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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4
votes
2answers
129 views

Find $I$ in $\frac{\overline{SIX}}{\overline{NINE}}=\frac23$

In $\frac{\overline{SIX}}{\overline{NINE}}=\frac23$ every letter denotes a UNIQUE digit,find $I$. Expanding the fraction in base $10$ we have: $300S+30I+3X=2020N+200I+2E$ , but this doesn't ...
0
votes
4answers
56 views

When to simplify a quadratic equation?

I had the following quadratic equation: $$38x^2 - 140x - 250 = 0$$ And before starting to solve it, I simplified it by dividing all terms by $2$: $$19x^2 -70x - 125 = 0$$ But when I solved it I got: $...
0
votes
2answers
68 views

proving no real roots exist

Prove that $x^8-x^7+x^2-x+15$ has no real roots. I did it by first assuming it has real roots and then applying Descartes rule of signs. We find that if there are any real roots, they all must be ...
1
vote
0answers
50 views

What is meant by a function being linear in two variables?

I'm trying to understand the Mangasarian condition in the context of dynamic optimization (see here p 8.12) and am not sure what exactly is meant by a function $f(x,u)$ being linear in $x$ and $u$. If ...
2
votes
1answer
15 views

Proportion questioning

2 firms make the following charges for renting a car over the weekend . Firm A - Has a fixed charge of $320, and Charged 50 cents per km for every km over 300 Firm B - has a fixed charge of $60 and ...
1
vote
2answers
43 views

Proving $(w-1)^m$ is purely imaginary.

I'm having trouble trying to prove this: Let $ m\in \mathbb Z$, m even and $w\in\mathbb C$ a primitive $2m$-th root of unity. Prove that $(w-1)^m$ is purely imaginary. What I've tried to do so ...
10
votes
3answers
447 views

Why are there two versions of a polar equation for a circle from geometric form

In class today we learned that a rectangular/geometric equation for a circle such as $x^2+(y-5)^2 = 9$ can be converted into a polar equation by reducing it to the quadratic equation $r^2-10r\sin \...
2
votes
1answer
75 views

If $f(f(x)) = f(x^2)$, then must there be some constant $c$ such that $f(x)=c$ for all values of $x$ in the domain of $f$?

Here is a problem from Rusczyk-Crawford's Art of Problems Solving: Intermediate Algebra textbook (Chapter 2 Review, problem 2.30). If $f(f(x)) = f(x^2)$, then must there be some constant $c$ such ...
4
votes
2answers
79 views

The sequence $(a_n)$ is given as $a_1=1, a_{2n} = a_n - 1, a_{2n+1} = a_n + 1$. $a_{2015}=$?

The sequence $(a_n)$ is given as $a_1=1, a_{2n} = a_n - 1, a_{2n+1} = a_n + 1$. What's the value of $a_{2015}$ Correct answer should be $a_{2015} = 9$. How? thing that came to mind was to see what $...
1
vote
5answers
62 views

Algebraic manipulation of a limit.

What are the algebraic manipulations and steps that makes the limit \begin{equation} \lim_{x\to2}\left(\frac{x^3-8}{x-2}\right) \end{equation} equal to \begin{equation} \lim_{x\to2}(x^2+2x+4) \end{...
1
vote
4answers
79 views

If $0 \le a \le 1$, then show that $xa + (1-a)y$ will always lie between $x$ and $y$.

If $0 \le a \le 1$, then show that $xa + (1-a)y$ will always lie between $x$ and $y$. I am sorry if this may seem like elementary question. I have tried many examples and they all seem to work. ...
2
votes
1answer
37 views

Plot of a function

What is the plot of: $$y=\frac{\beta(1-\alpha)x}{\alpha(1-\beta)+(\beta-\alpha)x}$$ with $0<\alpha<\beta<1$. How do I handle the parameters? How do I compute the derivatives to check for ...
3
votes
3answers
22 views

What is the logarithm of $(a-b)\delta_{ij}+b$?

