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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0
votes
1answer
18 views

Algebra - fraction problem

"The cooler in a car contains $8$ litres. The coolant fluid contains $\dfrac3{10}$ of glycol and rest is water. To increase the glycol content to $\dfrac35$ you drop some of the coolant fluid and fill ...
9
votes
4answers
619 views

Show that $\frac{2a_1^2}{a_1+a_2}+\frac{2a_2^2}{a_2+a_3}+…+\frac{2a_n^2}{a_n+a_1}\geq a_1+a_2+…+a_n$

Showing that $ \frac{2a_1^2}{a_1+a_2}+\frac{2a_2^2}{a_2+a_3}+...+\frac{2a_n^2}{a_n+a_1}\geq a_1+a_2+...+a_n$ holds for positive $a_i$s. I've tried adding $a_1+a_2, a_2+a_3,...,a_n+a_1$ respectively ...
3
votes
0answers
62 views

If $A = \frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\cdots+\frac{1}{\sqrt{999}}+\frac{1}{\sqrt{1000}}.$ Then $\lfloor A \rfloor$ is,

If $\displaystyle A = \frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\cdots\cdots\cdots+\frac{1}{\sqrt{999}}+\frac{1}{\sqrt{1000}}.$ Then $\lfloor A \rfloor$ is, where $\lfloor A\rfloor = A-\{A\}.$ ...
0
votes
2answers
29 views

Help with system of linear equations

I am in need of help solving for $x$ and $y$. $$ \begin{cases} 10x-8y=-5x \\ 5x=4y-20 \end{cases} $$ I've tried but don't really get what I am doing wrong. Thanks in advance.
0
votes
1answer
26 views

How do I compute the coordinates of the center and the radius of a circle.

$$9y^2+7x^2+35=8x−2x^2−36y $$ I am having trouble with this problem, and I managed to do some of it. but I got stuck here. My steps so far: $$9y^2+7x^2+35-8x+2x^2+36y = -35$$ $$(9y^2 + 36y)+(9x^2 ...
0
votes
0answers
10 views

Spacing of points in closed curves

Given an (implicitly defined by $F(x,y) = 0$) closed curve $C$ lying on $\mathbb{R}^2$ and the set of points inside $C$ be named $I$ for $I \in \mathbb{R}^2$, and $\sigma$ points, is it possible to ...
0
votes
3answers
30 views

Exponential inequality to Different Bases

How do I solve this exponential equation if I can't make 6 in base 2? $$ 6 - 2^x \geq 4^x $$ I know that the solution is $ x \in ]-\infty,1] $ because it just makes sense that $ 6-2=4$ I just don't ...
1
vote
1answer
41 views

Evaluation of $\sum_{i=0}^{100}\sum_{j=0}^{100}\binom{100}{i}\cdot \binom{100}{j}\cdot\binom{200}{i+j}^{-1}$

Evaluation of $\displaystyle \sum_{i=0}^{100}\sum_{j=0}^{100}\frac{\binom{100}{i}\cdot \binom{100}{j}}{\binom{200}{i+j}} = $ $\bf{My\; Try::}$ Let $\displaystyle i+j=n\;,$ Then Sum convert into ...
-1
votes
0answers
20 views

Finding the roots of fourth degree polynomial [duplicate]

$$ax^4 + bx^2 +cx + d = 0$$ How do I find just the real roots not even complex roots ?
0
votes
0answers
19 views

Do we recognize higher degree asymptotes beyond Horizontal and Oblique?

I am reading a textbook, and it talks about doing synthetic division in order to rewrite a function into the quotient $$R(x)=\frac{p(x)}{q(x)}= f(x) + \frac{r(x)}{q(x)}$$ Since $\frac{r(x)}{q(x)}$ ...
0
votes
1answer
26 views

Solve denominator so quotient is whole number?

