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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-6
votes
0answers
19 views

Need a solution to find the locus of an equation. [on hold]

Find the equation of the locus of a point which moves so that its distance from $(a,0)$ is equal to its distance from the $y$-axis. The answer is $$y^2 - 2ax + a^2 =0$$ Please can someone find the ...
1
vote
3answers
283 views

What did I do wrong?

A question I was faced with asked "For which $x$ is $\log_{10}(x)^{\log_{10}(\log_{10}(x))}= 10,000$?" My instincts tell me I can say $$\log_{10}(x)=10$$ and $$\log_{10}(\log_{10}(x))=4$$ However, ...
0
votes
0answers
22 views

Please help me check this derivative work

I have $$ J_{\theta}(X) = - \frac 1 m \cdot \left[ y \cdot ln( h_{\theta} (X ) ) + ( 1 - y) \cdot ln ( 1 - h_{\theta}(X) ) \right] $$ I need $\frac d {d\theta} J_{\theta}(X)$. I tried many time, and ...
1
vote
0answers
21 views

Rapidly Decreasing Functions

Can someone explain the notion of a rapidly decreasing function? Namely, a function in the Schwartz space: $$\mathscr{S}(\mathbb{R}^n):= \{ f \in C^{\infty} (\mathbb{R}^n) : ||f||_{\alpha, \beta} ...
0
votes
2answers
30 views

Splitting a segment with a ratio

I came across the homework question that I attempted to do. After looking at the answers, and getting it wrong I didn't understand why. I'm specifically lost at why we would get a fraction of 2/5 ...
0
votes
0answers
8 views

Determining Moving-Average Representation of AR(2) Process

Consider a stationary $AR(2)$ process given by $$X_{t} - X_{t-1} + 0.25X_{t-2} = 5 + a_{t}$$ where $a_{t} \sim WN(0,1)$ (white noise). I am interested in obtaining the causal representation of ...
1
vote
0answers
31 views

Ordinary differential equation­

$$\dfrac{dy}{dx}-\dfrac{\tan y}{1+x}=(1+x)e^x\sin y$$ I tried $\sin y=t$ but failed. It seems to immune to methods I know of or I am just unable to make the right substitution... Wolfram alpha ...
0
votes
0answers
29 views

Convergence of norms

I have this space $H_{0,p}^1=\lbrace u\in AC([0,+\infty),\mathbb{R}),u(0)=u(+\infty)=0, \sqrt{p} u'\in L^2(0,+\infty)\rbrace $ endowed with the norm $||u||^2=\int_0^{+\infty} p(t) u'^2(t) dt$ ...
2
votes
1answer
39 views

The complex equation

In solving $|z|i +2z =1$, I seem to be constantly getting two solutions while both answer key and Wolfram claim to be only one. What am I doing wrong? Let's share the fun: $(\sqrt{x^2 +y^2}) i +2x ...
1
vote
1answer
38 views

Quadratics Word Problem

The path of a football flying through the air can be modelled by a quadratic equation. The football reaches the ground after 12 seconds in flight and is kicked from a height of 1 meter. The parabola ...
1
vote
1answer
32 views

Find value of $x$ for: $(1/3)(1-x) \geq 2(x-3)$

Find what value of $x$ satisfy: $(1/3)(1-x) \geq 2(x-3)$ First I multiplied both sides by $3$ so that $1/3$ became $3/3=1$. So I tried to find $x$ this way: $(1-x) \geq 6(x-3)$. I tried solving it ...
1
vote
2answers
58 views

Free action by cyclic group.

Let $G$ be a group acting on a set $X$. If $g\in G$ has no fixed points, prove or disprove the cyclic group $\left \langle g \right \rangle$ acts freely on $X$. edit: Can also assume $g$ has finite ...
1
vote
0answers
16 views

finding the symmetric point

let there be $4$ points. $A(-1,1,1), B(2,0,-1), C(1,3,-2), D(-2,-1,0)$. the $4$ points are not on the same line. the plane which goes through the points $A$ and $B$, and which is also paralel to the ...
1
vote
3answers
32 views

Given the equation of the parabola, for what values of $t$ will the quadratics have $0$, $1$, or $2$ solutions?

Given the equation of the parabola, for what values of $t$ will the quadratics have $0$, $1$, or $2$ solutions? Equation: $$0 = 3x^2 + tx + 10$$ Can you please explain the answer in simple terms, ...
0
votes
1answer
46 views

Probability and coin tosses

Taking a Probability & Statistics class this term and trying to get my head wrapped around how I calculate coin tosses with specific out comes in mind. We're using the nCr and nPr functions on our ...
0
votes
0answers
31 views

Apps for making geometric shapes [on hold]

Is there any apps for making geometric shapes? I need to make shapes like rhombus and equilateral triangles.
-1
votes
0answers
13 views

Cardinality of the following set of functions on $\mathbb R$ [duplicate]

