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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8
votes
2answers
396 views

Equality of two iterated square roots

Solve for $x$: $\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+\dots}}}}=\sqrt{x\sqrt{x\sqrt{x\sqrt{x\dots}}}}$ My attempt: The L.H.S is equal to $\dfrac{1+\sqrt{4x+1}}{2}$ and R.H.S equals $x^2$ Equating both ...
2
votes
4answers
2k views

How do I show that f is strictly decreasing on (0, infinity)?

I have been asked to define $f: (0, \infty) \to (0, \infty)$ by $f(x) = \frac 1 x$ a) How do I show that f is strictly decreasing on $(0, \infty)$? I realize that I have to show that $f'(x)<0$, ...
1
vote
4answers
342 views

What does "sin$\theta > 0$ mean here?

The question is: If $\tan$ $\theta$ = -$\frac{8}{15}$, and $\sin$ $\theta$ > $0$, find $\cos$ $\theta$. What I did was draw a triangle on the unit circle with sides 8, 15 and therefore ...
1
vote
1answer
115 views

Real Analysis: proving basic properties from order properties of real number system

I have the order property: $a \leq b \wedge c \in \mathbb{R} \Rightarrow a + c\leq b +c $. Now, how do I prove that: $a\le b \Rightarrow a \leq b + c \ \ \forall \ \ c \geq 0$ Edit 1: If I had ...
0
votes
2answers
57 views

Exact value of a trigonometric ratio

I was asked to find the exact value of $\tan 240^\circ$. On my calculator, I type $\tan 240^\circ$, and then square the value to get a final answer of $\sqrt3$. However, the textbook answer says the ...
0
votes
2answers
51 views

How would I go about showing that f is one-to-one?

How would I go about showing that f is one-to-one? (Excuse the image link, I am not savvy enough to use mathematical symbols on here!) Define $f: \mathbb R \setminus\{-\frac{3}{2}\}\to ...
2
votes
1answer
60 views

Solving an exponential

I was practicing some exponential equations. I made up the following equation and cannot seem to solve it: $3^{2x}+5x=12$. I noticed most equations do not have the $x$ in the 5. I keep getting ...
0
votes
1answer
44 views

Find the time spend walking down a hill, given the time spent walking up the hill and the up/down walking patterns

Lance walks up a hill. For every 40 minutes of walk he takes 10 minutes to rest. When he walks down the hill, he instead takes 5 minutes of rest for every 40 minutes of walk. Lance walks downhill at a ...
1
vote
2answers
141 views

Simplifying difference trig expression

Rewrite the following expression as a simplified expression containing one term: $$\cos (\frac{\pi}{3}+\varphi) \cos (\frac{\pi}{3}-\varphi) - \sin (\frac{\pi}{3}+\varphi) \sin ...
3
votes
1answer
274 views

How to solve the trigonometric equation $\cos (\pi\theta/\beta) - \cos(2\pi\theta/\beta)=0$?

I have a question regarding a problem I've been attempting to solve. It is an acceleration equation: $$a = ...
4
votes
4answers
41 views

Prove that $ \frac {12^{x-2}.4^{x}} {6^{x-2}} = 2^{3x-2} $

Can someone please help me with this question? $ \large \frac {12^{x-2}.4^{x}} {6^{x-2}} = 2^{3x-2} $ My steps so far: $ \large \frac {4^{x-2}.3^{x-2}.4^{x}}{3^{x-2}.2^{x-2}} = 2^{3x-2} $ $ \large ...
-1
votes
2answers
34 views

Another simplification problem with algebraic indices

Could someone please help me with some steps for how to solve this question? $ \frac {3^{n} + 3^{n+2}} {3^{n-1} - 3^{n-2}}$ The answer is 45 apparently. I need to simplify to get to this answer. If ...
3
votes
1answer
382 views