I have a simple equation. road_length = ROADLENGTH / ROADSPACING The problem is, I really need road_length to be a whole number because it's used in FOR loop in ...
0
votes
1answer
48 views

A minimum Value Sum [on hold]

The minimum value of $\sqrt{x^4 - x^2 - 24x + 145} + \sqrt{x^4 - 23x^2 - 2x + 145}$ can be expressed in the form ($a\sqrt{b}$), where $a$ is an integer, $b$ and is not divisible by the square of any ...
1
vote
0answers
27 views

Generalization of a Diophantine Equation Problem

I've been working a lot with Pythagorean triples and sums of squares that are themselves squares, specifically interlocking sums (where one square is part of two or more sums). As part of my work I ...
0
votes
2answers
25 views

Solving triangles with trig, word problem

Engineers want to measure the distance from P to Q, but the span from P to Q is across the tip of a lake. So they select a point R on land and find that the distance from R to Q is 100 feet and from R ...
2
votes
1answer
30 views

Trigonmetric sum of inverses

Prove that: $$\sum^{45}_{k=1}\frac{1}{\cos1^\circ-\cos(87+4k)^\circ}=\frac{1}{2\sin 1^\circ}$$ Numerically, this is accurate comparing the lhs and rhs. Some ideas: We can transform the question ...
1
vote
1answer
55 views

Prove that $(1+2+3+\cdots+n)^2=1^3+2^3+3^3+\cdots+n^3$ $\forall n \in \mathbb{N}$. [duplicate]

Prove that $(1+2+3+\cdots+n)^2=1^3+2^3+3^3+\cdots+n^3$ for every $n \in \mathbb{N}$. I'm trying to use induction on this one, but I'm not sure how to. The base case is clearly true. But when I add ...
0
votes
3answers
30 views

Range of values of $t$ for which $ 2\sin t = \frac{1-2x+5x^2}{3x^2-2x-1}\;,$

Calculation of Range of values of $t$ for which $\displaystyle 2\sin t = \frac{1-2x+5x^2}{3x^2-2x-1}\;,$ where $\displaystyle t \in \left[-\frac{-\pi}{2}\;,\frac{\pi}{2}\right]$ $\bf{My\; Try::}$ ...
1
vote
4answers
42 views

Proof by Induction Divisibility.

$6^n-5n+4$ is divisible by 5 for all positive integers $n$. $n >=1$ Prove By Induction My attempt is as follows: $n=1$ $6^1-5(1) +4$ $=5$, Therefore 5 is divisible by 5 so $n=1$ is true ...
1
vote
2answers
30 views

Algebra problem with fractions

"In a musical class, the students either played piano or violin as head instrument. By a concert, the students got to choose whether they would do a solo or pair-performance. A piano player can only ...
2
votes
3answers
34 views

Simplify - Help

I am supposed to simplify this: $$(x^2-1)^2 (x^3+1) (3x^2) + (x^3+1)^2 (x^2-1) (2x)$$ The answer is supposed to be this, but I can not seem to get to it: $$x(x^2-1)(x^3+1)(5x^3-3x+2)$$ Thanks
0
votes
3answers
37 views

Derivative of Equation

So i have this problem with the function: $$U(x)=\frac{A^2}{x^2} - \frac{A}{x}$$ I need to find the derivative of $U$ to find the min and max values. It says in the problem that $A$ is a positive ...
3
votes
2answers
62 views

Can the choice of epsilon be arbitrary in epsilon-delta proofs?

I've been reading Spivak's chapter on limits and something that I don't feel I understand entirely is how the epsilon is decided upon. It makes sense to me in the context of $\,|f(x)-L|<\epsilon$ ...
-3
votes
2answers
26 views

Show that $a c\equiv b c\pmod m,\;a. b, c, m \in \mathbb Z$ and $m \geq 2 $ does not imply $a\equiv b \pmod m$ [on hold]

Show that $a c\equiv b c\pmod m$ with $a. b, c, m \in \mathbb Z$ and $m \geq 2 $ does not imply $a\equiv b \pmod m.$
3
votes
4answers
68 views

How many zeros does $f(x)= 3x^4 + x + 2 $ have?