Just now I came across the expression similar to: $x_{ij} = (a-b)\delta_{ij}+b$ The author then somehow converts this expression, into: $\ln x_{ij} = (\ln a-\ln b)\delta_{ij}+\ln b$ This comes ...
0
votes
2answers
43 views

Solution of irrational equations

I need some help solving these equations: $ \sqrt{2x+1} - \sqrt{x+8} > 3$ and $ \sqrt{3x^2 - 5a^2} = 2a - x$ Thank you in advance! :)
3
votes
3answers
84 views

Why does $n \geq 2$ imply that $\frac n 2 < n$?

It has been a while since I did math proof in school, and I just can't figure out why $$n \geq 2 \text{ implies that } \frac n 2 < n$$ Anything would help! Thanks.
4
votes
6answers
120 views

Prove that $ 1+2q+3q^2+…+nq^{n-1} = \frac{1-(n+1)q^n+nq^{n+1}}{(1-q)^2} $

Prove: $$ 1+2q+3q^2+...+nq^{n-1} = \frac{1-(n+1)q^n+nq^{n+1}}{(1-q)^2} $$ Hypothesis: $$ F(x) = 1+2q+3q^2+...+xq^{x-1} = \frac{1-(x+1)q^x+xq^{x+1}}{(1-q)^2} $$ Proof: $$ P1 | F(x) = \frac{1-(...
0
votes
0answers
21 views

How would you solve an equation in two variables multiplied between each other?

In particular, how would you solve such an equation: $$ \alpha(1-\beta)x_2-\beta(1-\alpha)x_1+(\beta-\alpha)x_1x_2=0$$ with $0<\alpha<\beta<1$ and $x_1,x_2\in[0,2]$. Obviously one solution is ...
0
votes
1answer
20 views

Deal-Grove Model: How do we arrive at $F_{1}=F_{2}=F_{3}=F=\frac{C^{*}}{\frac{1}{k_{s}}+\frac{X_{ox}}{D}+\frac{1}{h}}$

Seems like a trivial question, but I'm confused. What is the step by step way to combine three equations in the Deal-Grove model: $$F_{1}= h(C^{*}-C_{o})$$ $$F_{2}= \frac{D(C_{o}-C_{i})}{X_{ox}}$$ $$...
0
votes
1answer
39 views

Parameterise linear combination of cosines

How do I parameterise the following implicit surface? $$ \cos x + \cos y + \cos z = 0 $$ Motivation for this problem comes from attempting to find stable motion for an object balanced on one point. ...
1
vote
2answers
50 views

Expanding a factorial

Can you explain me how we got this identity? $$\frac{1}{(3n)!}$$ the same as $$\frac{(3n)!}{(3n+3)!}$$ I have been trying to expand, but didn't get the same. Thanks.
0
votes
1answer
19 views

Determine all intervals of numbers $x$ satisfying the following inequalities.

i) $(x-5)^2 (x+10)\leq 0$ ii) $(x-5)^4 (x+10) \leq 0$ My answer : i) $(-10)\leq x \leq (5)$. ii) $(-10)\leq x \leq (5)$. Can you check my answer?
0
votes
0answers
18 views

Solving a cubic equation by factorization

I have the cubic equation $G^3 - (C - C_0)G - \frac{F}{\eta}$, with the constants $C, C_0, F, \eta \in \Re$. I want to find the solutions, so I tried factorising the equation, but I have after ...
0
votes
1answer
36 views

Volume of an igloo

AN igloo is a hemispherical structure built of blocks of ice by the Eskimos. THe interior diameter of the floor of an igloo is 6. Explain whether an Eskimo who is 1.7m tall could stand up straight ...
1
vote
4answers
61 views

If $9 ≥ 4x + 1$, which inequality represents the possible range of values of $12x + 3?$

If $9 ≥ 4x + 1$, which inequality represents the possible range of values of $12x + 3$? I've been trying to do SAT prep, and I came across this question. It allowed me to show an explanation and it ...
0
votes
0answers
28 views

How to return unique values by adding 5 variables

In the below image, I need to replace the Return values with the most appropriate sets of numbers so that when I add one Type, one Sub-Type, 0-2 User options, and the age together, I get the most ...
0
votes
1answer
27 views

Factorising an expression with two variables.