Finding exact values of trig functions

Find exact value of each trigonometric function of $\theta$ if $\tan\theta=-1/5$ and $\sec \theta >0$ I know that $\cot \theta=-5,$ right? Secant and cosine are positive in the fourth ...
1
vote
3answers
37 views

Simplifying expressions with algebraic indices

I am having a ridiculous amount of trouble solving questions of this style - this one in particular, I know the answer is $6^{2n-4}$ but I can't get there!!!! $ \frac {36^{2n}.6^{n+2}} {216^{n+2}}$ ...
0
votes
4answers
68 views

Rewriting trigonometric expression in terms of $\cot x$

Rewrite the following expression in terms of $\cot x$: $$\frac{1}{1-\cos x}-\frac{\cos x}{1+\cos x}$$ I usually show my work on this site but I'm really lost about this problem. Any help ...
1
vote
4answers
149 views

Solutions for the given system with fractions

I have to solve in $\Bbb{R}$ the following system : $$ \ \left\{ \begin{array}{ll} \frac{y}{x}+\frac{x}{y}=\frac{17}{4} \\ x^2-y^2=25 \end{array} \right.$$ For this one I am ...
1
vote
2answers
1k views

Using sketch to find exact value of trigonometric expression

Use sketch to find exact value of $\tan (\cos^{-1}\dfrac{5}{13})$ I drew a right triangle with angle $\theta$ and sides $12,5,3.$ If $\cos \theta=\frac{5}{13},$ then $\sin \theta = \frac{12}{13}$ ...
2
votes
2answers
67 views

Solve the inequality $(1/2)^x-(1/2)^{-1-x}\ge1$ for real $x$

I have to solve in $\Bbb{R}$ the following inequality : $$ \left(\frac{1}{2}\right)^{x} - \left(\frac{1}{2}\right)^{-1 - x} \ge 1 \qquad(E) $$ So far I have : For $x=0$ this inequality if not ...
0
votes
1answer
142 views

Work = rate * time problems

Normally when we work with these problems, for example, Student A writes 2 essays each hour. Student B writes 3 essays each hour. If they worked together without interfering each other, how long ...
0
votes
1answer
107 views

If $x^2+1/x^2=83$, find the value of $x-1/x$

If $x^2+1/x^2=83$, find the value of $x-$$\frac1x$ I tried the following, As we know, $(a+b)^2=a^2-b^2+2ab$ Therefore, $(x^2+1/x^2)^2$$=(x^2)^2-(1/x^2)^2-2(x^2*1/x^2)$ ...
2
votes
2answers
55 views

Solving $\frac{\sqrt{108x^{10}}}{\sqrt{2x}}$

Simplify $$\frac{\sqrt{108x^{10}}}{\sqrt{2x}}$$ $\dfrac{\sqrt{108x^{10}}}{\sqrt{2x}}= \dfrac{(108x^{10})^{1/2}}{(2x)^{1/2}}$ The $1/2$ exponent cancels $\implies \dfrac{108x^{10}}{2x}$ ...
2
votes
1answer
41 views

Writing statement as algebraic expression

Write "a number decreased by the sum of the number and eight" as an algebraic expression. Let $x$ represent the number. Then simplify. algebraic expression: $x-(8+x)$ simplified: $-8$ Is ...
1
vote
2answers
512 views

A tricky running problem

I'm having trouble with the following problem: Tom starts running towards a park which is at $800$m from him at speed $20$ m/s. Kate who starts running with Tom at $25$ m/s goes back and ...
1
vote
2answers
48 views

If $4x^2+y^2=40$ and $xy=-6$, find the value of $2x+y$.