How many zeros does this function have? $$f(x)= 3x^4 + x + 2 $$
-3
votes
2answers
59 views

Help me solve $-4p+21=43$ [closed]

$$-4p+21=43$$ I have no idea what it is
1
vote
1answer
23 views

Area of $\triangle ABC$ whose sides $a,b,c$ satisfy $0\leq a \leq1;1\leq b \leq2;2\leq c \leq3$ is

The maximum area of a triangle whose sides $a,b,c$ satisfy $0\leq a \leq1;1\leq b \leq2;2\leq c \leq3$ is $\bf{My\; Try::}$ Area of $\displaystyle \triangle ABC = \frac{1}{2}ab\sin C = ...
11
votes
2answers
111 views

How to prove $\large\sqrt[\pi]{e} < \sqrt[\pi]{\pi}<\sqrt[e]{e}< \sqrt[e]{\pi}$

I was given a challenge of sorting the following numbers. $\Large\sqrt[\pi]{e} < \sqrt[\pi]{\pi}<\sqrt[e]{e}< \sqrt[e]{\pi}$. After some work I was able to figure out the order. How can one ...
3
votes
2answers
67 views

Why is the derivative of a polar function $dy/dx$ and not $dr/d\theta$?

I don't understand. If $r = 2\cos(\theta)$ then why is the derivative: $dy/dx$? I have a "hypothesis," By the polar equation are you really describing a curve in the cartesian plane? So is that ...
2
votes
5answers
56 views

Problem involving cube roots of unity

Given that $$\frac{1}{a+\omega}+\frac{1}{b+\omega}+\frac{1}{c+\omega}=2\omega^2\;\;\;\;\;(1)$$ $$\frac{1}{a+\omega^2}+\frac{1}{b+\omega^2}+\frac{1}{c+\omega^2}=2\omega\;\;\;\;\;(2)$$ Find ...
1
vote
2answers
36 views

Calculus Problem I need help

An efficiency study of the morning shift at a certain factory indicates that an average worker arriving on the job at 8:00 A.M. will have assembled $f(x)= -x^3 + 6x^2 +15x$ television set $x$ ...
-2
votes
1answer
57 views

Solving $\frac{4P}{3} = 4P^{1/3}$ [closed]

Could I have some pointers on solving following equation for $P$ $$\frac{4P}{3} = 4P^{1/3}$$ Thanks!
0
votes
6answers
96 views

Finding the roots of $x^2+(3+5i)x+(7+11i)=0$

how can I solve following equation analytically $$x^2+(3+5i)x+(7+11i)=0$$ I need the roots as follow $x=a+bi$
-1
votes
0answers
45 views

Show that $\frac{\tan x + \cos y }{ \tan x \cos y} = \tan y \cos x $

Show that $$\frac{\tan x + \cos y }{ \tan x \cos y} = \tan y \cos x $$ I was given this problem due to me losing to my teacher in a game of cards and he expects it done tomorrow but honestly ...
-4
votes
0answers
24 views

help with algebra/precal [closed]

http://imgur.com/RCHo581,KevRAvO,pQTIbpZ Stumped by these questions. Help is greatly appreciated!
0
votes
0answers
25 views

General solution for N apples?

