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
votes
0answers
18 views

Accuracy of line intersecting algorithem decrase with large precisions

from the above pic I found the value of x from equation of line p1-p2 and perpendicular line from point a to the Line(p1,p2) .The intersecting point is X ,but the accuracy is less see the result ...
2
votes
3answers
555 views

Looking for examples of first countable, compact spaces which is not separable

Could someone give me some classical examples of first countable, compact spaces which is not separable? However, other examples are also welcome. Any help will be appreciated.
2
votes
1answer
44 views

Simplify: $\ln(x^2 − 4)− \ln(x − 2)− \ln 2$

Simplify: $$\ln(x^2 − 4)− \ln(x − 2)− \ln2$$ $$\ln\dfrac{x^2 − 4}{x − 2}− \ln2$$ $$\ln(x + 2)− \ln2$$ $$\ln(x + 2)/2$$ I got this far, is there any other way to simplify it, or do I stop here?
13
votes
2answers
497 views

Intuitive ways to get formula of cubic sum

Is there an intuitive way to get cubic sum? From this post: combination of quadratic and cubic series and Wikipedia: Faulhaber formula, I get $$1^3 + 2^3 + \dots + n^3 = \frac{n^2(n+1)^2}{4}$$ I think ...
3
votes
3answers
44 views

Trigonometry Question: find Value of…

Find value of $3 + \cos2x + \cos4x + \cos6x - 4\cos x\cos2x\cos3x$. I tried with $\cos A + \cos B$ identity but it was not simplifying.... Help..
1
vote
2answers
49 views

If $a_i>o$ then $(a_1a_2\cdots a_{2^n})^{1/2^n}\leq \frac{a_1+a_2+\cdots+a_{2^n}}{2^n}$

I need help to prove this inequality, I have no idea how to proceed with the inductive step: $$a_1,a_2,\ldots,a_{2^n}>0 \Longrightarrow(a_1a_2\cdots a_{2^n})^{1/2^n}\leq ...
1
vote
1answer
27 views

find a $B_{n,j}$ such that $|A_{n,j}-L_j| \leq B_{n,j}$ $\forall n,j$ and $\sum_{j=0}^{\infty}B_{n,j}$ converges

We have $A_{n,j}= 3(-1)^j2^{n-j+1}\frac{(2(n-j)-4)!}{(n-j)!(n-j-2)!}\binom{j+2}{2}\frac{n^\frac{5}{2}}{8^n}$ and $L_j=(-\frac{1}{8})^j\binom{j+2}{2}\frac{3}{8\sqrt{\pi}}$ So I know $\lim_{n \to ...
2
votes
4answers
26 views

Heaviside Unit Step Function

Convert to heaviside function: $$f(t) = \begin{cases}e^t ,& 0 \leq t \leq 1 \\0 ,& t > 1\end{cases}$$ My attempt: $f(t) = U(t) e^t - U(t-1) e^t $ I think my solution is not right because ...
5
votes
1answer
84 views

A number related to the roots of a quartic polynomial is a root of a cubic polynomial

So here is the problem, $a$ and $b$ are two distinct real roots of $f(x)=0$ where $f(x)=x^4-6x+3$, show that $(a+b)^2$ is a root of $g(x)=x^3-12x-36$. I have tried many methods, such as substitution, ...
6
votes
6answers
382 views

How to calculate this $\sqrt{3\sqrt{5\sqrt{3\sqrt{5\cdots}}}}$

I didn't know how to calculate this: $$\sqrt{3\sqrt{5\sqrt{3\sqrt{5\cdots}}}}$$ Please help me. Thanks.
6
votes
3answers
50 views

Prove that $u(x,t)=\int_{-\infty}^{\infty}c(w)e^{-iwx}e^{-kw^2t}dw\rightarrow 0$ if $x\rightarrow \infty$

