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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1
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2answers
37 views

Does it span $\mathbb{R}^3$?

I have a T/F question and I think I know the direction to go, but I am not sure. It states: $\{[17,6,-4]^t,[2,3,3]^t,[19,9,-1]^t\}$ does not span $\mathbb{R}^3$. Let me get this straight. It SPANS ...
1
vote
1answer
38 views

Existence of injective function in a manifold with special atlas

I am trying do the following question: Let $M$ be a $n$-dimensional smooth manifold that admits an atlas with only two charts. Show that there exists an injective smooth map ...
0
votes
1answer
33 views

Application Closed Graph Theorem to Cauchy problem

Consider $E:=C^0([a,b])\times\mathbb{R}^n$ and $F:=C^n([a,b])$ equipped with the product norms. Consider $$ u^{(n)}+\sum_{i=0}^{n-1}a_i(t)u^{(i)}=f $$ with $$u(t_0)=w_1,\dots,u^{(n-1)}(t_0)=w_n \\ ...
9
votes
4answers
830 views

If $n = 51! +1$, Then find no of primes among $n+1,n+2,\ldots, n+50$

If $n = 51! +1$, Then find no of primes among $n+1,n+2,\ldots, n+50$ Really speaking, I don't have any clue ...
1
vote
3answers
78 views

Find the remainder of $\frac{1! +2!+\, \dots\, +95! }{15}$.

I think I found my answer but I am looking for better ones
0
votes
2answers
60 views

closed formula for a series

The first column below is n (iterator), the second is the value, I'm trying to understand if there is a closed formula that I can use for obtaining the value given a fixed n $$n=0: a$$ $$n=1: ...
0
votes
2answers
68 views

I Don't Understand Error Bounds

I understand they're supposed to give us a limit on how off our approximation of an integral can be, but I don't understand how the formula gives that. What does the second derivative have to do ...
0
votes
1answer
57 views

Calculus Problem general polynomial limit to infinity [duplicate]

I have to solve the following problem for homework for a calculus class. I really have no idea where to start, does anyone have any hints?: let n be a positive integer greater than 0. Let P(x) be a ...
-4
votes
1answer
44 views

Graph all vertical and horizontal asymptotes [closed]

Graph all vertical and horizontal asymptotes of the function $$f(x)=\frac{-10x-11}{-4x-2}.$$ Please help me figure out how to start this problem.
1
vote
1answer
54 views

Showing $ (R ∪ R^{-1})^∗ = R^∗ ∪ R^{−1∗} $ is false by giving a counterexample.

Show that $$ (R ∪ R^{-1})^∗ = R^∗ ∪ R^{−1∗} $$ is false by giving a counterexample. I tried the following, but every time it keeps coming out as true (instead of false): If $R = \{(a,b), ...
0
votes
1answer
31 views

Solution to SDE using Itô calculus

So if I have the following generator and an initial condition: $$A(f)(x) = \alpha x f'(x) + f''(x) \\ X_0 = x \in \mathbb{R}^+$$ I've been asked to find $X_t$ and assume that $\alpha$ is a constant. ...
0
votes
2answers
49 views

Proof the expession $\log_{12}{18}*log_{24}{54} + 5(\log_{12}{18}-log_{24}{54})=1$

I am trying to proof the following expression (without a calculator of course). $\log_{12}{18}*\log_{24}{54} + 5(\log_{12}{18}-\log_{24}{54})=1$ I know this isn't a difficult task but it's just ...
1
vote
2answers
54 views

How do you simplify an algebraic expression?

Please explain how to simplify an expression that is similar to this one $\displaystyle\frac{a+3}{6}+\frac{a-4}{4}+\frac{a+2}{-3}$
2
votes
5answers
72 views

For the polynomial

For the polynomial, -2 is a zero. $h(x)= x^3+8x^2+14x+4$. Express $h(x)$ as a product of linear factors. Can someone please explain and help me solve?
3
votes
9answers
144 views

Find the exact value of $\sin (\theta)$ and $\cos (\theta)$ when $\tan (\theta)=\frac{12}{5}$

So I've been asked to find $\sin(\theta)$ and $\cos(\theta)$ when $\tan(\theta)=\cfrac{12}{5}$; my question is if $\tan (\theta)=\cfrac{\sin (\theta) }{\cos (\theta)}$ does this mean that because ...
2
votes
1answer
19 views

Finding the range

Let $a < b$ and $f: (a, b] \to R$, $f(x) = 5 - x$. What is the range of the function? How do you find the range of an equation with unknowns?
0
votes
1answer
27 views

How do I form these quadratic equations

A car travels a distance of $1200 \ km$ at a speed of $a \frac{km}{h}$, while a train travels the same distance at $(a – 20) \frac{km}{h}$. If the time taken by the train is $5$ hours more than the ...
0
votes
3answers
35 views

