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
295 views

Maximum inequality

Can anyone show that ...
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1answer
74 views

Manipulating Indices on Series

I have a series: $$ \sum_{n_1-l_1=0}^{\infty}\sum_{n_2-l_2=0}^{\infty}\sum_{n_3-l_3=0}^{\infty}a_{n_1-l_1,n_2-l_2,n_3-l_3}r^{n_1-l_1}s^{n_2-l_2}t^{n_3-l_3} $$ Which is equal to another series: $$ ...
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1answer
243 views

What is the name of these equations?

$xy=0$ $ax +by +cxy +d=0$ $ax +by +cz +dxy +eyz +gxyz=0$ I made myself the examples, sometimes I face these equations and I do not know how to resolve them, all equations whose unknowns have ...
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2answers
67 views

How do I boost lower percentages more than higher ones?

I have a set of percentages {1%, 12.5%, 25%, 50%, 100%} I want to find a way to 'pull up' all of these percentages to be higher in a way that is more advanced than an average mean like ((x+100)/2) ...
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1answer
100 views

Exponential growth on discrete quantities

I'm familiar with exponential growth, however I'm not sure how deal with situations where my quantities are discrete and rounding errors come into play. To be concrete, say I got $N$ items arranged ...
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3answers
1k views

Binary expansion Unique

I am trying to show that how the binary expansion of a given positive integer is unique. According to this link, http://www.math.fsu.edu/~pkirby/mad2104/SlideShow/s5_3.pdf, All I see is that I can ...
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4answers
231 views

How can I show that this family of curves may be described by this differential equation?

I have a homework problem in which I wish to show that the family of curves given by $x^2 + y^2 = c x$ where $c$ is an abitrary constant may be described by the differential equation $\frac{dy}{dx} ...
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1answer
115 views

Calculating the percent of a person's life that a certain time takes up?

A while ago I read an article on Cracked.com about why you wouldn't want to be immortal. Among many other reasons was that time would speed up for you after so many years and it would make every ...
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1answer
121 views

Would this be the most elegant way to approach an elementary system of equations?

In the document here, I have solved problem $58.2$ on page $636$. My solution is below: Now, I hope that this is not a duplicate (and I suspect it isn't, since this seems to be a slightly obscure ...
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1answer
541 views

What is $f(x)$ divided by $(x-a)$?

This is an exercise from Spivak's Calculus: Prove that for any polynomial function $f$, and any number $a$, there is a polynomial function $g$, and a number $b$, such that $f(x)=(x-a)g(x)+b$ for ...
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3answers
242 views

How many solutions has $z^\pi = 1$?

I know that for $z \in \mathbb C$ and some natural $n\geq 1$, the equation $z^n = 1$ has exactly $n$ solutions. But what if I say $n$ need not be natural, e.g. $$ z^\pi = 1.$$ I mean the equation ...
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3answers
254 views

If $f(x)$ and $f(x)-x$ have only one real root, then $f(f(x))-x$ has only one real root.

First edition was: Let $f(x)$ be a polynomial such that $f(x)$ and $f(x)-x$ have only one real root. How to prove, without derivatives, that $f(f(x))-x$ also has only one real root? Second edition: ...
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3answers
102 views

how can I prove that when $3x+2y\leq 5$, then $x > 1$ implies $y < 1 $?

Basically what I have is that $x,y \in \mathbb{R}$ and that $3x + 2y \leq 5 $, so what I need to prove is that $x > 1 \rightarrow y < 1$ How would you prove this? In some way i know that if I ...
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1answer
141 views

Invertible functions and fixed point iteration

Consider two curves $y=f(x)$ and $y=h(x)$, both are one-to-one (invertible). Define $g(x)= f^{-1}(h(x))$ (inverse of $h(x)$). Assuming the scheme $X_{n+1}= g(X_n)$ converges to a fixed point $x^*$, ...
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3answers
329 views

Logarithms of the form $x=e^y$

I have the following math problem: The number of people in a town of 10,000 who have heard a rumor started by a small group of people is given by the following function: $N(t) = ...
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1answer
174 views

Converting to partial fractions

I would like to see a step by step conversion of this fraction to two partial fractions. $\frac{x^2}{1-x^2}$
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2answers
209 views

Can this be solved by induction? (number of ways of cutting a rod into pieces)

I am reading an algorithm example. The example is about Rod cutting. The idea is that a steel rod can either be sold as it is, or be cut into integral pieces and ...
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6answers
393 views

How could we solve $x$, in $|x+1|-|1-x|=2$?

