Linear, exponential, logarithmic, polynomial, rational, and trigonometric functions, conic sections, binomial, surds, graphs and transformations of graphs, equation- and system-solving, and other symbolic-manipulation topics.

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1answer
26 views

How to make a comparison between two variables like $4\sqrt{\frac{b}{a}}$ and $\frac{2}{a}+2b$?

Both a and b are positive numbers. And how to make a comparison between two variables like $4\sqrt{\frac{b}{a}}$ and $\frac{2}{a}+2b$? I tried subtraction and division, but both failed.
-1
votes
1answer
14 views

Matrix for rounding to the nearest whole number

http://imgur.com/nwYFNhz Having trouble with part b. Is there a way to get a matrix calculation to round to the nearest integer?
2
votes
2answers
41 views

What is the % operator? Why is 4 % 3 = 1?

Sorry this is a dumb question. I'm trying to relearn math: $4 / 3 = 1.333....$ So, why is 4 % 3 = 1? I have 4 cats. I cut them evenly into three groups, meaning each group contains 1.3333th of a ...
1
vote
0answers
30 views

Complex quadratic question

I am having trouble solving this problem below. I've tried a number of methods but don't seem to be getting any closer to a solution. Find the possible values of x $$(\frac{3x^2 - 2x - 3}{2x^2})^2 ...
2
votes
1answer
17 views

Reciprocals of interval union length

Let $I_1,I_2,\ldots,I_n$ be nondegenerate intervals in $[0,1]$. What is the minimum of $\sum_{1\leq i,j\leq n}\frac{1}{|I_i\cup I_j|}$, where the sum is over pairs of intervals that are not disjoint? ...
0
votes
0answers
23 views

formula of $an+b$ to except the first item. is this possible?

Well, I feel kinda dumb because I can't solve this. I want a formula of $an+b$ kind to describe any following number, but not the first one, where's $a$ represents a cycle size, $n$ is a counter ...
3
votes
0answers
28 views

Is it possible to simplify a nested radical in the form $\sqrt[3]{\sqrt[3]{A}-B}$ into $\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}$

I'm wondering if there is a way to simplify any nested radical in the form $\sqrt[3]{\sqrt[3]{A}-B}$ into $\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}$. Examples such as ...
0
votes
1answer
22 views

Difference of Squares Oddity

I was practicing factoring polynomials and ran across a problem I'd never seen before. $$x^{2m} -36y^2$$ I know this is a difference of squares but I'm not sure how to handle the '$m$' and the ...
0
votes
1answer
22 views

Bombelli's solution to a cubic

On p151 of my edition of Ian Stewart's "The Problems of Mathematics", he describes early work with imaginaries and Cardano's noting that Tartaglia's formula for solving a cubic, when applied to: $$ ...
5
votes
2answers
360 views

A polynomial of degree 3 that has three real zeros, only one of which is rational.

Find a polynomial of degree 3 that has three real zeros, only one of which is rational. My answer: $(x - \sqrt{2})(x - 3)(x - \pi)$. Is this correct? It does have two irrational zeros, but I'm not ...
1
vote
1answer
40 views

Solve $2x\cos x+(1-x^2)\sin x=0$

I can't solve: $$2x\cos x+(1-x^2)\sin x=0$$ The solution must be $(k-1)\pi<x_k<(k-1)\pi+\frac{\pi}{2}$ for $k=2,3,4,\ldots$ Any hint? Many thanks!
0
votes
0answers
18 views

Solving an integer equation (equi-energy transition)

In chemistry, we came across an equation as follows: $$\frac{Z_1^2}{n_1^2}-\frac{Z_1^2}{n_2^2}=\frac{Z_2^2}{n_3^2}-\frac{Z_2^2}{n_4^2}$$ We were supposed to assume that this implied that ...
2
votes
8answers
55 views

Is there a simple, intuitive way to see that $f(x)=x-\sqrt{x^2-1}<1$ if $x>1$

Is there a simple intuitive way to show that $f(x)=x-\sqrt{x^2-1}<1$ if $x>1$? I sense it could be done more simple than this: 1 - take the derivative $f'(x)=1-\frac{x}{\sqrt{x^2-1}}<0$ if ...
0
votes
1answer
25 views

