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
4answers
144 views

Series Question: $\sum_{n=1}^{\infty}\frac{n^2}{(4n^2-1)^3}$

How to compute the following series: $$\sum_{n=1}^{\infty}\frac{n^2}{(4n^2-1)^3}$$ I tried to use partial fraction ...
3
votes
7answers
687 views

How come the function and the inverse of the function are the same?

What is the inverse of the function: $$f(x)=\frac{x+2}{5x-1}$$ ? Answer: $$f^{-1}(x)=\frac{x+2}{5x-1}$$ Can one of you explain how the inverse is the same exact thing as the original equation?
0
votes
1answer
77 views

How to simplify $3^{(2\log_335)}$

$3^{2\log_35}$ How do I simplify this? This is what I have done so far: $2\log_35=\log_35^2=\log_3(25)$ $3^{\log_3(25)}$ What do I do from here? And the answer is one of these mixed solutions: ...
1
vote
2answers
74 views

How would you find $\tan(3\cdot\arctan(x))$?

How would you find $\tan(3\cdot\tan^{-1}(x))$ as quickly as possible? I don't really understand how to use $\tan(2x)$ or any other identity for that matter to solve this.
1
vote
3answers
232 views

Simplifying $(1/x-1/5 )/( 1/x^2-1/25)$

How do I get $\frac{5x}{x+5}$ from simplifying the following? $$\frac{(\frac{1}{x}-\frac{1}{5} )}{( \frac{1}{x^2}-\frac{1}{25})}$$ My work: I multiplied the top and bottom by the LCD: $25x^2$ (to ...
0
votes
1answer
207 views

Finding the general solution for complex trig equations.

$4\tan^2 x-3=0$ $ \tan^2 x =\frac34$ $x = \tan^{-1}(\pm \frac{\sqrt{3}}{2}$) therefore general solution $x= \tan^{-1}(\pm \frac{\sqrt{3}}{2}) + n\pi$ where $n$ is an element of all real ...
1
vote
1answer
183 views

How to find the values of constants when there is one stationary point, no stationary point, and determining the maximum number os stationary points.

b) values of x is when f'(x) = 0 c) how do i solve this without using common sense and knowing that if a=0 there will be no turning points/inflections d)how do i solve this? e) max number of ...
0
votes
2answers
36 views

Re-writing a a differential function

I don't understand the concept of this... how do I derive a an equation written in terms of a function? How do I differentiate f(function inside) ...?
2
votes
1answer
60 views

Why does $ \frac {a}{b}$ of $c$ mean $ \frac {a}{b} \cdot c$ [closed]

When someone writes "$ \frac {a}{b}$ of $c$", why is the preposition "of" interpreted as multiplication of $c$ by $a/b$?
1
vote
2answers
81 views

real solution of eqn. in $\sin x+2\sin 2x-\sin 3x = 3,$ where $x\in (0,\pi)$.

The no. of real solution of the equation $\sin x+2\sin 2x-\sin 3x = 3,$ where $x\in (0,\pi)$. $\bf{My\; Try::}$ Given $\left(\sin x-\sin 3x\right)+2\sin 2x = 3$ $\Rightarrow -2\cos 2x\cdot \sin ...
12
votes
6answers
21k views

Can the square root of a real number be negative?

Can the square root of a real number be negative? Dealing with the questions of functions in eleventh class my maths teacher says that square root of a real number is always positive. How is it ...
2
votes
2answers
70 views

Multiplying an infinite sequence

This is a "challenge" problem in the current text I'm studying. When simplified, the product, ...
1
vote
1answer
314 views

Find Coordinates on a track

Charlie and Alexandra are running around a circular track with radius 60 meters. Charlie started at the westernmost point of the track, and, at the same time, Alexandra started at the northernmost ...
0
votes
1answer
38 views

Expanding the sum of two natural valued terms.

I was looking at the IMO 2013 problems and I was trying to solve the first problem. Prove that for any pair of positive integers $k$ and $n$, there exist $k$ positive integers ...
0
votes
2answers
52 views

Finding coordinates on a circle

So this problem I am have difficulty with. I think where I am going wrong is how to calculate the initial theta. Do I just use pi/2 because in the pictures it show to angle theta off the 90 degree ...
2
votes
4answers
342 views

How to solve this weird inequality?