Consider the following set $W$ = The set of constant functins on $\mathbb R$. $X$ = The set of polynomial functins on $\mathbb R$. $Y$ = The set of continous functins on $\mathbb R$. $Z$ = The set ...
0
votes
0answers
23 views

Great Common Division with Continued Fractions

If I have this GCD equation: $$89=16\cdot5+9\\ 16=9\cdot1+7\\ 9=7\cdot1+2\\ 7=2\cdot3+1\\ 2=1\cdot2+0$$ Then my continued fraction will be: $[5: 1, 1, 3, 2]$ But if I will have this GCD equation: ...
0
votes
0answers
32 views

What is a bi-rhombus? [on hold]

Can anyone tell me what a bi-rhombus is? I need it for my school project. If possible could you explain the relationship between a equilateral triangle, a rhombus and a bi-rhombus. This is how my ...
0
votes
1answer
22 views

Solving for the rate at which water is pumped into a conical tank using related rates.

Water is leaking out of an inverted conical tank at a rate of $12000.0$ cubic centimeters per min at the same time that water is being pumped into the tank at a constant rate. The tank has height ...
-2
votes
1answer
28 views

Baye's theorem may be required. [on hold]

A message is sent which consists of $n$ binary symbols $0$ and $1$. Each symbol is distorted with a small probability $p$ (is changed to the opposite). To be on the safe side the message is repeated ...
0
votes
1answer
60 views

Prove the reflexivity of $\subseteq$.

My professor gave me a list of exercises, I've been able to figure out what mechanism I should exploit to prove them, but I'd like to know if it's good. Until now we've been taught a little logic and ...
-1
votes
2answers
75 views

How to evaluate the limit $\lim\limits_{x \rightarrow \infty} x\ln(x)$

How can I determine if the limit exists for the following? $$\lim_{x \rightarrow \infty} x\ln(x) $$
1
vote
2answers
23 views

Steps to creating 3 plane equations with 3 lines of intersection

I was wondering if anyone can give me pointers on to how to mathematically create 3 plane equations that meet in 3 lines. In other words, each plane intersects one another in a straight line (so it ...
0
votes
1answer
23 views

Question regarding a wording of an exercise related to Noetherian topological space

The exercise states "If $X$ is a Noetherian topological space, show that the union of any subset of the connected components of $X$ is always open and closed in $X$." Does the question mean "If I ...
3
votes
2answers
52 views

Evaluate integral by completing the square and doing trigonometric substitution

$\int \frac{1}{(x-2)\sqrt{x^{2}-4x+3}} dx$ is my problem Complete the square $\int \frac{1}{(x-2)\sqrt{(x-2)^{2}-1}} dx$ I know I'm probably supposed to use $ \frac{d}{dx}\operatorname{arcsec}(u) = ...
1
vote
2answers
36 views

How to solve $3 - 2 \cos \theta - 4 \sin \theta - \cos 2\theta + \sin 2\theta = 0$

I have got a bunch of trig equations to solve for tomorrow, and got stuck on this one. Solve for $\theta$: $$3 - 2 \cos \theta - 4 \sin \theta - \cos 2\theta + \sin 2\theta = 0$$ I tried using ...
0
votes
2answers
24 views

Find range of values

Find the range of values of the constant $a$ at which the equation $x^3 - 3a^2x + 2 = 0$ has $3$ different real number roots. I took the derivative and found that $x = -a, a$ Then I solved for $f(a) ...
1
vote
3answers
130 views

Deriving the sum-to-product identities

I've been asked by my textbook to derive the "sum-to-product" identities from the "product-to-sum" identities. I've attempted to to do this but i've met a dead end, and i'm quite confused. Using ...
1
vote
1answer
30 views

Understanding an algorithm

I want to understand the above algorithm. My solution says that the algorithm should return $0$ if $n$ is a prime or 1. Otherwise it returns the smallest (positive) non-trivial divisor. Lets ...
0
votes
3answers
51 views

Maxwell's Equations Divergence Question

$$ \left\{ \begin{align} \text{div } \textbf{E} & =0, \\ \text{div } \textbf{H} & =0, \\ \text{curl } \textbf{E} & = \frac{-1}{c} \frac{\partial \textbf{H}}{\partial t}, \\ \text{curl } ...
0
votes
1answer
37 views

Maxwell's Equations Curl Question

$$\left\{ \begin{align} \text{div } \textbf{E} &=0, \\ \text{div } \textbf{H} & =0, \\ \text{curl } \textbf{E} & = \dfrac{-1}{c} \dfrac{\partial\textbf{H}}{\partial t}, \\\text{curl } ...
0
votes
0answers
43 views

How to find an example

I want to find a function $f\in C^1([0,+\infty)\times\mathbb{R},\mathbb{R})$ such that $f(t,0)=0$ $f(t,u)\leq \alpha u+\beta$, $\alpha<\lambda_1,\beta\geq 0$ $f(t,u)\geq C_1 |u|^{\sigma}$ where ...
3
votes
1answer
182 views