Of 6000 apples, every 3rd is too small, every 4th is too green, every 10th is too bruised, and the rest are perfect. how many are too small, green, bruised, and perfect. what if there were N apples? ...
2
votes
3answers
81 views

Combinatorics question about choosing non consecutive integers

The problem is as follows: How many ways are there to pick $6$ of the first $20$ positive integers such that no $2$ of them are consecutive? At first glance, this seems like a fairly ...
0
votes
2answers
39 views

Properties of exponents $e^{2.5-.5t}$

How does $e^{2.5-.5t}$ = $e^{2.5}$($e^{-.5t}$) I thought that $e^{2.5-.5t}$ = $e^{2.5/.5t}$. does the variable t make my algebra incorrect?
1
vote
2answers
19 views

Properties of exponents with Euler's number and logarithims

How does $80e^{ln(5)t/(20)}$ equal $80(5)^{t/20}$?
0
votes
1answer
17 views

Find the range of a complicated function

I need to find the range of the following function : $$f(x,y) = \sqrt[4]{\frac{4x - 3y + 5}{3y-4x + 13}}$$ So my thoughts about it are first the bottom part $( 3y - 4x + 13 )$ must be greater than ...
0
votes
1answer
13 views

mathematical model

The question: $g$ varies directly with $f$ and inversely with $c$ and the square of $d$. So we have to setup the equation given that information. Looking at my notes a bit it seems like it might be ...
4
votes
0answers
147 views
+250

How can I construct a solution for this system of many inequalities?

Let there be types $\omega\in\{0,1\}^n$ drawn according to some probability distribution. Suppose that these types are relayed through some imperfect message service. Specifically, any type $\omega$'s ...
0
votes
2answers
70 views
+50

Find the locus of points M the difference of the squares of whose distances from two given points A and B is equal to a given value c.

Find the locus of points M the difference of the squares of whose distances from two given points A and B is equal to a given value c. For what values of c does the problem have a solution? I am ...
6
votes
2answers
62 views

Minimum of $ay+az+bz+bx+cx+cy$ with $ab+bc+ca=xy+yz+zx=1$

Let $a,b,c,x,y,z\in\mathbb{R}^+$, and $ab+bc+ca=xy+yz+zx=1$. What is the minimum value of $ay+az+bz+bx+cx+cy$? When $a=b=c=x=y=z=\dfrac{1}{\sqrt{3}}$, the desired value is $2$. When ...
5
votes
5answers
163 views

Show: $(x+y)^4 \leq 8(x^4 + y^4)$ Using Cauchy-Schwarz Inequality

I wish to show the following statement: $ \forall x,y \in \mathbb{R} $ $$ (x+y)^4 \leq 8(x^4 + y^4) $$ What is the scope for genralisaion? Edit: Apparently the above inequality can be shown ...
0
votes
1answer
20 views

Substituting Formulas

Substituting the formula for height of a tree in the formula for volume of a tree, the new formula for volume becomes ________________. A)V = (1/3)πr(kr2/3) B)V = (1/3)πr2 C)V = (1/3)πkr3 D)V = ...
0
votes
2answers
20 views

System of Equations Given One Equation

7=3x+2y-z How many more equations would you need to solve x, y, and z? In which variables can the additional equations be? Give examples of equations that would help solve these variables. (Hint: ...
0
votes
1answer
10 views

Distance/measurement question! Help!

There is a question in a book that I am trying to solve. "A man usually rides his bike 1 kilometers per hour, yet the wind slows him to 6.76 kilometers for 26 minutes and 5.55 kilometers for 10. How ...
-1
votes
4answers
54 views

How can we find factorials in decimal form? [duplicate]

I've heard of factorials such as $5!$ and $3!$, which work like this: $5!=5\times4\times3\times2\times1=120$ and $3!=3\times2\times1=6$. At least this is what we get. Also, surprisingly, $0!=1$, but ...
-4
votes
2answers
31 views

Find its simplest form [closed]

Given $$ \psi(x)=x^2+5 $$ Simplify $$ \frac{\psi(x+h)-\psi(x)}{h}, h\not =0 $$
-1
votes
1answer
17 views

sketch the following functions stating the domain and range of each [closed]

sketch the following functions stating the domain and range of each:and sketch if possible a. $y = \sqrt{9-x^2}$ b. $y = \sqrt{x-1}$ c. $y = |2x|$ d. $y= 1/x-4$ e. $y= |2x|-1$ please help me ...