I have the following problem: Be the equation: $$u(x,t)=\int_{-\infty}^{\infty}c(w)e^{-iwx}e^{-kw^2t}dw$$ Show that $u\rightarrow 0$ as $x\rightarrow \infty$, even when $e^{-iwx}$ does not falter ...
6
votes
4answers
449 views

For what values of m are the roots of $x^2 +2x+3 = m(2x+1)$ real and positive

I am only able to show that to be real, $m <-1$ or $m\geq2$ Don't know how to finish solution Answer is $2 \leq m < 3$ So far: After expanding and factorising, $x^2 + 2(1-m)x + (3-m) = 0 $ ...
0
votes
0answers
23 views

Solve the given differential equation by using Green's function method

I am really struggling with the concept and handling of the green's function. I have to solve the given differential equation using Green's function method $\frac{d^{2}y}{dx^{2}}+k^{2}y=\delta ...
0
votes
2answers
25 views

Doubts on locus and its equation

Find the equation to the locus of a point which is col-linear with points M(a,0) and N(0,b) Answer is:- x/a + y/b How i tried to find the solution:- P is a point whose assigned coordinates are (x,y) ...
5
votes
1answer
46 views

Is S a group under matrix addition

Another matrix question! Let $$S=\{A \in M_2(\mathbb{R}):f(A)=0\}\text{ and }f\left(\begin{bmatrix}a&b\\c&d \end{bmatrix}\right)=b$$ Is S a group under matrix addition. Either prove that ...
1
vote
1answer
23 views

How is the power rule applied to whole numbers

For the following function, how does the $+1$ become $0$ when finding its derivative via the power rule? Original function: $f(x) = 6x^2 − 4x^{-1} + 5x^{-2} − 2x + 1$ Derivative: $f '(x) = 12x + ...
0
votes
0answers
54 views

$H_I^n(R)=0$ and $H_I^n(M)\neq 0$ [on hold]

Find R and M as an R-module such that $H_I^n(M)\neq 0$ and $H_I^n(R)=0$, where I an ideal of R and $n\in N$. I found it in Cohen Macaulay rings. there`s nothing to find.
1
vote
2answers
74 views

Polynomial Division - “Define the largest natural number…” [on hold]

Would someone mind helping me with this question? The more detailed possible so I can have 100% of understanding. Thanks. Question: Define the largest natural number m such that the polynomial ...
1
vote
4answers
54 views

Trigonometric functions of the acute angle

Find the other five trigonometric functions of the acute angle A, given that: \begin{align} &\text{a)}\ \ \sec A = 2 \\[15pt] &\text{b)}\ \ \cos A = \frac{m^2 - n^2}{m^2 + n^2} \end{align} ...
2
votes
1answer
162 views
+50

An arithmetic sequence whose members do not contain the digit ‘9’

There is a non-constant arithmetic progression made of natural numbers only; none of them contains the digit $9$. Prove that such an arithmetic progression has no more than $72$ terms.
2
votes
1answer
20 views

finding parallel sides from a irregular decagon?

Is it possible to find out that which of two sides are parallel in this irregular decagon.If,it is yes;then how can I proceed? I have tried with "Consecutive Interior Angles".but can't come to a ...
0
votes
1answer
45 views

Help with math steps, chain rule.

I'm trying to to understand the math steps to go from Eqn. (1) to Eqn. (2). $$\tag{1} q(x,t)=\frac{-V_t(1+\delta f(c,g))}{P(x,t)}\cdot \left(\frac{dP_o}{dt}\right)$$ $$\tag{2} \frac{-V_t ...
0
votes
1answer
31 views

Total derivative proof [on hold]

The wikipedia article does not prove it http://en.wikipedia.org/wiki/Total_derivative Neither the top articles in google search. Could somebody help me proving it? I've found this: ...
0
votes
1answer
36 views

Trigonometry Question - Tough one [on hold]

If in triangle ABC, sin A sin B sin C + cos A cos B = 1. Then find the value of sin C.
0
votes
1answer
14 views

