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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1answer
90 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
34 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 ...
6
votes
3answers
246 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 ...
3
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1answer
39 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
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0answers
57 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
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3answers
54 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
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2answers
56 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
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8answers
110 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.
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2answers
34 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
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0answers
26 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
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2answers
68 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
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2answers
27 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
45 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
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2answers
35 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 $|a| = \sqrt{x^2+y^2}$), ...
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0answers
21 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
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2answers
78 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
14 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
68 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
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2answers
28 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
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3answers
103 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
28 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
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9answers
2k 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
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1answer
40 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
48 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
76 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
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1answer
28 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
51 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
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3answers
33 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
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0answers
43 views

On the matrix representation of a composition of Möbius transforms

Let the Möbius 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
54 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
32 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 ...
1
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3answers
200 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
62 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
50 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
40 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
121 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
63 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 ...
1
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1answer
36 views

To find nature of roots of $Ax^{4} + Bx^{3} + Cx^{2} + Dx - E $

To find nature of roots of $$f (x) = Ax^{4} + Bx^{3} + Cx^{2} + Dx - E $$ Where $A, B, C, D, E$ are all positive. After applying Descartes' Rule of signs to $f(x)$ there is one sign change , so ...
0
votes
2answers
33 views

Factoration / Simplification / Leibniz Polynomial

Simplify $\frac{(b-c)\times a^n - (a-c)\times b^n + (a-b)\times c^n}{(a-b)(a-c)(b-c)}$ for $n>2$. The answer is $(a+b+c)^n$, but I can't seen to get it. Can someone help me? Thanks
2
votes
1answer
54 views

Holder type inequality

If $A$ is a symmetric and positive semidefinite matrix is it true that $$\sum_{i,j=1}^n A^{i,j}x^iy^j \leq \sqrt{\left(\sum_{i,j=1}^n A^{i,j}x^ix^j\right)\left(\sum_{i,j=1}^n A^{i,j}y^iy^j\right)},$$ ...
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votes
4answers
57 views

Prove $|a - b|< c$ if and only if $b - c < a < b + c$.

Prove $|a - b|< c$ if and only if $b - c < a < b + c$. It is a task from real analysis and I am failing the class I tried doing it on a quiz, but I got it incorrect.
0
votes
2answers
84 views

Why does $\lim_{x\to a}\; \frac{e^x - e^a}{x-a} = e^a$

My attempt: $$\lim_{x\to a}\; \frac{e^x - e^a}{x-a} = $$ $$\frac{e^a - e^a}{a-a} = $$ $$\frac{e^a(1 - 1)}{a(1 - 1)} = $$ $$\frac{e^a}{a}$$ My textbook says the correct answer is $e^a$. How do I get ...
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votes
2answers
96 views

Precalculus Word Problem: setting up linear equation

This is a math question, I got $18 I wonder if anyone can get the same answer and explain how. Beatrice went shopping with #120. She bought items in five stores. In each store she spent ...
1
vote
1answer
25 views

How to find a term in an arithmetic progression given relationships among the terms?

The title is bad, but I was unable to think of better one, I apologize for this. I have this system: \begin{align*} a_2 + a_3 + a_4 + a_5 & = 34\\ a_2 * a_5 & = 52 \end{align*} I have to ...
0
votes
2answers
67 views

Forming equations for exponential growth/decay questions

Problem Dry cleaners use a cleaning fluid that is purified by evaporation and condensation after each cleaning cycle. Every time the fluid is purified, 2.1% of it is lost. The fluid has to be topped ...
0
votes
1answer
74 views

$f(x)$ as a difference of two increasing functions

Let $f(x)$ be a continuous function. Find $g(x)$ and $h(x)$ - two increasing functions, which difference equals $f(x)$, e.g. $f(x)=g(x)-h(x)$. Examples: $\arctan(x^3-9x)$ $\frac{1}{1+(\sin x)^2}$ ...
2
votes
0answers
88 views

Impossible System of Equations

This is from a competition: DMM Olympiad, Ural State University P4 I don't understand what the question means exactly (the first part, i.e. "exclude $x$ or $y$ from..." part). Does it mean "write $x$ ...
25
votes
2answers
438 views

How to prove $\sum_{n=0}^{\infty} \frac{1}{1+n^2} = \frac{\pi+1}{2}+\frac{\pi}{e^{2\pi}-1}$

How can we prove the following $$\sum_{n=0}^{\infty} \dfrac{1}{1+n^2} = \dfrac{\pi+1}{2}+\dfrac{\pi}{e^{2\pi}-1}$$ I tried using partial fraction and the famous result $$\sum_{n=0}^{\infty} ...
1
vote
1answer
40 views

The speed of learning and prior

If I know $$\frac{\alpha}{\alpha+\beta}<\frac{\lambda}{\lambda+\gamma}$$ can I know the sign of $$\frac{\alpha+1}{\alpha+1+\beta}<\frac{\lambda+1}{\lambda+1+\gamma} $$ And the sign of ...
-1
votes
1answer
75 views

give direct proof of the fact $a^2 - 5a + 6$ is even for any integer [duplicate]

I know this is true but I don't know how to prove it. I have worked it out for the integers from $1$ to $10$ but this is not direct proof, is there a formula I need?