$4^{3a-1}-5^{2b-3}=0$ find a in terms of $b$

If $4^{3a-1}-5^{2b-3}=0$ then find a using $b$. My Attempt:we know that $2^{6a-2}=5^{2b-3}$ with this way we can find a value for $a$ and $b$ if both sides are zero if we can find another value for ...
-1
votes
3answers
43 views

Proof of a logarithm identity

I would like to know how to prove the following log identity: $x^{\frac{\log(\log(x))}{\log(x)}} = \log(x)$
1
vote
1answer
32 views

Expanding $\frac{\Gamma(n)}{\Gamma(n-k)}$ as a polynomial

I want to expand $\frac{\Gamma(n)}{\Gamma(n-k)}$ as a polynomial, where $\Gamma$ is the gamma function. For $k\in\mathbb{N}$, it can be "simplified" as ...
2
votes
3answers
67 views

Find the inverse function of $f(x) = \dfrac{x}{1-x^2}$

The function $F : (-1, 1) \to \mathbb {R}$ is defined by $F(x) = \dfrac{x}{1-x^2}$. Example 5 page 106 at Munkers' Topology says that its inverse is $$G(y) = \dfrac{2y}{1+(1+4y^2)^{\frac12}}.$$ I ...
1
vote
2answers
28 views

Find all real numbers m for which equation $z^3+(3+i)z^2-3z-(m+i)=0$ has ..

Problem : Find all real numbers m for which equation $z^3+(3+i)z^2-3z-(m+i)=0$ has atleast a real root. No idea of approach : I am not getting idea how to approach a cubic polynomial in complex ...
5
votes
5answers
423 views

Proving an algebraic identity

Prove: $$(a + b + c)(ab + bc + ca) - abc = (a + b)(b + c)(c + a)$$ Problem: I am not sure how to proceed after expanding the brackets on the RHS. I am not sure if I also expanded correctly. My ...
0
votes
2answers
44 views

Questions about proof of inequality $(x^p-1)/p <> (x^q-1/q) $

I am working through a proof of the following inequality, where x,p,q are positive, and p and q are integers. $$ \frac{(x^p-1)}{p} \neq \frac{(x^q-1)}{q} $$ Which gives $$ \tag1 ...
1
vote
2answers
22 views

Maximum Height is giving me negative

Hey guys for this parametric equations its giving me negative Question is: A dart is thrown from a point 5 feet above the ground with an inital velocity of 58 ft/sec and angle of elvation of 41∘. ...
4
votes
1answer
41 views

Greatest Integer Function

OK so it's been a while since I've done this math. I'm familiar with graphing greatest integer functions as this $[3x]$. I've bumped into this problem and I can't quite figure out how to graph it. ...
1
vote
2answers
14 views

How to calculate duration of event at different speeds

Specifically I want to figure out the formula which will tell me: how long will it take to watch this video (normal length $L$) at speed $x$. I think this will be asymptotic, no matter how fast you ...
-1
votes
0answers
22 views

conics, ellipse etc [on hold]

A ball is thrown from $3$ ft. Equations: $x = 69t, y= 3+40t-16t^2$. How long is the ball in flight and how far it travel? Answer: Eliminating the parameter and solving the quadratic ...
0
votes
2answers
21 views

Moving vectors to the left and the right of a product

Suppose that $A$ and $B$ are $1\times n$ row vectors and $x$ is a $n\times 1$ column vector. I have an expression $$ (Ax)^2B'B $$ which is an $n\times n$ matrix. Question: Is it possible to write ...
0
votes
1answer
29 views

How to deal with identities in which one expression, but not the other, evaluates to “undefined” in particular instances

I haven't touched trigonometry for awhile and whilst flicking through an old set of notes I came across the following expression: $$\csc(x) \cdot \sec(x) \cdot \sin^2(x)$$ I'm aware that this can ...
0
votes
2answers
28 views