How could we solve $x$, in $|x+1|-|1-x|=2$? Please suggest a analytical way that I could use in other problems too like this $ |x+1|+|1-x|=2$ and of this genre. Thank you,
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3answers
133 views

Is this proof about $a^3>a \rightarrow a^5>a $ correct?

Basically what I need is to know if this proof is correct what I need to prove is: if $a^3 > a $ then $a^5>a$ so, what i did was this: $a^3 a^2 > a a^2$ $a^5 > a^3$ because $a^5>a^3$ I ...
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5answers
635 views

I want to study mathematics ahead of high school, but I found that I'm rusty on the elementary stuff

Next week, I'm beginning the 2nd semester of 9th grade in the country's leading Comp Sci High School. (the profile is actually Math-Comp Sci, but this HS focuses more on Comp Sci, whereas other ones ...
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1answer
159 views

maximize product of two numbers formed with the digits $1$ to $9$

You are given the digits from $1$ to $9$. You can form two numbers by concatenating them, for example, $975123$ and $864$, and then take the product of the two resulting numbers. Find how to maximize ...
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1answer
223 views

How to solve equations with absolute value and using the Archimedean property

I'm trying to learn Real Analysis on my own, but I found that i'm a bit rusty with the elementary stuff. How do I solve equations like $|x| + |x+1| = 1$ and $|x-1| + |x+1| = 2$? I don't want the ...
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1answer
1k views

Help writing proof for $\sqrt{a^2 + b^2} \neq \sqrt[3]{a^3 + b^3}$

Like the title says I'm attempting to write a existence proof showing "that there exists no non-zero real numbers a and b such that $\sqrt{a^2 + b^2} = \sqrt[3]{a^3 + b^3}$". I'm having trouble ...
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0answers
233 views

Bloom of Thymaridas

I'm interested in learning more about the Bloom of Thymaridas, a description of which can be found here. Obviously the mathematics behind the identity is not particularly deep from a modern ...
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3answers
110 views

How to derive the share of a prize given scores

I am curious to find out reasonable ways of dividing a prize among $n$ people in the following situation: To make the example specific, we have $6$ people in total who are going to share a prize of ...
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2answers
574 views

Upper bound for the partial sum $\sum k \lg k$ via summation?

In this lecture of an introductory class to algorithms (video here, time 74:09), the professor cites the following as an upper bound: $$ \sum_{k=2}^n k \lg k \leq \frac{1}{2} n^2 \lg n - \frac{1}{8} ...
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2answers
234 views

Graphing Inequalities in Two Variables

When we have to graph an inequality in two variables, we usually graph the corresponding equality, i.e. the straight line on the coordinate plane, which divides the plane into two parts. Then, we use ...
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3answers
93 views

How do I interpret this mathematical model?

"A pharmaceutical company firm manufactures a liquid form of penicillin in bulk at a price of £200 per unit. The total production cost £$C$ for $x$ units is: $$ C = 500,000 + 80x + 0.003x^2 $$ and ...
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1answer
1k views

distance of two point (latitude and longitude) on earth to meters without using $\cos,\sin, \mathrm{harversine}$ formula, etc.

I'm getting the distance between two locations (lat/long) using Pythagoras theorem. my data look like this (I use microdegrees because I have limitations) point1: -34608420,-58373160 point2: ...
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3answers
276 views

Does $\log(ab)^n$ equal $(\log(a)+\log(b))^n$ or $n\log(a)+n\log(b)$?

I think this might be a case of slight ambiguity in notation, but here goes: On a test question, I was required to expand the expression $\log (ab)^n$. Since the logarithm is a function, I reasoned ...
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1answer
239 views

creating sliding scale

I'm trying to figure out a formula to create a sliding scale. I've created an index and now I want to assign a fee based on the index.The fee should be a maximum of $1 for those with the highest ...
2
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1answer
63 views

solving equation of circles

I have $2$ points on circle from which I want to find the center of circle. I also know the r. I am trying to solve $2$ equations but not able to find the centre $( C_x,C_y)$. Can someone check what I ...
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2answers
129 views

