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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1
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2answers
52 views

Neither $\log x$ nor $\exp(x)$ are rational functions [on hold]

(a) Prove that $\log x$ cannot be expressed in the form $f(x)/g(x)$ where $f(x)$ and $g(x)$ are polynomials with real coefficients. (b) Prove that $e^x$ cannot be expressed in the form $f(x)/g(x)$ ...
0
votes
1answer
38 views

Calculating Average Rate of Return

I am having a hard time on my homework with understanding how to calculate the average rate of return for an investment. After calculating the average rate of return is explained to me, I want to ...
3
votes
3answers
198 views

what is the sum of this?$\frac12+ \frac13+\frac14+\frac15+\frac16 +\dots\frac{1}{2012}+\frac{1}{2013} $

What is the sum of $$\frac12+ \frac13+\frac14+\frac15+\frac16 +\dots\frac{1}{2012}+\frac{1}{2013} $$
0
votes
0answers
22 views

$\sum$ of binomial coefficients inequality

Let $m,n$ be positive integers with $m>n$. When is it true that $$m\cdot 5^{m-1}\cdot 3+\binom{m}{3}\cdot 5^{m-3}\cdot 3^3\cdot 2+\cdots +\binom{m}{2k+1}\cdot m^{m-2k-1}\cdot 3^{2k+1}\cdot ...
0
votes
1answer
47 views

Mid level algebra headscratcher. [on hold]

I have 4 numbers. Base = 100 , Start = 0.4 , Count = 4 , Multiplier = 0.64448 I do 100 × 0.4 = 40 Then 4 times: 40 × 0.64448 = 25.7788368 25.7788368 × 0.64448 = 16.61371067 16.61371067 × ...
0
votes
1answer
18 views

characteristic equation of transfer function

$$\frac{K}{s(s+1)(s+5)}$$ Find the characteristic equation of this transfer function. The book gives this answer: $$\frac{K}{s(s+1)(s+5)} +1=0$$ or $$s^3 +6s^2 +5s +K =0.$$ I don't get how the ...
2
votes
1answer
223 views

Reducing a fraction by dividing top and bottom

I am trying to cancel out to reduce this: $$\frac{ 6xh + 3h^2 + 5h }{ h }$$ Is it possible to cancel out the h's to become this; $$\frac{ 6x + 3 + 5 }{ h }$$ While the $3$ goes into the $6x$? So ...
6
votes
2answers
513 views

Does anyone know of any good ways to get good at algebra without as much“grind” as doing hundreds of questions a night

I'm coming to a point in college where I can't avoid my math classes any longer. I need to get better at algebra so I don't flunk out of the class when I take it, however I've never been able to get ...
-1
votes
1answer
45 views

Finding two sided bounds on $(x+y)/(xy)$ given inequalities for $x$ and $y$

Given $\dfrac{1}{6} < x < \dfrac{1}{2}$ and $\dfrac{1}{7} < y < \dfrac{1}{3}$, can we determine bounds for $\dfrac{x+y}{xy}$?
0
votes
1answer
34 views

Solve $\frac{(x - 1)^3(x + 1)^8}{(x + 2)^4} > 0$

Solve the inequality $$\frac{(x - 1)^3(x + 1)^8}{(x + 2)^4} > 0$$ A) $X<1$ B) $X>1$ C) $X>-1$ D) $X<-1$ E) $X>-2$
0
votes
3answers
37 views

Finding the tangent line to the graph of $f(x)=(x+2)^{3/5}$ at $x=-2$

Does the graph of the function $f$ have tangent line at the given points? If yes, what is the tangent line? $f(x)=(x+2)^{3/5}$ at $x=-2$ solution: yes, $x=-2$ The derivative I found: ...
1
vote
2answers
32 views

Finding (or rather expanding) the product $(5-xy)(3+xy)$

Given the product $(5-xy)(3+xy)$ I tried the following, As we know, $(x+a)(x+b)=x^2+(a+b)x+ab$ Here $x$ is $xy$. But $xy$ has two signs$-$ and $+$. How do I solve this.
-7
votes
1answer
135 views

Please Solve the following Problem and choose the correct answer. [on hold]

If $ a^2 + b^2 +c^2 +3=2(a-b-c) $, then the value of $ 2a-b+c $ is: (A) $4$ (B) $0$ (C) $2$ (D) $3$
17
votes
8answers
3k views

Negative Exponents, Why? [closed]

Why is it that when a negative exponent is present, the root and the exponent are both automatically placed under "1" and the exponent is made positive? Why does that make sense?
20
votes
1answer
253 views

Show that $x^{35}+\dfrac{20205}{2+x^{17}+\cos^2x}=100$ has no root $x\in \mathbb{R}$

Show that $x^{35}+\dfrac{20205}{2+x^{17}+\cos^2x}=100$ has no root $x\in \mathbb{R}$. By plotting graph I have seen that there are no roots for $x$. Can somebody prove it theoretically?
0
votes
1answer
35 views