Conic sections checking my answers

A ball is thrown from $3$ ft. Equations: $x = 69t, y= 3+40t-16t^2$. How long is the ball in flight and how far it travel? Answer: Eliminating the parameter and solving the quadratic ...
3
votes
1answer
63 views

Exponent analog to the factorial function

Triangular numbers can be discovered by taking any number $n$, and adding $$\sum_{i=0}^n i = n + (n - 1) + (n - 2) ... 1 = \frac{n(n + 1)}{2}$$ These numbers can be generalized by putting any real ...
0
votes
2answers
16 views

Image: proof re fractional exponents

Can someone help me prove that $(a^m)^{1/n} = (a^{1/n})^m$ per the textbook excerpt captured in the image?
0
votes
2answers
76 views

algebra homework question [on hold]

Let $$f(x)= \frac {1}{e^{1-2x}+1}$$Find $$f\left(\frac {1}{2009}\right)+f\left(\frac {2}{2009}\right)+f\left(\frac {3}{2009}\right)+ \cdots +f\left(\frac {2008}{2009}\right)$$
0
votes
1answer
21 views

Help Computing integral of continuous function

Consider the Dirichlet function $F:R \rightarrow R$ given by $f(x):=\begin{cases} x &\text{if } x < 1 , \\{}\\ x+1 &\text{if } 1<= x <= 2, \\{}\\ -x+5 &\text{if } 2<x ...
-1
votes
0answers
43 views

The irreducibility of $a^{4n}+b^{4n}$ [on hold]

How to prove that $a^{4n}+b^{4n} $, for any natural number $ n $, is irreducible over the rationals?
0
votes
3answers
27 views

Continous function

There are two functions g(x)= ($(2x+1)^{1/2}$-$1$)/x , where x is not equal to zero = 1 , x=0 h (x) = $x^9 - 6x^8 -2x^7 + 12x^6 +x^4 -7x^3 + 6x^2 + x-7$ ...
5
votes
3answers
50 views

Factorization of $x^5+x$

I need to make the decomposition in $\mathbb{R}$ of: $$x^5+x$$ Here my steps: $$x(x^4+1)$$ $$x\big[ (x^2)^2+1 \big]$$ $$x\big[(x^2+1)^2-2x^2\big]$$ how should I proceed?
1
vote
1answer
39 views

Evaluating the arc length integral $\int\sqrt{1+\frac{x^4-8x^2+16}{16x^2}} dx$

Find length of the arc from $2$ to $8$ of $$y = \frac18(x^2-8 \ln x)$$ First I find the derivative, which is equal to $$\frac{x^2-4}{4x} .$$ Plug it into the arc length formula ...
1
vote
0answers
14 views

Number of terms in multivariate polynomial

We know that the number of terms in a univariate polynomial of degree n is n+1. But what about if there are multiple variables: for eg: for variables $x,y$ polynomial of degree 2 will have: ...
0
votes
0answers
29 views

Solving an inequality systematically

A question states: "Find all $n >3$ such that $$ \frac{1}{n^{1.1}}<\frac{1}{n \ln n}" $$ Here's my step: $$ n^{1.1}>n \ln n $$ $$ n^.1>\ln n $$ $$ n >(\ln n)^{10} $$ Setting $(\ln ...
0
votes
1answer
19 views

Show inequality is correct by simplification

This may be a bit of a silly question, but I was wondering if there is any way to show with some simplification (without using a calculator) that clearly $32-16\sqrt2 < 8\sqrt2$ (since it ...
1
vote
2answers
46 views

build absolute value equations know solution

We have absolute value equations with unknown coefficients: $$|x + a| = b$$ and we know the solutions: $$x = 11 \text{ and } x = 5$$ We need to find $a$ and $b$. From $$11 + a = b \\ 5 + a = -b$$ we ...
1
vote
1answer
15 views

simplification question on Diff. Eq. Solution

Can anyone explain this simplification from y^-2 to y? If you distribute the x^4 through, you obtain y^-2 = (2/x) + Cx^4. This leads to y^2 will equal the reciprocal of what I just wrote. Where in ...
0
votes
1answer
34 views

