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
votes
2answers
129 views

Is $x/x$ continuous at $0$? [closed]

Just wondering, while studying limit, if $x\over x$ is continuous at $0$. $f(0)={0 \over 0}$ ,, but $x/x=1$. In this case, is it continuous at $0$?
1
vote
2answers
43 views

Combinatorics using a geometric diagram

How can I do this without trial-and-error? It has something to do with a triangle and summing the next row?
2
votes
1answer
43 views

Polynomial prove exercise

$P(x)=x^n + a_1x^{n-1} +\dots+a_{n-1}x + 1$ with non-negative coefficients has $n$ real roots. Prove that $P(2)\ge 3n$ I don't have an idea how to do that, I'm in 4th grade high school, you don't have ...
0
votes
2answers
36 views

Sum of the coefficients of the expansion

Find the sum of the coefficients of the expansion: $$\frac{(1+x)\cdot(2+x^2)\cdot(3+x^3)...(103 + x^{103})}{103!}$$ The answer says let $x=1$, is this the way to go? Why not let $x=0$ ??
1
vote
5answers
113 views

$x=yx$. Can this statement be true when we don't know that $y=1$?

I am dealing with an equation which is saying that $yx=x$. On the other hand it is telling us that $\frac{x}{x}=1$ which connotes that $x=x$. Is it not absurd to say that $x=x=yx$ when we don't know ...
7
votes
2answers
160 views

How to find the inverse arc in the configuration space

The following Figure shows the function from configuration space (Torus) to operational space (Annulus). There is a naturally defined continuous function from configuration space $(\theta_A, ...
1
vote
3answers
34 views

Prove that $a^2+b^2+c^2+d^2+e^2 > a(b+c+d+e)$

Prove that $a^2+b^2+c^2+d^2+e^2 > a(b+c+d+e)$ Seems to be easy but, cannot see the method right now. Tried adding known things like $a^2+b^2>=2ab$ and so on with other letters.Maybe I didn't ...
1
vote
7answers
58 views

Logarithms with an answer that is a fraction

How does log base $16$ of $32$ equal $1.25$? If we divide $32/16=2$ but then if we divide $2/16$ it doesn't come out to a whole number unlike with log base $2$ of $4$ where $4/2=2$ and $2/2=1$ I am ...
1
vote
1answer
25 views

Application of Dimensional Analysis Problem

It is given that the radius $R$, in meters, of the expansion of a liquid in the soil is given by $t$ (time elapsed since the liquid was released), the mass $M$ of the liquid released and of the ...
-1
votes
3answers
38 views

simplify the equation [closed]

I need help simplifying this equation. It is a fraction just in case the way I formatted it doesn't turn out right. $$ \frac{(4x + 3)^{1/2} − (x + 6)(4x + 3)^{−1/2}}{(4x+3)} $$
1
vote
1answer
43 views

Expected value of prime lottery ticket

Below is a problem I think that I have solved correctly, but cannot seem to get the correct answer. Any help would be greatly appreciated. You pay $\$13.00$ for a ticket. When you buy a ticket, ...
2
votes
2answers
33 views

How to go about solving this inequality question?

$\cos(3x-\pi/3) \leq (1/2).$ Here is what I have done so far... Let $3x-\pi/3 = X$. So I need to solve $\cos(X) \leq 1/2$. Which is all $X$ from $\pi/3$ to $5\pi/3$, so-- $\pi/3 \leq X \leq 5\pi/3 ...
5
votes
4answers
221 views

Why is $\frac{\sqrt{x+1}-1}{x}$ equal to $\frac{1}{\sqrt{x+1}+1}$?

I'm working with the expression $$\frac{\sqrt{x+1} - 1}{x}.$$ According to Wolfram Alpha "alternate form" section (http://www.wolframalpha.com/input/?i=%28%28x%2B1%29%5E1%2F2-1%29%2Fx) it is equal to ...
0
votes
2answers
40 views

To find inverse of function [closed]

Given $ f(x) = \begin{cases} 2x, & \text{if $x\in[0,1]$} \\ 8 - 2x, & \text{if $x\in [2,3)$} \end{cases} $ Then how to find inverse of f ?
3
votes
1answer
31 views

If $ j , k , n$ are consecutive integers and $jn$ has last digit $9$, what is the last digit of $k$?

$ j , k , n$ are consecutive integers such that $0 < j < k < n$ and the units (ones) digit of the product $jn$ is $9$, what is the units digit of $k$? SAT Question. I don't know if we are to ...
2
votes
0answers
35 views

Periodicity of a sum of periodic functions?

The sum of two periodic functions is periodic if: a) Both periodic functions are continuous b) If the ratio of their fundamental periods is rational Can someone explain why the first ...
1
vote
3answers
38 views

Translate a point on a circumference

If I have a point $A$ on the circumference of a circle with origin $O$ and radius $r$, how would I find the coordinates of point $B$, which is also on that circumference, but is $d$ units away from ...
1
vote
2answers
49 views

solving nonlinear equations

Suppose I have the following two nonlinear (degree two) equations: $y = x^2$ $y = 8 – x^2$ By solving these two equations, the possible values for $x$ and $y$ are: $x = –2, +2$ and $y=4$. Note ...
4
votes
8answers
102 views

factor the following expression $25x^2 +5xy -6y^2$

How to factor $$25x^2 +5xy -6y^2$$ I tried with $5x(5x+y)-6y^2$. I'm stuck here. I can't continue.
0
votes
2answers
25 views

Is this a solution to the equation $a|bx|+c=0$?