If the function $f(x) = 3x^2 + 5xy -2y^2 - 5x + 4y + p$ can be expressed as a product of two linear functions, find p. Hence express $f(x)$ as a product of two linear functions.
-1
votes
0answers
62 views

Solve $x*(x^x)=a$ for $x$

I tried with Lambert W-function but it seems it can not solve it. $x*x^x = a$ $a$ is constant find $x=f(a)$
2
votes
4answers
130 views

Common tangents to circle $x^2+y^2=\frac{1}{2}$ and parabola $y^2=4x$

I'm having trouble with this. What i do is say $\epsilon: y=mx+b$ is the tangent and it meets the circle at $M_1(x_1,y_1)$, i equate the $y$ of the tangent with the circle: $y=\pm \sqrt{1/2-x^2}$ and ...
0
votes
1answer
23 views

Exponential decay + a recurrence relation

I'm not sure if I get this right, some pointers could be helpful. Say you have to take 60m of some sort of medication at midnight. It has a blood half-life of 6 hours. Meaning that after 24 hours 3....
4
votes
2answers
88 views

How would you find the roots of $x^3-3x-1 = 0$

I'm not too sure how to tackle this problem. Supposedly, the roots of the equation are $2\cos\left(\frac {\pi}{9}\right),-2\cos\left(\frac {2\pi}{9}\right)$ and $-2\cos\left(\frac {4\pi}{9}\right)$ ...
0
votes
2answers
75 views

Solve for y in $2^{y + 3} = 5^{y}$

Solve for y in $2^{y + 3} = 5^{y}$. I know that \begin{align} \begin{split} 2^{y + 3} = 5^{y} \\ (y + 3) log 2 = y log 5 \\ \frac{y+3}{y} =\frac{log 5}{log2} \end{split} \end{align} and then I got ...
0
votes
5answers
335 views

Squaring both sides in an epsilon-delta proof

I am sorry if this is a silly question for all of you experienced mathematicians. I won't go through the whole problem since I am only concerned about one part. Why can't I do the following: $ |\sqrt ...
1
vote
7answers
80 views

Why is $\sin(\arccos(x))$ a semicircle with radius 1?

It was unexpected to see that from looking at the equation. Is there an intuitive explanation for why it's a perfect semicircle?
-5
votes
1answer
36 views

Graph the function [closed]

Graph the function $$ f(x)=\begin{cases} -x-2, & -2<x\le -1 \\ -x^2, & -1<x\le 1 \\ x+2, & 1<x\le 2 \end{cases} $$
0
votes
1answer
78 views

For what real values of $c$ does $x-\ln {(1+e^x)}=c$ hold for some $x$?

For what real values of $c$ does $x-\ln {(1+e^x)}=c$ hold for some $x$? Rewriting the equation as $\ln {e^x}-\ln {(1+e^x)}=c$, I get $e^c=\frac{e^x}{1+e^x}$. Not sure how to proceed from here.
5
votes
2answers
515 views

Re-write a quadratic equation in another form?

$x^2 + \sqrt{2}x = \frac{1}{2}$ I need to find the real solutions for this equation and write it in this form: $$\frac{-\sqrt{A} \pm B}{C}$$ So when I work the problem out with the quadratic ...
0
votes
1answer
32 views

Precalculus questions: Domain, range, and composition of functions

Directions: evaluate each of the functions at the indicated value of $x$. construct each of the functions, then find the domain. If $f(x)=\{(3,5),(2,4),(1,7)\}$, $g(x)=\sqrt{x-3}$, $h(x)=\{(3,2),(4,3)...
0
votes
2answers
49 views

Finding $\lim_{L \to \infty} \exp{\frac{T}{L}}\sum_{i=1}^L[ \exp{iA + (i-1)B}]$

I am working on a problem and I have come up with a formula that I would like to simply. WLOG, it looks like the following: $\exp{\frac{T}{L}}\sum_{i=1}^L[ \exp{iA + (i-1)B}]$ Here, $A,B, T$ are ...
0
votes
1answer
46 views