Quadratics with unknowns

If $5x^2 – t = 4x$, and $x$ and $t$ are both positive real numbers. What is $x$ equal to? How do you find $x$? Is there a specific formula?
6
votes
4answers
255 views

question on limits and their calculation

In taking each of the limits $$\lim_{x\to -\infty}\frac{x+2}{\sqrt {x^2-x+2}}\quad \text{ and } \quad \lim_{x\to \infty}\frac{x+2}{\sqrt {x^2-x+2}},$$ I find that both give the value $1$, although it ...
0
votes
2answers
24 views

Calculate the sum of triangle's medians squared if hypotenuse is 2

Given a right triangle with sides a,b and a hypotenuse c=2, calculate the sum of trianle's squared medians i.e. if medians are x,y, and z, calculate $x^2+y^2+z^2$ The only thing i thought of is ...
3
votes
1answer
110 views

When is $\lim_{b\to a} \int_a^b f(x)dx=\int_a^af(x)dx=0$

An elementary question on Riemann - Integration: Under what conditions on $f$ is the following true: $$\lim_{b\to a} \int_a^b f(x)dx=\int_a^af(x)dx=0$$ If $f$ is bounded in $[a,b]$, then this is ...
0
votes
0answers
25 views

Product of stochastically independent random variables

Let $X, Y, Z$ be three stochastically independent random variables that are quadratic integrable (quadratintegriertbar is the German term, I didn't find a exact translation). No which statements hold ...
0
votes
1answer
26 views

Average Speed using variables

A cyclist travels $m$ km at $v$ km/h and $n$ km at $u$ km/h. What is the average speed of the cyclist? Is there a formula for calculating average speed?
0
votes
1answer
49 views

inverse of a function which contain logarithm

$f(x)=2^{x}(x-1) $is a bijective function . What will be the inverse of $f$ ? For this we want to find another function $g$ with composition of both give identity function. So I want to separate $x$ ...
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votes
1answer
29 views

Help required for algebra 2 math homework [closed]

If two persons together have Rs. 12100. If 4/5th of first person's amount is equal to 2/5th of second person's amount. how much the amount they individually have?
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votes
2answers
68 views

I need help solving this related rates equation.

I need help answering the following question and I'll show you what I have. ! $$x=20,y=\sqrt{2100},z=50, \frac{dy}{dt}=30$$so differentiating $(20)^2+y^2=z^2$ $$2y\frac{dy}{dt}=2z\frac{dz}{dt}$$ And ...
3
votes
1answer
83 views

Derivative of the parameter

I have the equation$\begin{cases} x'(t)=x(t)+y(t) \\y'(t)= \mu y^2(t)+x(t)\end{cases}$ Cauchy problem $\begin{cases} x(0)= 1 + \mu \\y(0)=-2\end{cases}$ . I must calculate $\frac{\partial ...
30
votes
5answers
3k views

Is it okay to reverse engineer proofs in homework questions?

In a linear algebra text book, one homework question I received was: Prove that $\mathbf{a \cdot b} = \frac{1}{4}(\|\mathbf{a + b}\|^2 - \|\mathbf{a - b}\|^2)$. Where $\mathbf{a}$ and ...
0
votes
1answer
38 views

Prove that These Families of Level Curves are Orthogonal

From p. 79 in Brown's and Churchill's "Complex Variable and Application": Let the function $f(z) = u(x, y)+iv(x, y)$ be analytic in a domain $D$, and consider the family of level curves $u(x, y) = ...
2
votes
1answer
57 views

Determining whether or not a vector is a linear combination of a give matrix

$$ A= \begin{bmatrix} 1 & 0 & 5\\ -2 & 1 & -6\\ 0 & 2 & 8 \end{bmatrix} ,b= \begin{bmatrix} 2\\ -1\\ 6 \end{bmatrix} $$ The problem asks to determine whether or not vector $b$ ...
2
votes
3answers
61 views

Finding an area of a triangle inside of a triangle, given certain areas of other triangles, and area ratios.

I'm studying for the Waterloo Math Contest (Galois, Gr. 10) taking place in April of 2015 and I am preparing by looking at previous problems and solving them. This is question 4(c) on the 2010 Galois ...
1
vote
1answer
59 views

How to get $(\frac{x^2}{2}+\frac{1}{2x^2})^2$ from $1+(\frac{x^2}{2}-\frac{1}{2x^2})^2$?

How can I get $(\frac{x^2}{2}+\frac{1}{2x^2})^2$ from $1+(\frac{x^2}{2}-\frac{1}{2x^2})^2$? The book lists the former as the solution to that step. This is part of an arc length problem, and I think ...
1
vote
0answers
46 views

Divisibility is not definable over $\mathbb{N}$ with coprimality relation

I am asked to show that the divisibility relation "|" is not definable over $(\mathbb{N},\perp)$, where "$\perp$" is the coprimality relation. I am pretty sure that I should use the following: ...
2
votes
1answer
53 views

How to calculate this integral using Rodrigues' formula?