$\frac{x-1}{x+1} < x$ Thanks! I did the following. $\frac{x-1}{x+1} - x< 0 /-x$ $\frac{x-1 - x(x+1)}{x+1} < 0$ $\frac{-x^2-1}{x+1} < 0$ What to do next?
0
votes
1answer
203 views

Find the height of the dam given angles of a triangle

The top of a dam has an angle of elevation of 1.3 radians from a point on a river. Measuring the angle of elevation to the top of the dam from a point 155 feet farther downriver is 0.8 radians; assume ...
2
votes
2answers
102 views

Algebra Manipulation Contest Math Problem

The question was as follows: The equations $x^3+Ax+10=0$ and $x^3+Bx^2+50=0$ have two roots in common. Compute the product of these common roots. Because $x^3+Ax+10=0$ and $x^3+Bx^2+50=0$ it means ...
3
votes
3answers
404 views

$p(x)=0$ with real coefficient has purely Imaginary roots.Then the equation $p(p(x)) = 0$ has

If the Quadratic equation $p(x)=0$ with real coefficient has purely Imaginary roots.Then the equation $p(p(x)) = 0$ has $\bf{OPTIONS::}$ $(a)\;\; $ Only purely Inaginary Roots. $\;\;\;\;\;\;(b)$ ...
1
vote
1answer
38 views

Find values of the parameter a so that equation has equal roots.

$x^2+2a\sqrt{a^2-3}x+4=0$ My final result was 2 and -0.5. Was it correct?
0
votes
3answers
670 views

Which point on the line $ 2x-3y+3=0$ is nearest to the point $(-1,9)$

Which point on the line $2x-3y+3=0$ is nearest to the point $(-1,9)$. I've tried to use the distance formula... (2×(-1)-3×9+3)/√(4+9) = 26/√13 The right answer: (3,3) How can I get it?
0
votes
0answers
41 views

Silly fraction question

I've got $$ MPK'=\frac{1-d}{1+r+x} $$ But I need it in the form $$ MPK'=d+r+x $$ How do I bring those up top?
-1
votes
2answers
45 views

How to solve $\frac12 \sec^2 \frac x2 = 1$ under restricted domain?

solve: $$\frac12 \sec^2 \left(\frac x2\right) = 1$$ and domain $x: (-\pi,\pi) \cup (\pi,3\pi)$. sec^2 (x/2) = 2 sec^2 (x/2) can be re-written as tan(x/2)^2 + 1, therefore tan^2(x/2) + 1 = 2 ...
0
votes
1answer
58 views

Root of equation, solvability

I was trying to solve the following equation for t $$(P\cdot l \cdot \exp(-l\cdot t) + R \cdot l \cdot \exp(-l \cdot t))/t + (P \cdot \exp(-l \cdot t) + R \cdot (\exp(-l \cdot t) - 1))/t^2 = 0 $$ ...
0
votes
4answers
200 views

Multiple choice question on rates of change (or so I thought)

If I were to find the resistance of the component (see image below), I would either find the equation of the curve and use differentiation or I'd draw a tangent at $V_2$ and then find the reciprocal ...
1
vote
5answers
100 views

Proof of basic properties

Can anyone provide proof of properties such as: $$a(b+c) = ab+ac$$ $$(a+b)^2 = a^2+2ab+b^2$$ And exponent rules: $$a^n \cdot a^m = a^{n+m}$$ $$(a^n)^m = a^{n \cdot m}$$ For $a, b, c \in ...
1
vote
3answers
150 views

Explaining something to the half

I'm a private tutor in my free time, teaching some basic high school mathematics and I've often been asked: ''Why is something to the half equal to the root of that something?''. And I'm having ...
1
vote
3answers
162 views

Solution Set of $\sqrt{x+1}+\sqrt{x-1}=1$

If $x$ is real, then find the solution set of $\sqrt{x+1}+\sqrt{x-1}=1$.
0
votes
1answer
1k views

Convert standard form of hyperbola to function form

I understand the concept of converting an equation of a hyperbola from general form into standard form, however I need to do the opposite. The equation is the following: ...
3
votes
2answers
219 views

Proving Injectivity $x + \sin(x)$

I'm trying to prove injectivity of a particular function (without calculus), but I've come across a bit of a problem. The function is: $$f(x) = x+\sin(x)$$ I started by (abiding by common standards) ...
0
votes
2answers
143 views

Compound interest coumpounded n time per year formula. $A=P\left(1+\frac{r}{n}\right)^{nt}$ intuition behind it.

I know that the compound interest formula for the interest compounded annually is given by $$A=P(1+r)^t$$ I know the intuition behind it. But why the compound interest formula for the interest ...
0
votes
1answer
35 views

How to convert a value that belongs to a range to its equivalent in another range?

Let's say we have the range $[0, 1]$ and the value $0.7$ that belongs to that range, how can I convert that value to its equivalent in the range $[0.8, 1]$? (or any other arbitrary range) Could you ...
0
votes
2answers
46 views

How do you get from this to this formula?