Transform/approximate this expression to avoid undefined value

I have an expression like this $$\sum_i^n\log\frac{x_i}{y_i}+\alpha\sum_i^nx_i\log\frac{x_i}{\beta}$$ A potential problem is that $x_i$ and $y_i$ may take value $0$ for certain $i$, hence making ...
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2answers
92 views

Inverse Equation Largest Possible Set

I have a question I'm not sure how to work out (because I don't really understand it). Please help me; I have spent a lot of time on this question and tried to work backwards from the answer but ...
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2answers
152 views

Trying to figure out the pattern in this series (2)

I could use some help figuring out what the next row is in the pattern below (and what the rule is generally for each number in the series). I put it in a triangle because I'm pretty sure it has ...
2
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1answer
286 views

How to calculate percentage of comment lines in a code?

I have a file within which I have 6 lines of code and 8 lines of comment. What's the formula to calculate how much percent of the whole file comments have?
3
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1answer
3k views

Relationships among the terms “slope”, “parameter”, and “coefficient”?

In $y=mx$, is $m$, are there different implications of referring to $m$ as a "slope", a "coefficient", a "parameter"? Or perhaps the "slope coefficient" or "slope parameter"? For context, I am ...
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3answers
486 views

finding point on line 1 unit next from start point

I have a line passing through points P1(x1,y1) and P2(x2,y2). Can I find next point on the same line thats 1 unit away from Point P1(x1,y1)? If yes how can I find? I just draw line at these points ...
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6answers
11k views

What is 'zero of multiplicity'?

What is a 'zero of multiplicity'? I looked up this definition on Mathwords.com: How many times a particular number is a zero for a given polynomial. For example, in the polynomial function $f(x) ...
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2answers
63 views

Need help modifying this binomial expansion

The binomial theorem states that $$(A+B)^n=\sum_{k=1}^{n}{n \choose k}A^{n-k}B^k$$ I need help expressing the following summation as a modified binomial expression: $$\sum_{k=1}^{n}n!{n-1\choose ...
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1answer
42 views

Question about a scetch in the book and vector length?

The book says that RQ and SP are the same. On the drawing I can see that the length is not the same. If two vectors are equal shouldn't the length be equal too? The book also says that AC is twice ...
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4answers
4k views

Whats the rule for putting up a plus-minus sign when taking under root?

Given a simple equation.... $\ (x+1)^2 =21 $ if we take the under root of both sides , we get $\ x+1 = \pm \surd 21 $ why dont we get a $ \pm $ on the left hand side ?
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1answer
137 views

Function defined by rules

This has to be the stupidest question for the whole week but, I watched this video: http://www.khanacademy.org/video/introduction-to-limits--hd?playlist=Calculus and the guy defined a function ...
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2answers
137 views

Mapping of Inverse functions.

Let $\mathsf{f(x)}$ be a differential and an invertible function, such that $\mathsf{f''(x)>0}$ and $\mathsf{f'(x)>0}$. Prove that$$\mathsf{ f^{-1}\left(\frac{x_1 + x_2 ...
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3answers
662 views

software for algebraic simplifying expressions

I have many huge algebraic expressions such as: $$\frac{8Y}{1+x}-\frac{(1-Y)}{x}+\frac{Kx(1+5x)^{3/5}}{2}$$ where $\ ...
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1answer
248 views

What is the process in calculating values on the Unit Circle?

I realize that this question is incredibly basic for this website, but I really need help. I took this image from MathIsFun.com: It's a picture of the Unit Circle. On the outside, in purple, are ...
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3answers
4k views

Finding the center and radius of a circle given a general degree 2 equation

I am trying to find the center and radius of the circle with equation $x^2 + y^2 -6x + 10y + 9 = 0$
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6answers
52k views

Finding an equation for a circle given two points

No idea how to do this, I used to have these conic shapes committed to memory but I forget them already. I am supposed to find an equation for the circle that has center $(-1, 4)$ and passes through ...
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2answers
152 views

Finding an equation for a line with given criteria

I have no idea how to do this at all, I am trying to study before I take calculus again. I am supposed to find equations for the line that passes through the point $(2, -5)$ and: Has slope $-3$. is ...
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2answers
3k views

Inequality; when we switch signs? From basic to complicated

Let's say I have $$a > b$$ Now we know that the inequality sign will switch if: We take the reciprocal of both sides Multiply $-1$ to both sides. But, are there more? What if we do more ...