Find the relation of quantity and price, assuming it is linear

How would I find the numbers in between if: 1 item = $75 40 items = $600 For example: What would 2 items be? 14? 26? 37? etc... What would the formula be that is used to determine the answer?
30
votes
7answers
1k views

Pi Estimation using Integers

I ran across this problem in a high school math competition: "You must use the integers $1$ to $9$ and only addition, subtraction, multiplication, division, and exponentiation to approximate the ...
0
votes
1answer
41 views

Solving a linear equation from a mixture word problem

Ok so I already know how to throw a word problem into a mixture solution table. In this situation I am going to go ahead and throw the equation at you guys and see if you can help me figure out this ...
-2
votes
5answers
117 views

What is $x^{1/2} + x^{-1/2}$

Would these two terms cancel out and be zero or can they not be put together since technically the exponent is different? EDIT: I am working on $x^{1/2} + x^{-1/2}$ not $x^{1/2} = x^{-1/2}$
1
vote
2answers
56 views

Paper square diagonal

A paper square has diagonal length 6 in. two folders are made along lines perpendicular to the diagonal and through trisection points A and B. By what percentage has the visible area of the paper ...
1
vote
3answers
334 views

The number of real roots of $2x^5+8x-7$

Is there a (theoretical) method to tell the number of real roots of $$2x^5+8x-7=0?$$ Note, without using calculator.
2
votes
3answers
60 views

Solving equation involving logarithm

How do I find $x$ in the equation $x - \ln x = 1.9$? Next I have $x - 1.9 = \ln x$ we are learning about logarithm but as I tried to take $\ln$ of both side it leads me to nothing. Could you help ...
2
votes
2answers
52 views

Expand $(x-7)^2 + x^2 = (x+2)^2$ algebraically

$(x-7)^2 + x^2 = (x+2)^2$ $(x-7)(x-7) + x^2 = (x+2)(x+2)$ $x^2 -7x -7x + 49 + x^2 = x^2 + 4x + 4$ $x^2 + 18x - 45 = x^2 + x^2$ From that point on, everything I do is incorrect. I don't know what ...
0
votes
2answers
35 views

The number of stamps in ziko's collection is 75% of the number of Devid 's collection

The number of stamps in Ziko's collection is 75% of the number of Devid 's collection.If Devid has 80 more stamps than Ziko ,how many stamps do they have altogether.
0
votes
1answer
34 views

How do I form these quadratic equations

A car travels a distance of $1200 \ km$ at a speed of $a \frac{km}{h}$, while a train travels the same distance at $(a – 20) \frac{km}{h}$. If the time taken by the train is $5$ hours more than the ...
2
votes
1answer
29 views

Finding the range of $f(x) = 5-x$

Let $a < b$ and $f: (a, b] \to R$, $f(x) = 5 - x$. What is the range of the function? How do you find the range of an equation with unknowns?
0
votes
2answers
98 views

Algebra clock problem

An absent-minded watch repairman connected the hour hand to the minute hand pinion and the minute hand to the hour hand pinion and set the clock at 6AM which was the correct time then. How soon after ...
0
votes
1answer
8 views

Left & Right Area Approximation Using Y-Axis - Method Alternatives

Is there a simpler way of solving this then calculating x1(h)+x2(h)+x3(h)+x4(h) by using the given y values (in this case h, the height is one, because the length of each rectangle is one) ...
3
votes
1answer
105 views

Is $(\sqrt{-x})^2$ equal to the $\sqrt{(-x)^2}$?

I got these graphs from Wolfram Alpha. Can anyone please explain why the result differs by the order of operations for square roots? I thought the rule only applied to division, multiplication and ...
-3
votes
1answer
21 views

Fourth roots of a certain complex number [on hold]

Find the fourth roots of $81(\cos 320^\circ + i\sin 320^\circ )$. Write the answer in trigonometric form. \begin{array} \text{a.} & 3(\cos 160^\circ + i \sin 160^\circ ); & &3(\cos ...
2
votes
1answer
152 views

The roots of $x^3+4x-1=0$ are $a$, $b$, $c$. Find $(a+1)^{-3}+(b+1)^{-3}+(c+1)^{-3}$

This is a question in A level Further Pure mathematics pastpaper Nov 2010. The roots of the equation $x^3+4x-1=0$ are $a$, $b$ and $c$. i) Use the substitution $y=1/(1+x)$ to show that the equation ...
-1
votes
1answer
48 views

Word problem: father is aged three times more than his son…

Father is aged three times more than his son. After 8 years, father would be two and half times of his son's age. After further 8 years, How many times would father be of son's age? Thanks a lot in ...
0
votes
2answers
17 views

Average Value - Graphs

long method: Determine an equation for each and solve using average value formula alternative methods? How could you prove the average value to be C over an interval [a,b] if you are given a ...
0
votes
1answer
8 views

Related Rates of Change - Cylinder Question

A cylindrical tank with radius 5 cm is being filled with water at rate of 3 cm^3 per min. how fast is the height of the water increasing? I dont want this question solved, but please help me correct ...
4
votes
4answers
58 views

How does $x^3 - \sin^3 x$ become $x^3 + \frac{1}{4}\sin{3x}-\frac{3}{4}\sin x$?

