$ $\neq$ $0$, $b,$ and $c$ be complex constants. Show that the quadratic equation $az^2+bz+c=0$ has one or two roots. My thoughts: Let $a=a_1+ia_2,$ $b=b_1+ib_2,$ and $c=c_1+ic_2$. I also ...
0
votes
1answer
24 views

Calculating optimum values of $u$ and $m$ from $\mathbb V(\bar {y_2}\prime)=\frac{S_2^2(n-u\rho^2)}{n^2-u^2\rho^2}$

I have to find optimum sample size in sampling on two occasions. Suppose that the samples are of the same size n on both occasions. In selecting the second sample, $m$ of the units in the first ...
1
vote
1answer
40 views

How to show $\sqrt[3]{X-i}\notin \mathbb{C}(X,\sqrt[3]{X+i})$

I'm trying to show $\sqrt[3]{X-i}\notin \mathbb{C}(X,\sqrt[3]{X+i})$. But this is harder than I expected. Is there any easy way to show this?
3
votes
2answers
2k views

Show that every group of prime order is cyclic

Show that every group of prime order is cyclic. I was given this problem for homework and I am not sure where to start. I know a solution using Lagrange's theorem, but we have not proven ...
1
vote
1answer
24 views

To use Vieta's formula for complex constant solution or not?

Let $b$ and $c$ be complex constants such that $z^2$ + $bz$ + $c$ = $0$ has two different real roots. Show $b$ and $c$ are real. I think I need to be using Vieta's formula, however I have solved it ...
0
votes
1answer
23 views

Small question about limit at $+\infty$ to $-\infty$

please the definiton of $\displaystyle\lim_{u\rightarrow+\infty}G(t,u)=-\infty$ is: $\forall M>0 ,\exists R>0 $ such that $|u|\geq R \Rightarrow G(t,u)\leq M$ or $G(t,u)\leq -M$ ? please ...
2
votes
0answers
60 views

Problem in functional analysis.

I heard of this problem that caught my attention and I am curious now thus I would appreciate if I could have a hint or a solution. Let $(x_n)$ a sequence in a normed space $X$ such that ...
-1
votes
1answer
42 views

How many coloured pencils? [on hold]

Drawing pencils cost $8$ cents each and coloured pencil cost $11$ cents each. Two dozen assorted pencils cost $\$2.16$. How many coloured pencils are there?
1
vote
1answer
38 views

solving logarithmic equations by expressing in terms of exponents

Is it always valid to solve logarithmic equations by raising both sides as powers of a common base? As in: $$ln(x) = ln(y) $$$$ e^{ln(x)} = e^{ln(y)} $$$$ x = y $$$$ where \quad x,y ∈ ...
8
votes
4answers
242 views

$x<y$ then $x^3<y^3$

I'm looking for a proof to the following theorem: For any $x,y\in R$: $x<y \Rightarrow x^3<y^3$ I'm trying this approach: Let $z = x^3 - y^3 = (x-y)(x^2+xy+y^2) = z_1 z_2$ where $z_1 ...
2
votes
3answers
57 views

Find the 325th term of the series 7,16,25,34…

One of my friend gave me the series 7,16,25,34,43... I figured it out easily that the sum of digits is 7 in each case. How can I find the 325th term of this series? Also is there any trick/formula to ...
7
votes
2answers
387 views

Integration by parts

Ok I need to evaluate the following function and then prove by taking the derivative of my answer to check: $$\int e^{ax}\cos(bx)\,dx$$ where $a$ is any real number and $b$ is any positive real ...
4
votes
2answers
115 views

How to calculate this improper integral $\int_0^{+\infty} e^{-(ax+\frac{b}{x})^2}\mathrm{d}x$?

How to calculate this improper integral $$ \int_{0}^{\infty}{\rm e}^{-\left(ax\ +\ b/x\right)^2}\,{\rm d}x\ {\large ?} $$
5
votes
3answers
106 views

How to $\int_{0}^\infty {\sin^3(x)\over x}dx$

How to evaluate : $$\int_{0}^\infty {\sin^3(x)\over x}dx$$ I don't know how to do it. I tried to finish it using integration by parts, but it doesn't work? Can someone tell me how to evaluate the ...
0
votes
0answers
41 views

Face Boundary and bipartite question classification

Is this question wrong? Let G be a connected planar graph with a planar embedding where every face boundary is a cycle of even length. Prove that G is bipartite. Consider a graph of 2 squares ...
4
votes
1answer
78 views

Find intermediate fields of $\mathbb{Q}(\sqrt[3]{2}, \sqrt{3},i) \, | \, \mathbb{Q}(i)$

This is the problem I am facing: Compute the intermediate fields of the extension $K | \mathbb{Q}(i)$ where $K = \mathbb{Q}(\sqrt[3]{2}, \sqrt{3},i)$ and find the intermediate fields $M$ such ...
0
votes
0answers
21 views

Nonlinear Maps with additivity or homogeneity

Examples of linear maps from $\phi :R^2 \to R$ that has homogeneity but is not linear. Example of a function $\phi : C \to C$ that is additive but is not linear. All the examples I have found for ...
4
votes
1answer
826 views

A Problem in Evans' PDE

Problem 7 in §6.6 states as follows: Let $u\in H^1(\mathbb{R}^n)$ have compact support and be a weak solution of the semilinear PDE $$-\Delta u+c(u)=f\,\,\text{ in } \mathbb{R}^n,$$ where ...
0
votes
4answers
63 views

Find the basis for the subspace of the set of polynomials of degree less than five?

Let U = {p $\in P_4(F): p(2) = p(5) = p(6)$. Find a basis for U. I know how to do this problem if I were given p(2) = p(5). Set the two equal to each other and solve for one of the coefficients. I ...
1
vote
1answer
560 views

Is the intersection of the function of two sets a subset of the function of the intersection of two sets?

Let X and Y be sets and let f: X --> Y be a function from X to Y. If A and B are subsets of X, is it true that f(A) intersect f(b) is a subset of f(A intersect B)? If so, prove your answer; ...
0
votes
1answer
31 views

Finding and proving a basis for $W=\{f(x) \in P_2[\mathbb{R} ]:f'(x) +xf(0) = 0 \}$

I'm having a trouble proving/finding a basis for $W= \{f(x) \in P_2[\mathbb{R}]:f'(x) +x \bullet f(0) = 0 \}$. I'm supposing $\{ x, 1 \}$ is a basis for W because any vector in $P_2[\mathbb{R}]$ gets ...
0
votes
5answers
63 views

Bicycles: probability question?

I know this is quite easy but I would appreciate the help. In a survey, children were asked if they owned a bicycle. The results collected were: $46$ more pupils said ‘No’ than said ‘Yes’. ...
0
votes
1answer
19 views

How to find angle between line and plane?

The normal of the plane is [-5,8,-14] and the direction vector of the line is [2,4,3]. I know that after using the equation cosθ=u⃗ ⋅v⃗ ||u⃗ ||⋅||v⃗ |, I must subtract 90º by the found angle in order ...