How to find $\tan{\theta}$ when $\theta=\arctan⁡{(8/3)}$

Basically I'm trying to find the exact value of $\tan{\theta}$ when $\theta = \arctan{(8/3)}$. I'm not exactly sure where to start. I know that $\arctan$ is the inverse of $\tan$, but I can't really ...
1
vote
1answer
24 views

Work and Time Problem, machines

Three accounting machines and 2 operators are finish in one day the work done by 10 clerks in two days. How many machines would be required to do in one day the work done by 40 clerks in one day? I ...
0
votes
0answers
33 views

Give rational with axioms and properties

A careful solution of $4(x+2)=11$ is given below. Give the rationale for each step from the ten real number rules $$\{\mathrm{AC},\mathrm{AA},\mathrm{A0},\mathrm{AI},\mathrm{MC},\mathrm{MA},\mathrm{...
1
vote
1answer
14 views

Optimal Way To Get Unique Results Given Two Options of Selecting

sorry for the vague title. I've got this problem I've been working on solving. It is as follows: Imagine there are a an infinite amount of gumballs with 12 different colors with an equal chance of ...
1
vote
2answers
55 views

Find all solutions of $\left[\ln(\sin^{-1}(e^x))\right]^5=\ln(\sin^{-1}(e^x))$

The question is: Find all solutions of $\left[\ln(\sin^{-1}(e^x))\right]^5=\ln(\sin^{-1}(e^x))$, where $x$ is real. Give the solutions in exact form. What I have done $$\left[\ln(\sin^{-...
0
votes
4answers
107 views

Sum of all real numbers in $]-60,60]$ [closed]

I can't solve this problem. Find the sum of all real numbers in $]-60,60]$, Is the answer zero or 60?? And of course why..
1
vote
0answers
53 views

How would you denest this radical using Ramanujan's Cubic Identity?

The identity states that given the cubic $y=x^3+ax^2+bx+c$, you have this equation: $\sqrt[3]{u+x_1}+\sqrt[3]{u+x_2}+\sqrt[3]{u+x_3}=\sqrt[3]{w+3\sqrt[6]{d}}$ where $$u=\frac {ab-9c+\sqrt{d}}{2(a^2-3b)...
2
votes
3answers
39 views

absolute value proof with properties and axioms

I'm in precalculus and totally new to this 'proofs with axioms' concept, and it's stumping me beyond belief. I get the concept, I just don't understand how to start or where to implement the proofs. ...
9
votes
1answer
120 views

Intuitive ways to get formula of binomial-like sum

Is there an intuitive way, though I am not sure how to find a conceptual proof either, to establish the following identity: $$\sum_{k=1}^{n} \binom{n}{k} k^{k-1} (n-k)^{n-k} = n^n$$ for all natural ...
2
votes
1answer
49 views

Algebraic problem solving technique

In a "seven-eleven" (7-11) store, a customer selected four items to buy. The check-out clerk says that he multiplied the costs of the items and obtained exactly $7.11$, the very name of the store! The ...
3
votes
4answers
206 views

Solution to this complex number equation

Solve $z^5 +32 =0$ My attempt : $$z^5 = -32$$ Multiply the powers on both sides by $\frac{1}{5}$ we get $$z = 2 * (-1)^\frac{1}{5}$$ Now I'm stuck at this step I don't know how to ...
3
votes
3answers
77 views

Prove $\sum_{i=1}^{n}\frac{a_{i}^{2}}{b_{i}} \geq \frac{(\sum_{i=1}^{n}a_i)^2}{\sum_{i=1}^{n}b_i}$

So I have the following problem, which I'm having trouble solving: Let $a_1$ , $a_2$ , ... , $a_n$ be real numbers. Let $b_1$ , $b_2$ , ... , $b_n$ be positive real numbers. Prove $$ \frac{a_{1}^{2}...