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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3answers
51 views

population change

According to the 2000 U.S. Census, the U.S. total population was 281,421,906. The 2010 Census indicated the total population was 308,745,538. What was the percent change in the U.S. population from ...
0
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1answer
40 views

Normalize a vector to be between -1 and 1

I have an acceleration vector in m/s^2 and I am going to use an algorithm that assumes these values are between -1 and 1. I have searched the web and found formulas to get it between 0 and 1. ...
0
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2answers
100 views

Value of $f(x,y,z)= x+y+z$ if $z$ decreases as $x+y$ increases

We have three continuous variables $x,y,z$ and we know their behavior. 1) When $x+y$ increases, $z$ decreases. 2) When $x+y$ decreases, $z$ increases. We have a function $f(x,y,z)= x+y+z$. What ...
2
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2answers
86 views

If all the signs are negative in an $(a + b + c)^2$ bracket, can I just make them all positive?

I have to do the expansion $$(-y - z - x^2 - y^2 - z^2)^2$$ Can I say that this is $$(y + z + x^2 + y^2 + z^2)^2$$ as all the signs are the same inside the brackets and so multiplying two ...
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1answer
167 views

A can complete a project in 20 days and B can complete the same project in 30 days.

A can complete a project in 20 days and B can complete the same project in 30 days. If A and B start working on the project together and A quits 10 days before the project is completed, in how many ...
0
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1answer
32 views

If $f(x) = \log_ax$, show that $\frac{f(x+h)-f(x)}{h} = \log_a(1+\frac{h}{x})^{1/h}$

If $f(x) = \log_ax$, show that $$\frac{f(x+h)-f(x)}{h} = \log_a\left(1+\frac{h}{x}\right)^{1/h},$$ where $h\neq0$.
6
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1answer
111 views

a problem on a special root of $x^{11}-1=0$

I have came across the following problem if $\alpha$ be a special root of the equation $x^{11}-1=0$ , then prove that $$(\alpha+1)(\alpha^2+1)......(\alpha^{10}+1)=1$$ totally stuck on it. how to ...
3
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1answer
78 views

Given two ratios $\frac{p_i}{q_i}$, what is $\frac{p_1+p_2}{q_1+q_2}$ in their terms

I am ashamed to say that I cannot figure this one out: I am given two ratios $\dfrac{p_i}{q_i}$ where $i=1$, $2$. (We just know the ratios and not the numbers $p_i, q_i$. What I mean by this is ...
4
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3answers
281 views

There isn't a product operation that is commmutative on $ \mathbb{R}^{n} $ that satisfies all the field axioms for $ n \geq 3 $.

This proof is broken down into simple easy algebra and vector questions. I would like to discuss different answers and approaches. Please see pg 162-163 on books.google.ca/books?isbn=0387290524 ...
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1answer
56 views

Calculating the time to travel 100 miles

A train has an acceleration and deceleration of 10 miles per hour squared, and it has to travel 100 miles. It has a max speed of 100 miles per hour. How do you determine it's maximum speed in that ...
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2answers
131 views

Polynomial roots

How is it possible to construct two distinct third degree polynomial equations with real coefficients and roots $2$ and $2+ i$? Isn't the only possibility $p(x)=(x-2)(x-2-i)(x-2+i)=0$? Am I losing ...
12
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8answers
19k views

Calculus book recommendations (for complete beginner)

Well I have not started calculus yet but I am really keen to. I would love if you suggest some books. Points to be noted: I really don't like the way textbooks are written so please no "textbooks" ...
2
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3answers
100 views

How does $\frac{1}{2} \sqrt{4 + 4e^4} = \sqrt{1 + e^4}$

My understanding would lead me to believe that: $$\frac{1}{2} \sqrt{4 + 4e^4} = \frac{1}{2}(2 + 2e^4) = 1 + e^4$$ But it actually equals: $\sqrt{1 + e^4}$ Can you explain why?
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2answers
176 views

Solving System of Equations - precalc question

I need help to solve for the values of $x, y, z.$ $x+2y+z=6$ $2x-y+3z=-2 $ $x+y-2z=0$
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1answer
103 views

Finding The Contour Maps Of A Function Of Two Variables

I am given the function $f(x,y) = \ln|y-x^2|$, and am suppose to find the contour maps. Let $z = c = f(x,y)$. $c = \ln|y-x^2| \rightarrow e^c = e^{\ln|y-x^2|} \rightarrow e^c = |y-x^2|$ I know I ...
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1answer
58 views

Simplifying equation from an aggregate one

I have got the 4th order coefficient of $\epsilon$ from the equation (9) from the paper. I got : $$\phi_4+ \ddot \phi_4+\omega_2 \ddot \phi_2 - \Delta \phi_2+g_2 \phi_2^2+2 \phi_1 \phi_3+ 3g_3 ...
0
votes
1answer
106 views

I want to find a closed form of the sum $\sum_{i=1}^n x^{i^2}$

Is there any closed form of the following sum? $\sum_{i=1}^n x^{i^2}$, where $x$ is a variable. Calculating the sum for a first few $n$ does not give any pattern.
1
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2answers
191 views

How to derive formula for $\sin(A-B)$ from formula for $\sin(A+B)$?