If $4x^2+y^2=40$ and $xy=-6$, find the value of $2x+y$. I tried the following, As we know, $(a+b)^2=a^2+b^2+2ab$ Therefore, $(4x^2+y^2)^2=(4x^2)^2+(y^2)+2(4x^2*y^2)$ $=16^4+y^4+8x^2y^2$ What ...
4
votes
1answer
49 views

Inequality in four variables which sum up to 4

The positive real numbers $x,y,z,t$ satisfy $x+y+z+t=4$. Is the inequality $$x\sqrt{y}+y\sqrt{z}+z\sqrt{t}+t\sqrt{x}\leq4$$ true for all $x,y,z,t>0$?
2
votes
1answer
151 views

Given $n$, find smallest number $m$ such that $m!$ ends with $n$ zeros

I got this question as a programming exercise. I first thought it was rather trivial, and that $m = 5n$ because the number of trailing zeroes are given by the number of factors of 5 in $m!$ (and ...
3
votes
2answers
119 views

Simplifying an expression with algebraic indices

I don't know why but I just can't simplify this expression and its driving me mad!! Could someone please help me with the steps? $\frac {2^n 9^{2n+1}}{6^{n-2}}$ I know the answer is 4 x $3^{3n + 4}$ ...
3
votes
3answers
370 views

Using the chain rule with a composite function

I'm a little confused on this homework problem and I could use some explanation if anyone has seen something like it before. The question is: Use the Chain Rule to find $\frac{dy}{dt}$ at $t = 9$ ...
3
votes
1answer
247 views

How is equation art created [duplicate]

A friend of mine sent me a Wofram Alpha link to a parametric curve that creates a detailed drawing of a game character. The parametric equation that draws it is about ~9 screens tall on my ...
2
votes
1answer
103 views

Injective map from real projective plane to $\Bbb{R}^4$

Consider the mapping $\Bbb R^3\rightarrow\Bbb R^4$ given by $$f(x,y,z)=(x^2-y^2,xy,xz,yz)$$ which passes to the quotient and can therefore be viewed as a map from the projective plane $\Bbb ...
0
votes
0answers
28 views

A system of absolute value equalities

Background: I'm trying to show that the transformation $T:\Bbb R^n\to\Bbb R^n$ defined by $T(x_1,\dots,x_n) := (|x_2-x_1|,|x_3-x_2|,\dots,|x_1-x_n|)$ is (or is not, this is out of curiosity only) ...
1
vote
2answers
230 views

A train traveled at a constant rate of $f$ feet per second. How many feet did it travel in $x$ minutes?

I know the correct answer is (D), however, I don't understand why (D) is correct. Can someone explain it to me; I struggle with math?
2
votes
3answers
122 views

The meeting of Cars

Three cars, A, B and C move towards north in a particular straight track (consider the length of the tract infinite). Another car D comes from a certain distance towards south. The car A meets B at 8 ...
1
vote
1answer
86 views

Expressing an formula in term of another one

I have this formula $$-\frac 1\lambda\left[\lambda D+1+W_{-1}\left(-r\exp(-\lambda D-1)\right)\right]$$ with $r$ , $\lambda$ and $D$ >0. Where $W$ is the Lambert W function ...
9
votes
1answer
234 views

finding the value of $f(2001) $ if…

if $f (\frac{x}{y}) =\frac{f(x)}{y} $ and $f(2000)=1$ ; then what's the value of $f(2001)$. I tried hard but can't figured out anything. please help me, how can I proceed?
2
votes
2answers
51 views

Maximum and minimum of $z=\frac{1+x-y}{\sqrt{1+x^2+y^2}}$

Find the maximum and minimum of the function $$z=\frac{1+x-y}{\sqrt{1+x^2+y^2}}$$ I have calculated $\frac{\partial z}{\partial x}=\frac{1+y^2+xy-x}{(1+x^2+y^2)^{\frac{3}{2}}}$ $\frac{\partial ...
-1
votes
3answers
113 views

Transforming a polar equation into a Cartesian equation?