I have the formula : $$3×4^{n-1}×1×\left({1\over 3}\right)^{n-1}$$ And I would like to know how to get to this one (which is equal) : $$3× \left({4\over 3}\right)^{n-1}$$ How can I do that ?
3
votes
1answer
79 views

Proving no polynomial $P(x)$ exists such that $P(a) = b$, $P(b) = c$, $P(c) = a$

If $P(x)$ is a polynomial with integer coefficients and $a, b ,c$ are three distinct integers, then show that it is impossible to have $P(a) = b$, $P(b) = c$, $P(c) = a$.
0
votes
2answers
37 views

Explicit functional relationship

I have an implicit relationship between dependent variable $y$ and independent variable $x \in \mathbb{R}$ which read as follows: $$ \frac{(y-1)^{\alpha + 1}}{y} = \exp{(\beta x)}$$ Here $\alpha, ...
0
votes
3answers
52 views

Finding $a,b$ with two equations

$$ 16a + 4b + 1=37 \\ 64a + 8b + 1=25 $$ I am having trouble find the values of $a$ and $b$ in the equation above.
1
vote
2answers
108 views

How do you simplify $ \frac{\tan \theta \cos \theta}{\sec \theta} $?

How do you solve/simplify this? I am having trouble solving for the correct answer. We did this in class and I am getting a different answer than what the teacher said it was. $$ \frac{\tan ...
0
votes
1answer
34 views

How to derive geometric mean

Suppose that $(1+r_g)^n = (1+r_1)(1+r_2)$, how should I derive the formula such that $r_g = (r_1r_2)^{1/n}$. I am trying to prove that $r_g = (r_1r_2)^{1/n}$.
1
vote
1answer
38 views

How is this step completed?

User Did, did this step in his answer to my previous question: $$\sum_{k=0}^n{n\choose k}(zp)^kq^{n-k}=(q+pz)^n.$$ How is it done? Is it simply an identity, or something more?
32
votes
13answers
10k views

Algebra: What allows us to do the same thing to both sides of an equation?

I understand that the expressions on both sides of an equal sign are the same entity, and I know that when you modify one side, the other must be changed because it is referring to the same thing. ...
2
votes
3answers
1k views

How can I find the following product? $ \tan 20^\circ \cdot \tan 40^\circ \cdot \tan 80^\circ.$

How can I find the following product using elementary trigonometry? $$ \tan 20^\circ \cdot \tan 40^\circ \cdot \tan 80^\circ.$$ I have tried using a substitution, but nothing has worked.
0
votes
2answers
38 views

Given that $\log_2(x)=p$ and $\log_4(y)=q$, how do I evaluate $\log_x(4y)$?

Given that $\log_2(x)=p$ and $\log_4(y)=q$, how do I evaluate $\log_x(4y)$? There were some other questions like this and I applied this formula to them $\log_a(xy) = \log_a(x)+\log_a(y)$. However, in ...
0
votes
0answers
28 views

How to limit the result of an operation to a specific range?

Let's say I have $a = 1 / b$, and the highest possible value for $b$ is $1$ and the lowest is $0.1$. Because of this, the result range of the operation is from $1$ to $10$. I want to turn that from ...
17
votes
3answers
459 views

$ \tan 1^\circ \cdot \tan 2^\circ \cdot \tan 3^\circ \cdots \tan 89^\circ$

How can I find the following product using elementary trigonometry? $$ \tan 1^\circ \cdot \tan 2^\circ \cdot \tan 3^\circ \cdots \tan 89^\circ.$$ I have tried using a substitution, but nothing ...
4
votes
4answers
157 views

why do equations work and how do they relate to each other?

Ok, so I understand that an equation is something like 15 = 15 , and that the only criteria as far as I can tell for it being an equation is that both sides are equal to each other. I have a few ...
0
votes
1answer
684 views

8 less than triple a number is equal to -5 (I'm trying to find the unknown number)

It's a hard question... for my thinking! I've tried many solutions but haven't figured it out. (Let $M$ be the unknown number) I've tried: $M^3-8=-5$ Then $M^3-8+8=5+8$ Then I think the answer ...
0
votes
2answers
125 views

no. of ways to arrange $8$ sailors from which $3$ on one side and $2$ on other side.

(1) Out of $8$ sailors on a boat , $3$ can work only on one side and $2$ only on other side. Then the number of ways the sailors can be arranged on a boat , is (2) Passengers are to travel by a ...
2
votes
0answers
78 views

“Taxes and Option Prices” (question about Derivatives Markets by McDonald)

Thanks in advance for any help, and please tell me if there's anything I can do to make things clearer. I am having trouble understanding appendix 10.A to Derivatives Markets by Robert L. McDonald. ...
1
vote
2answers
50 views

Remainder theorem question

If $n$ is an integer, what is the remainder when $$3x^{2n+3}-4x^{2n+2}+5x^{2n+1}-8$$ is divided by $x+1$? How would we know what the value of $n$ is?
2
votes
1answer
76 views

How do you solve for $x$ in this equation?

$$ \frac{100}{9} = \frac{1 - \frac{1}{(1+x)^{12}}}{x} $$ I tried and tried, but can't seem to get $x$ into a form to isolate it or use a quadratic formula or imaginary numbers or something. I need ...