It has been a while since I have done any maths and am struggling with this question - Using the addition law $$\sin(A+B) = \sin(A)\cos(B) + \cos(A)\sin(B)$$ and the fact that $\sin(-\theta) = ...
0
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5answers
413 views

Determine the value of $k$ if $x-2$ is a factor of $2x^3-kx^2+5x-3$

I've divided using synthetic division but how would I solve for $k$?
1
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1answer
27 views

What are the $x$ intercepts of the equation $f(x)=-3(x+7)^4+48$

What are the $x$ intercepts of the equation $f(x)=-3(x+7)^4+48$? I know that I need to make $f(x)=0$ then solve for $x$, but how would I solve for the quartic?
3
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2answers
990 views

How can I solve trigonometric functions without a calculator?

I am basically looking for formulae that calculate trigonometric functions (both geometric and circular), because I want to write my own math functions for my game engine. I have found some that seem ...
1
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1answer
122 views

Equation of a surface (hyperellipsoid?) with specific conditions

Suppose I have two vectors $a,b \in \mathbb{R}^n$, and a scalar $\alpha$ where $0 < \alpha < 1$. If $d_a(x)$ is the distance from any point $x \in \mathbb{R}^n$ to point $a$, and $d_b(x)$ is the ...
0
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1answer
133 views

Inventory of Clocks and Frequency of Chimes

How do you determine the hours for which the clocks chime?
2
votes
5answers
124 views

Why when indices cancelled out leave 1 at top?

This is a really basic question, but I have just got interested in math and learning rules about powers/indices and this confused me a little. $\dfrac{a^3}{a^7}$ after they cancel out we get ...
2
votes
2answers
80 views

Show that $z^3 + (1+i)z - 3 + i = 0$ does not have any roots in the unit circle $|z|\leq 1$.

I need help with showing that $z^3 + (1+i)z - 3 + i = 0$ does not have any roots in the unit circle $|z|\leq 1$? My approach so far has been to try to develop the expression further. $$ z^3 ...
0
votes
1answer
34 views

Minumum of a function

What is minimum of the function $f(x,y)=\frac{2a}{\sqrt{xy}}$ where $a\in %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion $ and $x>0$, $y>0$?
4
votes
3answers
374 views

If $x^y + y^x = 84$ and $x>3$, find the value of $x$ and $y$.

How do we go about this questions? I see that taking log on both sides is not an option since the LHS consists of a sum and not a product. I tried differentiating it but without any boundary ...
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3answers
80 views

Solving $2|x+1|>|x+4|$

I'm trying to solve the following equations and inequalities for $x\in\mathbb R$: $$2|x+1|>|x+4|$$ I know I'm supposed to consider the intervals $(-\infty,-4), [-4,-1]$ and $(-1,\infty)$ but ...
1
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4answers
163 views

Simplifying $\{ x \in \mathbb{R} : x^2 - x - 6 \geq 0 \}$ when possible

Simplify the following interval notation when possible: $$\{ x \in \mathbb{R} : x^2 - x - 6 \geq 0 \}$$
3
votes
1answer
118 views

$0 = \left(\sqrt{p^2+m^2}-\sqrt{k^2+p^2+2\cdot k\cdot p\cos(\theta)}\right)^2 -k^2-m^2$ solving for $k$

This question is related to $\delta(f(k))$ concerning the Dirac-delta. OK I know this might seem trivial but the result is very very important to me so I want to check with you if my logic seems ...
4
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1answer
129 views

volume of cylinder

A cylindrical chocobar has its radius $r$ unit and height $h$ unit. If we wish to increase the volume by same amount either by increasing its radius alone or its height alone by the same number of ...
0
votes
2answers
166 views

upper bound for $e^{ax^2}$

I want to find a upper bound for $$e^{ax^2}\leqslant \: ?$$ "a" is a constant and $a\geqslant 0$ . x is a variable. I prefer to have a polynomial function or power function (like $ x^{k}$) is there ...
1
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1answer
45 views

ratio zero over zero uncertainty

I was just wondering if the following statement is correct. Assume I have a ratio $\frac{x-y}{z}$ for some variables $x,y,z$ and I also know that when $z=0$ I have $x\equiv y$. Does that imply that ...
0
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2answers
83 views