Solving this trigonometric equation

$$\sqrt{3} \cos x - 3 \sin x = 4 \sin 2x \;\cos 3x$$ I tried many things: opening $\sin 2x$, $\cos 3x$, simplifying LHS: $\cos(60^\circ+x)$. Nothing seems to work. Any hint?
1
vote
2answers
45 views

Find out the interval where Rolle's Theorem is applicable

Find out the interval for which the Rolle's theorem is valid for the function $f(x)=2x^3+x^2-4x+2$ My attempt : Supposing the interval is $[a,b]$, $f(a)=f(b)$ gives the equation ...
0
votes
1answer
16 views

Using differentials, estimate the difference in the deflection between the point midway on the beam and the point 1 10 ft above it

So I've been trying to figure out the problem for about an hour and I cannot figure it out. Question: To study the effect an earthquake has on a structure, engineers look at the way a beam bends when ...
3
votes
2answers
50 views

How can I find the volume of prism: $V = \frac{(a + b + c)Q}{3} $

In the book Handbook of Mathematics (I. N. Bronshtein, pg 194), we have without proof. If the bases of a triangular prism are not parallel (see figure) to each other we can calculate its volume by ...
0
votes
0answers
16 views

Variance- covariance matrix

Consider $H$ denotes hat matrix and $e$ denotes residual. In the book Applied regression Analysis by Draper/Smith, it is written that : $\mathbb V(e_i)$ is given ...
1
vote
2answers
57 views

How to evaluate the limit $\lim\limits_{x \rightarrow \infty} \frac{x}{\sqrt{x^2+1}} $

How can I determine if a limit exists for the following $$\lim_{x \rightarrow \infty} \frac{x}{\sqrt{x^2+1}} $$ By using L'Hopitals rule the function appears to flip flop back and forth
1
vote
3answers
69 views

Applications of calculus

We have the following formula for area $$A = r^2(\sinθ\cosθ-\sqrt{3}\sin(θ)^2)$$ We then need to find what value θ will give maximum area, so we differentiate to get; $$ ...
1
vote
1answer
27 views

How to show the cyclotomic polynomial is irreducible over $\mathbb{R}(T,\sqrt[n]{T})$

I'm trying to solve the following problem. Let $T$ be a transcendental over $\mathbb{R}$. Put $K=\mathbb{R}(T), n\geq3$. Let $L$ be the least splitting field of $X^n-T$. Then, calculate $[L:K]$. I ...
2
votes
3answers
87 views

$\ \sqrt{x+39}-\sqrt{x+7}=4 $

So I tried to solve this problem for x $\ \sqrt{x+39}-\sqrt{x+7}=4 $ I multiplied both sides ($\ \sqrt{m}\cdot\sqrt{n}=\sqrt{mn} $) $\ (\sqrt{x+39}-\sqrt{x+7})^2=16 $ $\ ...
1
vote
1answer
26 views

Find the triangular matrix and determinant.

I have a 4x4 matrix and I want to find the triangular matrix (lower half entries are zero). $$A= \begin{bmatrix} 2 & -8 & 6 & 8\\ 3 & -9 & 5 & 10\\ -3 & 0 & 1 & ...
0
votes
1answer
42 views

Nonlinear system of equations / factoring two-variable cubic over $\mathbb{R}$

About halfway through a homework problem, I end up with a three-way identity: $$\frac{uw}{v+w} = \frac{uv}{u+w} = \frac{vw}{u+v}$$ (I say I end up with..., but this is the method suggested by my ...
0
votes
0answers
32 views

Composition of injective linear maps.

I was looking at some solutions for my homework and I didn't understand this part: $S_1,\ldots,S_n$ are injective linear maps such that $S_1S_2 \dots S_n$ makes sense. Prove $S_1S_2 \dots S_n$ is ...
0
votes
1answer
64 views

How to find $\theta$ for $\tan\theta=-\frac{4}{3}$?

Given $\tan\theta = -\frac{4}{3}$, between $0\leq\theta\leq2\pi$, how can I find both values of $\theta$, with or without a calculator?
1
vote
0answers
26 views

Existence of Linear Maps and the Fundamental Theorem of Linear Maps.

Prove that there does not exist a linear map $T: \Bbb R^5 \to \Bbb R^5$ such that $\operatorname{range}(T) = \operatorname{null} (T)$. My proof goes like this: Suppose for the sake of contradiction ...
-1
votes
3answers
68 views

The chance to double 1000 points into 2000 points [on hold]

You own 1000 points. Your goal is to reach 2000 points, the only way you gain points is by gambling. You will always gamble 40 points, your chance of winning a 40 points gamble is 60%, how high is ...
11
votes
3answers
217 views

Suggestion for Computing an Integral

Let $$A=\left\{(x,y,z)\in \mathbb R^3:\dfrac{x^2}{2}+\dfrac{y^4}{4}+\dfrac{z^6}{6}\leq1\right\}.$$ Then I want to compute the following integral: ...
12
votes
1answer
135 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.