How to find plane that's equidistant from the origin

Objective: Give the equation of a plane that crosses the axes at points equidistant from the origin. How do I make sure that the points $A(1,2,-2)$, $B(-5,1,1)$, $C(4,-3,1)$ are equidistant from the ...
-2
votes
1answer
35 views

Equation of a plane equidistant from 3 points

Question: Given 3 point (point A, point B, point C), find an equation to a plane that crosses the axes at points equidistant to the origin P[0,0,0]. Are the following steps the right way to approach ...
6
votes
3answers
242 views

Integral $\int_1^{\sqrt{2}}\frac{1}{x}\ln\left(\frac{2-2x^2+x^4}{2x-2x^2+x^3}\right)dx$

Calculate the following integral: \begin{equation} \int_1^{\sqrt{2}}\frac{1}{x}\ln\left(\frac{2-2x^2+x^4}{2x-2x^2+x^3}\right)dx \end{equation} I am having trouble to calculate the integral. I ...
3
votes
1answer
81 views

any simple method to do integration?

$$\int_{-2}^{x^{2}-2x}e^{t}.e^{t^2} dt = ?$$ What i did is... on rewriting it , $$\int_{-2}^{x^{2}-2x}e^{t+t^2} dt=\frac{e^{t+t^2}}{t^2/2+t^3/3} $$ and then substituting limits is very long process ...
3
votes
1answer
88 views

A unfamiliar question

I'm sure asking this kinda problem is stupid but somehow I have never seen such problems before. $2{x}^2 + 3{y}^2 =0$ what is $3x+2y$?
1
vote
1answer
206 views

Dynamic Programming— Variable Width Bin (Equi-Depth) Histogram

Given some data, and a fixed number of bins (k)-- How can I design a Dynamic Programming algorithm that minimizes the largest difference between bin sizes? In other words, with a set number of bins ...
1
vote
1answer
72 views

combination of quadratic and cubic series

I'm an eight-grader and I need help to answer this math problem (homework). Problem: Calculate $$\frac{1^2+2^2+3^2+4^2+...+1000^2}{1^3+2^3+3^3+4^3+...+1000^3}$$ Attempt: I know how to calculate ...
0
votes
1answer
37 views

Solve initial value problem (C.S.I.R)?

The initial value problem is $$ \frac{\partial u}{\partial t} +x\frac{\partial u}{\partial x} = x, \ \ 0 \leq x \leq 1, \ \ t > 0 \ \ and$$ $$ u(x,0) = 2x \ \ has$$ a unique solution $u(x,t) \ ...
21
votes
5answers
9k views

If $a^2$ divides $b^2$, then $a$ divides $b$

Let $a$ and $b$ be positive integers. Prove that: If $a^2$ divides $b^2$, then $a$ divides $b$. Context: the lecturer wrote this up in my notes without proving it, but I can't seem to figure out ...
1
vote
1answer
37 views

Word Problem, Calculus estimation homework

Roger runs a marathon. His friend Jeff rides behind him on a bicycle and clocks his speed every 15 minutes. Roger starts out strong, but after an hour and a half he is so exhausted that he has to ...
0
votes
0answers
32 views

A question on limits

$$\lim_{h\rightarrow 0}\frac{2^{8\cos(h)}}{8h}\left [ sin^{8}(\pi/6+h))-sin^{8}(\pi/6) \right ]$$ MY ATTEMPT: for $\lim_{h\rightarrow ...
2
votes
6answers
242 views

How do I find the sum of the infinite geometric series?

$$2/3-2/9+2/27-2/81+\cdots$$ The formula is $$\mathrm{sum}= \frac{A_g}{1-r}\,.$$ To find the ratio, I did the following: $$r=\frac29\Big/\frac23$$ Then got: $$\frac29 \cdot \frac32= \frac13=r$$ and ...
3
votes
1answer
43 views