I was working on solving a problem in math class, and I was given this problem, $a|bx|+c=0$, as a challenge to solve. This is what I came up with. $$ a|bx|+c=0 \\ a|bx|=-c \\ |bx|=\frac{-c}{a} \\ ...
0
votes
0answers
18 views

Formula for roots of a polynomial, and nature of roots in detail, depending on the discriminant

I am searching for some authentic formula for finding roots of a cubic polynomial, if someone could provide me? I have to solve $$-a r^3 + r^2 - 2 m r + Q^2 = 0$$ for $r$. I am also interested in ...
0
votes
2answers
42 views

Simplifying the exponential expression $e^{-4\ln x +8\ln y +2}$ [closed]

I'm totally stuck on this. Tried numerous sites for a decent explanation but can't find anything. Simplify the expression $$e^{-4\ln x +8\ln y +2}.$$ Thanks in advance.
1
vote
2answers
16 views

Order of Inverse Operations

so this is a very simple question but I am having a tough time with it. So it's finals week and I'm studying up for an Algebra 2 final. The only part I am having trouble with is finding the inverse ...
0
votes
2answers
34 views

Solve for $m$ in $d^m = n$ [duplicate]

I believe the answer is $m = \lceil \sqrt[d]n \rceil$ or $\lfloor \sqrt[d]n \rfloor$. Can anyone help me?
0
votes
2answers
34 views

Finding $|a|$, a complex number, given a system of equations

$a$ and $b$ are complex numbers where $|2a - b| = 25$, $|a + 2b| = 5$, and $|a + b| = 2$. Using the information, find $|a|$. I tried using the magnitude formula (i.e. where $|a| = \sqrt{x^2+y^2}$), ...
0
votes
0answers
17 views

Combining fractions with powers of logarithm

How does this work? $${2^{x}x^2\over \ln(2)} + {2^{x+1}\over \ln^3{2}}- {2^{x+1}x\over (\ln(2)^2 }= {2^x(x^2\ln^2(2)-2x\ln(2)+2)\over \ln^3(2)} $$
-2
votes
0answers
32 views

Maxima and minima of 2 variable function with conditions

Let $a=2001$. Consider the set $A$ of all pairs of integers $(m,n)$ with $n\not=0$ such that $m<2a$ $2n\mid 2am-m^2+n^2$ $n^2-m^2+2mn\le 2a(n-m)$ For $(m,n)\in A$, let $$f(m,n)=\frac ...
2
votes
2answers
72 views

Is this a valid log operation?

I saw this and this 1st step looked fishy.... Bringing the 4 inside the ()'s....Is that valid? That last step also looks weird. the square just goes away? ...
0
votes
2answers
11 views

Finding the set of points of a polar coordinate

$\left\{ (r,\theta) : 2\le r\le 6,\frac{\pi}{3}\le\theta\le\frac{5\pi}{6}\right\}$, where $S$ stands for the set of points. What is the area of $S$? This is a bit confusing to me. How do I start ...
2
votes
2answers
24 views

Converting (7,5) Cartesian coordinates to polar coordinates

Find the point (r, $\theta$) in polar coordinates given the fact that when converted in Cartesian coordinates, the point is $(7,5)$. Use that to find the point $\left( 2r, \theta + \frac{\pi}{2} ...
1
vote
2answers
22 views

How to find the x intercepts

$\frac{4}{3} e^{3x} + 2 e^{2x} - 8 e^x$ I have some confusion especially because of the e how can I approach the solution? The solution of the x-intercept is 0.838 Many thanks
4
votes
3answers
101 views

Prove that $1\cdot f(1)+ 2\cdot f(2)+ …+ n\cdot f(n) \leq n(n+1)(2n+1)/6$ where $f(n+1) = f(f(n))+1$

Consider checking function $\mathbb{N}\to \mathbb{N}$ relationship $f(n+1) = f(f(n))+1$, for any positive integer $n$. Prove that $1\cdot f(1)+ 2\cdot f(2)+ ...+ n\cdot f(n) \leq n(n+1)(2n+1)/6$ for ...
0
votes
1answer
23 views

Find constants in expression of the form $y = ax^b$

So I have a real system that for a given setting, x, returns a value, y. These values appear to follow (with some limits) the form of $y = ax^{-b}$ - could also be expressed as $y = \frac{a}{x^b} $. ...
9
votes
9answers
1k views

examples of functions with vertical asymptotes in real life

As a math teacher, I tend to get the class involved by finding real-life applications of the math- with functions and vertical asymptotes I am having trouble finding simple enough (rational) functions ...
1
vote
1answer
20 views