Transform the polar equation to a Cartesian (rectangular) equation: $$r= \frac5{5cosθ + 6sinθ}$$ These equations really stump me, so if you could be more "heavy-handed" with the explanation, I'd ...
4
votes
2answers
245 views

Solution for this Logarithmic Equation

Recently I was going through a problem from the book Problems in Mathematics - *V Govorov & P Dybov* . $$(x-2)^{\log^2(x-2)+\log(x-2)^5-12}=10^2\log(x-2)$$ I tried solving by first considering ...
0
votes
2answers
729 views

Optimization problems: Finding the optimal path

I'm still trying to get the hang of optimization problems in calculus and I'm looking for a little help. I'm having trouble finding equations to model the following problem: I'm fairly sure I need to ...
2
votes
1answer
366 views

How to find the short run and long run cost functions, given the production function?

The production function of car is given by $f(x_1,x_2,x_3) = \sqrt{x_1}+\sqrt{x_2}+\sqrt{x_3}$ (assume competitive input and output markets). Find the short run cost function (let input 3 ...
2
votes
1answer
85 views

Help needed in verifying a trigonometric identity

I have the following identity: $$32\sin^{2}\left(\theta\right)\cos^{4}\left(\theta\right) =2 + \cos\left(2\theta\right) - 2\cos\left(4\theta\right) -\cos\left(6\theta\right) $$ I've tried ...
0
votes
1answer
69 views

The proportionality symbol

Given a question, Write down an equation that embodies the following relationship $$ a \varpropto b $$ Does the 'open' end of the $\varpropto$ symbol indicate that that's the side of the equation ...
3
votes
2answers
62 views

For $5$ distinct integers $a_i$, $1\le i\le5$, $f(a_i)=2$. Find an integer b (if it exists) such that f(b) = $9$.

Here's an interesting question I came across.The person who gave it to me told me that it should not take more than $3$ minutes to solve this question. But I could not find any definite solution :( ...
3
votes
2answers
118 views

What are some good questions for this trick, if $\frac{a}{b}=\frac{c}{d}=\frac{e}{f}=\dots=\alpha$ then $\alpha=\frac{a+c+e+…}{b+d+f+…}$?

I need some good algebra questions that are applications of this trick, often in a non obvious and elegant way: $$\text{If } \frac{a}{b}=\frac{c}{d}=\frac{e}{f}=\dots=\alpha \text{ then } ...
0
votes
2answers
134 views

how to solve the variable word equation

After covering a distance of 30Km with a uniform speed, there got some defect in train engine and therefore its speed is reduced to 4/5 of its original speed. Consequently, the train reaches ...
2
votes
1answer
54 views

Clarification regarding the Josephus problem in Concrete Mathematics (Knuth, et al)

In page 9 of Concrete Mathematics, regarding the Josephus Problem, they state that "each person's number has been doubled then decreased by 1". $J(2n) = 2J(n) - 1$, for $n \ge 1$ I don't quite ...
1
vote
1answer
56 views

Abstract Definition of $\log$

Suppose $l:(0,\infty)\to\mathbb{R}$ is continuous and satisfies (1) $l(e)=1$ and (2) $l(xy)=l(x) + l(y)$ for all $x,y\in \mathbb{R}$. I'd like to show $l=\log$ (the natural logarithm). I can prove ...
-1
votes
1answer
48 views

Another Motion Problem: Algebra

It takes a ship $6$ hours to travel downstream between two piers, and $8$ hours upstream. If the water flows at a speed of $2.5$ mph, at what speed will the ship travel in still water? It's a bit ...
0
votes
2answers
58 views

Motion Problem - Algebra

An ant crawls along the sides of a 45-45-90 triangle. It starts at the right angle and crawls at a rate per minute of 50cm, 20cm, and 40cm, seperately, on each of the three sides, at what average ...
2
votes
2answers
742 views

How long will it take to fill a water tank with two inlet pipes and one outlet?

E11/40. I can see here that I work the fill rate out as: $\large\frac13 + \frac14 - \frac18 =$ Overall fill rate of $\large\frac{11}{24}$ tank per hour. If I multiply $60$ minutes by ...