Area enclosed between half lines in polar space

I don't know if the anwser to my question is obvious because I cannot find any explanation anywhere on google. Question The blue region $R$ is bounded by the curve C with equation $r^{2} = ...
1
vote
1answer
23 views

Simple question (hopefully) on unitary method

In India we have an exam called NEST. I gave it today, and this was a question I encountered: Lactobacillus sp. and Streptococcus sp. are two bacterial species responsible for curdling milk. One ...
0
votes
1answer
27 views

Trig question, inequality

How can I find the following product using elementary trigonometry? Suppose $0 \lt x \lt \frac{\pi}{2}$ is an angle measured in radians. Use the trigonometric circle and show that $\cos(x) \le ...
-1
votes
1answer
12 views

Equation of a line with a positive gradient [on hold]

Two straight lines passing through the point (0,2) are tangent to the graph of the function y=1-x^2. Find the equation of the line with a positive gradient.
1
vote
1answer
24 views

Ambiguous Limits in Area Determination

I am to find the centroid of the area bounded by the curve $y=8x^3-24x+11$, the $x$-axis and the line $x=-1$. Now I know that the centroid requires me to find the area under the curve first. I have ...
1
vote
3answers
208 views

Find all values of $x$ at which the tangent line to the given curve has intercept $ y= 2$

Find all values of $x$ at which the tangent line to the given curve has intercept $y = 2$ I am confused about the $y$-intercept $2$ the function $$f(x) = \frac{(2x + 5)}{(x + 2)}$$ The derivative ...
1
vote
3answers
45 views

Diagonalization with the given eigenvalue and its vector

Let $-3$ be an eigenvalue of a $3\times3$ singular matrix $P$ and $$P\begin{bmatrix} 5\\ 3\\ -2 \end{bmatrix}=\begin{bmatrix} -20\\ -12\\ 8 \end{bmatrix}.$$ Then find whether $P$ is ...
-1
votes
2answers
26 views

Invertible Linear Maps Proof [on hold]

1) Suppose $V$ is finite dimensional and $S$, $T$, $U \in L(V)$ and $STU = I$. Show $T$ is invertible and $T^{-1} = US$. 2) Suppose $V$ is finite dimensional and $R$, $S$, $T \in L(V)$ are such that ...
1
vote
4answers
32 views

Number of distinct real roots with $e^{-x}$ in the equation

How to find the number of distinct real roots of the equation $$13x^{13}-e^{-x}-1=0$$ I know that we generally find number of real roots by observing number of sign changes in $f(x)$ and $f(-x)$ but ...
0
votes
1answer
13 views

Linear Operators Injectivity and Surjectivity

Suppose T $\in L(P(R))$ is such that T is injective and deg Tp $\leq$ deg p for every nonzero polynomial p $\in P(R)$. Prove that T is surjective and that deg Tp = deg p for every nonzero p $\in ...
7
votes
1answer
265 views

Galois over Galois

I am working on this exercise: If $E$ is an intermediate field of an extension $F/K$ of fields. Suppose $F/E$ and $E/K$ are Galois extensions, and every $\sigma\in Gal(E/K)$ is extendible to an ...
0
votes
1answer
22 views

How to know when a line is parallel to the xz-plane

What are some features of the equations of a line that is parallel to the xz plane, but does not lie on the plane, and is not parallel to any of the axes? So far all I got: -dot product of plane's ...
2
votes
1answer
20 views

number of ways to put 4 black,4 white,4 red balls in 6 different boxes

The question says:in how many ways we could put 4 black,4 white,4 red balls in 6 different boxes? boxes are distinguishable,black balls are identical,red balls are identical,and white balls are ...
1
vote
2answers
105 views

Determine the number of solutions of nonlinear system without solving.

$x^2-y^2+2y=0$, $2x+y^2-6=0$ I need to determine the number of solutions without solving it. There is a hint that a graph can help but I am still not sure how to go about this. Thanks