I was going through answers on this question and came across this answer and I was wondering how the user arrived at the first line where they state: $$f(x) \equiv x^3 - \sin^3 x = x^3 + {1 \over 4} ...
0
votes
3answers
54 views

Can anyone help me understand the simplification of $\frac{\sqrt 3 + \sqrt 2}{\sqrt 3 - \sqrt 2}\;$?

Can anyone help me understand the following simplification of the fraction? $$\dfrac{\sqrt 3 + \sqrt 2}{\sqrt 3 - \sqrt 2} = 5 + 2\sqrt 6$$ I cant understand how to simplify the left-hand side to get ...
2
votes
2answers
117 views

solving the inequality

I'm looking for hints on how to efficiently solve this inequality: $$\left( \frac {|x|-|1-x|}{|x|} \right)^{2x-1} \gt \left(\frac {|x|-|1-x|}{|x|} \right)^{8-x} $$
3
votes
2answers
69 views

Equality of real numbers whose squares sum to 1

I am helping a friend with homework and got stuck on the following problem. It doesn't seem like it should be that hard, I don't know why I'm having a hard time: For $a,b\geq0$, if $a^2+b^2=1$, then ...
3
votes
2answers
43 views

How do I find the inverse of a $\cos^2 \theta$?

This was originally a physics question, but the math is what is throwing my brain into loops. Basically, I need to find $\theta$: $$ \frac{7}{8}= \cos^2(\theta) $$
4
votes
4answers
141 views

Solving $a(x+2)=\pi-cy$ for $x$, arrived at an answer different from the one in the book

In an algebra review book, one exercise asked to solve for $x$: $$a(x+2)=\pi-cy$$ I arrived at the following: $$x=\frac{\pi-cy}{a}-2$$ The book stated the correct answer is: $$x=\frac{\pi-cy-2a}{a}$$ ...
3
votes
4answers
120 views

What is the the $n$ times iteration of $f(x)=\frac{x}{\sqrt{1+x^2}}$? [closed]

We were asked to determine the composition $f \circ f \circ f \circ...\circ f $, $n$ times, where $f(x)=\dfrac{x}{\sqrt{1+x^2}}$. Anyone has an idea?
4
votes
2answers
119 views

Which of these numbers is greater: $\sqrt[5]{5}$ or $\sqrt[4]{4}$?

I know that this is the question of elementary mathematics but how to logically check which of these numbers is greater: $\sqrt[5]{5}$ or $\sqrt[4]{4}$? It seems to me that since number $5$ is ...
4
votes
3answers
77 views

Solving $7^{2x}\cdot4^{x-2}=11^x$

I am trying to solve this equation but I am stuck little bit right now. This is how I did it: $$ 7^{2x}\cdot4^{x-2}=11^x\\ \text{log both sides}\\ \log(7^{2x}\cdot4^{x-2})=\log(11^x)\\ \log(7^{2x}) + ...
5
votes
2answers
253 views

Big-O notation, prove the following: $\sum\limits_{k=3}^n(k^2 - 2k)$ is $O(n^3)$.

Use the definition of Big-O notation to prove the following: $\sum\limits_{k=3}^n(k^2 - 2k)$ is $O(n^3)$. Can someone please give me some hints on how to expand $\sum\limits_{k=3}^n(k^2 - 2k)$?
6
votes
1answer
64 views

Substitution problem

My question is something I've been thinking about for some time now. Q: Why is it possible to make substitutions or change in variables ? I mean, how do I know which substitutions are allowed ? For ...
0
votes
1answer
18 views

Find equation of the straight line tangent to the curve at the point indicated

Find equation of the straight line tangent to the curve at the point indicated: $y=2x^2 -5$ at $(2,3)$ I think I have to use $y=m(x-x_o)+y_0$ etc but I'm not sure how to find the $m$! Thanks for ...
5
votes
7answers
684 views

When the numerator of a fraction is increased by 4, the fraction increases by 2/3…

When the numerator of a fraction is increased by $4$, the fraction increases by $2/3$. What is the denominator of the fraction? I tried, Let the numerator of the fraction be $x$ and the denominator ...
0
votes
1answer
31 views

Finding the exponent of $2$ such that $x \cdot 2^a$ is as close to $1$ as possible

How do I find an exponent of $2$ that when multiplied with another number would bring the result closest to the positive side $1$? Like this: $y = x \cdot 2^a$, where $y\ge 1$ has to be as small as ...
2
votes
3answers
35 views

Finding the perimeter of the room

If the length and breadth of a room are increased by $1$ $m$, the area is increased by $21$ $m^2$. If the length is increased by $1$ $m$ and breadth is decreased by $1$ $m$ the area is decreased by ...
0
votes
3answers
19 views

Finding the angles of a parallelogram.

In a parallelogram, one angle is $2/5th$ of the adjacent angles. Determine the angles of the parallelogram. I tried the following, Let the adjacent angles be $2x$ Let the other angle be $y$ ...