Parametric Problem

i have a question on parametric.. The question states A vector equation $(x,y) = (2,-1) + t(3,2)$. Write as a parametric equation. Show a table with x,y values. Sketch a picture of vector ...
0
votes
1answer
24 views

Algebra question - Formulas

Really having a hard time with this question. any help would be appreciated. Suppose that a cliff diver's height (in feet) after t seconds is given by the model $H = −16t^2+48t+28$ . Find the height ...
2
votes
4answers
49 views

how to verify $\frac{\sin(x)\cos(x)}{\cos^2(x)-\sin^2(x)}=\frac{\tan(x)}{1-\tan^2(x))}$? [on hold]

How would I verifty the following trig identity? $$ \frac{\sin(x)\cos(x)}{\cos^2(x)-\sin^2(x)}=\frac{\tan(x)}{1-\tan^2(x)} $$ I am not sure how to start.
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votes
1answer
37 views

Problem about water and time

You can fill a container with water from a hot water tap in $80$ minutes. The container can be filled with water from a cold water tap in $48$ minutes. How long does it take to fill the tub, if you ...
0
votes
4answers
37 views

Find particular solution to nonhomogeneous DE $y'+y=x^2+\sin{x}+\cos{x}$

I'm new to nonhomogeneous DE's and I have come across this DE: $$y'+y=x^2+\sin{x}+\cos{x}$$ which I'm supposed to provide a general solution to. However, I get stuck with the particular solution. The ...
-1
votes
2answers
41 views

Getting $\sin^2$ and $\cos^2$ values from $\sin^2 \alpha = \frac{1}{4} \cos^2 \alpha = \frac{1}{4}(1 - \sin^2 \alpha)$

How can you get from $$\sin^2 \alpha = \frac{1}{4} \cos^2 \alpha = \frac{1}{4}(1 - \sin^2 \alpha)$$ to $$\sin^2 \alpha = \frac{1}{5} \\or \\ \cos^2 \alpha = \frac{4}{5}.$$ Sorry, but I can't see ...
0
votes
3answers
33 views

Let $z \in C^*$ such that $|z^3+\frac{1}{z^3}|\leq 2$ Prove that $|z+\frac{1}{z}|\leq 2$

Problem : Let $z \in C^*$ such that $|z^3+\frac{1}{z^3}|\leq 2$ Prove that $|z+\frac{1}{z}|\leq 2$ My approach : Since : $(a^3+b^3)=(a+b)^3-3ab(a+b)$ $\Rightarrow ...
0
votes
0answers
22 views

Proving that if $m,n,p,q\in\mathbb{Z^+}, \sqrt[p]{m}\in\mathbb{R}\setminus\mathbb{Q}$ then $\sqrt[p]{m}+\sqrt[q]{n}\in\mathbb{R}\setminus\mathbb{Q}$

If $\sqrt{m}+\sqrt[q]{n}=r$ rational, the rationality of $\sqrt{m}$ is derived expanding $(r-\sqrt{m})^q$ using the binomial theorem: after rearrangement, isolating the terms containing odd powers of ...
1
vote
1answer
29 views

Let a,b,c be distinct non zero complex numbers with $|a|=|b|=|c|$ If each of …

Problem : Let a,b,c be distinct non zero complex numbers with $|a|=|b|=|c|$ If each of the equations $az^2+bz+c=0$ and $bz^2+cz+a=0$ has a root having modulus 1, then prove that : ...
0
votes
1answer
20 views

Let a be a positive real number and let $M_a=\{z \in C^* : |z+\frac{1}{z}|=a\}$ Find the minimum… [duplicate]

Problem : Let a be a positive real number and let $M_a=\{z \in C^* : |z+\frac{1}{z}|=a\}$ Find the minimum and maximum value of $|z|$ when $z \in M_a$ My approach : $|z+\frac{1}{z}|=a$ Squaring ...