$f(x) = ax^5-17x^4-15x^3+153x^2-122x-b$, solve for $ a, b$

The polynomial function $f(x) = ax^5-17x^4-15x^3+153x^2-122x-b$ has one of its zeroes at $x=5$ and passes through the point $ (1,-64).$ a) Find the values of $a$ and $b$ b) Determine all the factors ...
0
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2answers
87 views

Unit vector of a curve

Find a unit normal vector to the curve $y=\cfrac{4}{3}x-\cfrac{2}{3}$ at point $(2,2)$ Would like to know in detail how is this solvable, and how should i present my answer?
1
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4answers
150 views

Solving for $b$ in $25\left(\frac{\sqrt{10}-2\sqrt{5}}{50}\right) + 5b = \sqrt{5}$

What are the steps to get from: $$25\left(\frac{\sqrt{10}-2\sqrt{5}}{50}\right) + 5b = \sqrt{5}$$ to: $$b = \frac{\sqrt{5}}{5} + \frac{2\sqrt{5} - \sqrt{10}}{10}$$ Thanks.
0
votes
5answers
134 views

Determine all solutions to $|x+12|+|x-5|=15$

Determine all solutions to the following. $$ \lvert x+12\rvert +\lvert x-5\rvert =15.$$ How can I solve problems like this? Should I try graphing the function? Should I somehow consider various ...
2
votes
1answer
223 views

Is this equation hard to solve?

How do I solve the equation below for $x$? $A$, $B$, $a$ and $r$ are constants. \begin{equation} x + \frac{1}{1+Ar^{-x/a}} + B =0 \end{equation}
1
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1answer
55 views

Proportionality and Linearity

I have a very basic question that a friend and I are disagreeing on. Does (direct) proportionality between two terms a and b imply linearity between the two terms?
4
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3answers
259 views

Show that $2 xy < x^2 + y^2$ for $x$ is not equal to $y$

Show that $2 xy < x^2 + y^2$ for $x$ is not equal to $y$.
0
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2answers
124 views

Solving the equation $x^{ (\frac{x}{123}+11)} =123$

I came across this equation $$x^{ \left(\frac{x}{123} + 11 \right) } = 123 $$ All I could think of is to put $ \ln $ into the equation: $$ \begin{align} \ln\left(x^{ \left( \frac{x}{123} + 11 ...
1
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2answers
68 views

Bound for the solution of Non homogeneous equations

I have the following set of equations $x_1^2+x_2^2+x_3^2+....+x_n^2=1$ $x_1+x_2+x_3+....+x_n=1$ $-1 \leq x_i \leq 1$ Then what is the bound for the |$x_1|+|x_2|+|x_3|+....+|x_n$| The trivial ...
0
votes
0answers
161 views

Generating parabola from points applet

Does anyone know of an applet or something that generates a parabola (graph and/or equation) given three (unique, non-colinear) points? I'm going to be mentioning this fact to my students as an aside ...
5
votes
2answers
159 views

Proving basic math principles?

I've been thinking how to prove some of these basic "formulas" but most of them I don't know how: $$\frac{a}{b}\cdot\frac{c}{d}=\frac{ac}{bd}$$ exponentials such as $$(a^b)\cdot (a^c)=a^{b+c}$$ (I ...
1
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2answers
186 views

A question on minimum and maximum values

What are the (possible) minimum and maximum values of the rational algebraic expression $$\frac{AB}{(A + B)(A - B)},$$ if $A, B \in \mathbb{R}$ with $B < A$? Thank you!
4
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1answer
92 views

Show that $x$, $y$, and $z$ are not distinct if $x^2(z-y) + y^2(x-z) + z^2(y-x) = 0$.

Suppose that $x^2(z-y) + y^2(x-z) + z^2(y-x) = 0$. How can I show that $x$, $y$, and $z$ are not all distinct, that is, either $x=y$, $y=z$, or $x=z$?
1
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3answers
100 views

Dividing polynomials

When dividing variables, does each term in the numerator have to have a variable for it to be divided? For example if the problem is $$\frac{9x+8x^2+1}{x}$$ can it be simplified to $$9+8x+1?$$ Or ...
1
vote
1answer
929 views

Solving a quadratic diophantine equation in two variables

I have an equation in the following form: $$6mn+m+n=x$$ $$m,n,x\in\Bbb Z; \qquad0 < m,n$$ If I were given a value for $x$, how would I go about finding solutions to this equality for $a$ and $b$ ...
1
vote
1answer
106 views

Continued fractions with $n$ layers

Solve the equation $$x=2+\dfrac1{2+\dfrac1{...2+\dfrac1{2+\dfrac1x}}}$$ Where there are n layers in the fraction
3
votes
2answers
69 views

How to solve the non homogeneous equations

I am looking for the proof of the following I have the following equations $x_1^2+x_2^2+x_3^2+....+x_n^2=1$, $x_1+x_2+x_3+........+x_n=1$ $0 \leq x_i\leq 1$ for-all $i$ I believe that the only ...