I'm trying to get practice using Rodrigues' formula for Legendre Polynomials, but it's being quite confusing to manipulate that $n$-th derivative. Basically, I'm trying to calculate: $$\int_{-1}^1 ...
1
vote
2answers
43 views

Find a value of $x$ such that $f(x) = y$

Let $S_1$ be the sphere of radius $1$, centered at the origin. Let a be a number $> 0$. If $x$ is a point of the sphere $S_1$, then $ax$ is a point of the sphere of radius $a$, because $\|ax\| = ...
1
vote
1answer
61 views

Divide and Conquer (recurrence relation problem)…

The problem: (a) Use a divide-and-conquer approach to devise a procedure to find the largest and next-to-largest numbers in a set of n distinct integers. (b) Give a recurrence relation for ...
0
votes
1answer
34 views

Solve Itô integral with power

$$\int_0^t e^{Ws} W_s^r dW_s$$ where $W_s$ is Wiener process and r> in $\mathbb{Z}$ My first approach would be to use Ito's lemma, however, coming up with the function $g(t,x)$ is difficult The ...
2
votes
3answers
71 views

distribution of infinite sum of $\sum (2x_n -1)/2^n$

$\{X_n\}\sim\mathrm{Bernoulli}(\frac {1}{2})$ $$Y=\sum_{n=0} ^{\infty} \frac {2X_n -1}{2^n}$$ Find the distribution of $Y$ $X_n$ are independent
1
vote
5answers
52 views

Finding the range and domain of $f(x)=\tan (x)$

I am attempting to find the range and domain of $f(x)=\tan(x)$ and show why this is the case. I can seem to find the domain relatively well, however I run into problems with the range. Here's what I ...
3
votes
2answers
80 views

Are there any 3 natural numbers that satisfy $a^2+b^2=2z^2 $?

Are there any 3 natural numbers that satisfy $a^2+b^2=2z^2 $? This is a question that arised as I was trying to solve another question: Is there an arithmetic progression, of natural numbers in which ...
1
vote
2answers
36 views

How to find length of the sides of a triangle given the ratio of the sines of the sides?

Consider $\triangle ABC$. Let $\dfrac{\sin A}{\sin B} = \dfrac56$ and $\dfrac{\sin B}{\sin C} = \dfrac45$. Find $\dfrac{\vert AC\vert\cdot \vert AB\vert}{\vert BC\vert}$. If there is no definite ...
1
vote
0answers
44 views

Integration in polar coordinates?

Given $$ A=\begin{pmatrix} a & b \\b & c \end{pmatrix}, x=(x_1,x_2), (Ax,x)>0 $$ and $$(x,y)=x_1\cdot y_1+x_2\cdot y_2$$ I'm trying to prove that $$ \int_{-\infty}^\infty ...
1
vote
0answers
36 views

problem of computing limit

The problem is to prove the following for $n \geq 3$ $$u(0)=\frac{1}{n\alpha (n) r^{n-1}}\int_{\partial B(0,r)} g dS +\frac{1}{n(n-2)\alpha (n)} \int_{B(0,r)} (\frac{1}{|x|^{n-2}} - ...
1
vote
1answer
24 views

Pointwise and uniform convergence of sequence of functions

Let $(f_n)$ be a sequence of continuous functions on $\mathbb R$. If $(f_n)$ converges to $f$ pointwise on $\mathbb R$, then $$\lim\limits_{n\to ...
1
vote
1answer
49 views

is $(x-6)^2$ in $C_0^2$?

My math problem involves using a theorem that requires $f(x)=(x-6)^2$ to be in $C_0^2$. I'm trying to understand what $C_0^2$ means and how to check whether a function belongs to it. The course I'm ...
2
votes
3answers
94 views

How to calculate integral $I=\displaystyle\int_{-1}^{1}\dfrac{dz}{\sqrt[3]{(1-z)(1+z)^2}}$?

The integral is $I=\displaystyle\int_{-1}^{1}\dfrac{dz}{\sqrt[3]{(1-z)(1+z)^2}}$. I used Mathematica to calculate, the result was $\dfrac{2\pi}{\sqrt{3}}$, I think it may help.
1
vote
0answers
47 views

Number of branch points of holomorphic function on torus [closed]

Let $\Lambda\subseteq\mathbb{C}$ be the lattice $\{m+in: m,n\in\mathbb{Z}\}$ and let $X:=\mathbb{C}/\Lambda$ the associated complex torus. Consider the meromorphic function ...
1
vote
0answers
72 views

difference between n - 1 and n-1

I hope things are going well. I am doing some recursion stuff with databases and the text book I am reading has made it clear there is a difference between n-1 for example and n - 1. The difference ...
0
votes
0answers
43 views

Verification of a Combinatorial Identity

I was given a question and would like to see if I made any errors in my answer. The Question: My Answer: I noticed the following identity is very useful here: $\dbinom{n+1}{r}$ = $\dbinom{n}{r}$ ...
2
votes
0answers
79 views

Questions about central polygonal numbers $1, 2, 4, 7, 11, 16, 22, 29, 37, 46,\cdots$

Formula for Central polygonal numbers is $\frac{n(n+1)}{2} + 1$, if $n=1$ or $n$ is prime, we get the new sequence $A$: 2, 4, 7, 16, 29, 67, 92, 154, 191, ... It seems that all primes either is ...