Bernoulli's inequality variation

To prove: $(1+a_1)(1+a_2)\ldots(1+a_n)\geq\dfrac{2^n}{n+1}(1+a_1+a_2+\ldots+a_n)$ when $a_i\geq1$ This seems to be based on Bernoulli's Inequality (which can be proved by induction). Trying the ...
1
vote
1answer
33 views

Solving the equations $x_1= 4 x_2$ and $x_3= 5 x_2$, with the sum of all three being $150$

Here is the problem. A set X is partitioned into subsets x1, x2, and x3. The number of elements in x1 is 4 times the number in x2. And the number in x3 is 5 times the number in x2. If n(x)=150, ...
0
votes
2answers
33 views

Suppose that $a$ and $b$ are nonzero real numbers. Prove that if $a<\frac1a<b<\frac1b$ then $a<-1$

Suppose that $a$ and $b$ are nonzero real numbers. Prove that if $a<\frac1a<b<\frac1b$ then $a<-1$ I'm stuck on this one. Where does one begin?
3
votes
1answer
26 views

Is this a correct way to prove this?

I've just looked at this question and sketched a way to do it my head. When I looked at the answer it looked slightly more complicated than the way I did it so I just wanted to check whether this is a ...
0
votes
1answer
42 views

Square Roots with Exponents

I learned about Square roots and with exponents, but not this: The radius $r$ in millimeters of a platinum wire $L$ centimeters long with resistance $0.1$ ohm is $r = 0.059L^\frac 12$. How long is a ...
1
vote
3answers
27 views

One more formulae manipulation question

Just making sure I am right... Make c the subject of the formula: $ y = \frac{2a+b}{3c -d}$ so $ 3c -d = \frac{2a+b}{y}$ so $ 3c = \frac{2a+b}{y} +d$ so $ c =\frac{6a+3b}{y} + \frac{d}{3}$ If I ...
1
vote
0answers
27 views

On the matrix representation of a composition of Mobius transforms

Let the Mobius transform associated to the matrix $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$ be defined as $\mu_A:\mathbb C\to\mathbb C:z\mapsto\frac{az+b}{cz+d}$ provided $\det A\neq 0$. It is ...
1
vote
3answers
42 views

Solve for $y$ in $x=\sqrt{(y-1)/(y+1)}$

I always struggle with this: Express $y$ in terms of $x$ where $$x = \sqrt\frac{y-1}{y+1}$$ I know to square both sides and get $x^2 = \frac{y-1}{y+1}$ Then I'm thinking multiply both sides ...
0
votes
1answer
27 views

Algabreic manipulation with complex numbers

How does $(iwl + \frac{1}{iwc})^2$ equal to $(wl - \frac{1}{wc})^2$? Let me clarify. In physics there is the impedance which is a complex number Z = R + iwl + 1/iwc R, w, l, and c, are ...
0
votes
0answers
20 views

Parseval's Identity, problem with $|a_n|^2$

I'm trying to obtain the Fourier Series of this function: $$f(x)=\begin{cases} \pi -x, x\in [0, \pi]\\ \pi+x, x \in [-\pi, 0) \end{cases}$$ It is a even function, so I can write: ...
1
vote
3answers
84 views

(Infinite) Nested radical equation, how to get the right solution?

I've been tasked with coming up with exam questions for a high school math contest to be hosted at my university. I offer the following equation, $$\sqrt{x+\sqrt{x-\sqrt{x+\sqrt{x-\cdots}}}}=2$$ and ...
0
votes
1answer
44 views

Where did the $-1$ come from?

It's a very specific question: Let $f(x) = \sum_{n=0}^\infty x^{n+2} = \frac{x^2}{1-x}$ $$f'(x) = \sum_{n=1}^\infty (n+2)x^{n-1} = \sum_{n=1}^\infty nx^{n-1} + 2\sum_{n=1}^\infty x^{n-1} = ...
0
votes
1answer
44 views

Exponent Problem - How do I approach this question:$x^{m+2}\cdot x^{-2m}\cdot x^{m-5}$

Assume all variable exponents represent positive integers, and simplify each integer. $$x^{m+2}\cdot x^{-2m}\cdot x^{m-5}$$
0
votes
1answer
26 views

Utilizing scientific notation: change distance in light years to distance in miles

A light year, the distance light travels in 1 year, is approximately 5.9 x 10 ^12 miles. The Andromeda galaxy is approximately 1.7 x 10^6 light-years from our galaxy. Find the distance in miles ...
4
votes
4answers
114 views

How is $x^2+1=(1/{x^2})[1-{1}/{x^2}+{1}/{x^4}-{1}/{x^6}+\cdots]$?

The author of my book writes: $$x^2+1=x^2\left(1+\frac{1}{x^2}\right)$$ $$=\frac{1}{x^2}\left[1-\frac{1}{x^2}+\frac{1}{x^4}-\frac{1}{x^6}+\cdots\right]$$ I do not understand the last step. How did ...
2
votes
3answers
54 views

Rewrite a circle's equation to easily see centre and radius

$$x^{2}+y^{2}-5x-15y+30=0$$ I'm supposed to rewrite this equation so that you can easily see the centre and radius of the circle. I don't even know where to start. According to Mathematica the centre ...