Rational functions are ratios of two polynomials, for example $(x+5)/(x^2+3)$.

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Revenue cost profit [on hold]

A company that makes and sell memory chips establishes the followings : Revenue function, $R(x) = x (75 – 3x)$ Cost function, $C(x) = 125 + 16x$ Where $x$ is in millions of chips, and $R(x)$ and ...
4
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4answers
246 views

Limit of a rational function

Calculate the limit $$ \lim_{x \to 0} \frac{3x^{2} - \frac{x^{4}}{6}}{(4x^{2} - 8x^{3} + \frac{64x^{4}}{3} )}$$ I divided by the highest degree of x, which is $x^{4}$, further it gave $$ ...
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2answers
48 views

Integral of $\arcsin$ of a rational function, using integration by parts

I'm a class 12 student and this a question from my textbook: $$I=\int{\arcsin{2x\over 1+x^2}}\mathrm{d}x$$ I did it using integration by parts like this: $$I=\arcsin{\left(2x\over ...
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51 views

A simple-looking rational limit

Please help me compute: $$ \lim_{z\to 0}\frac{\sqrt{2(z-\log(1+z))}}{z} $$ I know the answer is 1 because I plugged it into Mathematica. Attempts with L'Hopital's Rule didn't work. This a step in an ...
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52 views

Find all values $c$ such that $(x+1)/(x^2+2cx+4)$ has domain R

Word for word: Find all values of $c$ such that $f(x)=\frac{x+1}{x^2+2cx+4}$ has a domain R I really don't know where exactly to start. I'm not sure what it means by "a domain R"
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1answer
37 views

Why is this rational expressions indeterminate when evaluated?

I have this rational expression to evaluate, $$ {{3a-3}\over {4a(a-1)}} \text { if } a=1. $$ I understand that if you substitute 1, both the numerator and denominator would turn out 0, thus making ...
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2answers
50 views

Integration of the rational function $ 4/( 1+4t^2) $

So it's August so my memory of math is a little rough right now. I was wondering if someone could help me with integration with a fraction involved? For example: $$\int_0^{1/2} \frac 4{1+4t^2} \, ...
2
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1answer
33 views

Is there any rational map from the nonsingular Segre quadric surface in $\mathbb{CP}^3$ to a nonsingular surface of degree greater or equal to 4?

Is there any rational map from the nonsingular Segre quadric surface in $\mathbb{CP}^3$ to a nonsingular surface in $\mathbb{CP}^3$ of degree greater or equal to 4? Someone told me that the answer is ...
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5answers
149 views

Why is $f(n) =\frac{n(n+1)(n+2)}{(n+3)}$ in $O(n^2)$?

Let: $$f(n) = n(n+1)(n+2)/(n+3)$$ Therefore : $$f∈O(n^2)$$ However, I don't understand how it could be $n^2$, shouldn't it be $n^3$? If I expand the top we get $$n^3 + 3n^2 + 2n$$ and the biggest ...
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67 views

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

(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 $f(x)/g(x)$ ...
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49 views

Integral of an exponential of rational function

I have an integral of the form $\int_{a}^{b} \text{exp}\left(\frac{\lambda}{\rho^2 m + \sigma^2_u}\right) \frac{1}{m^2}\text{exp}\left(-\frac{\lambda}{m}\right) dm$. Can this integral be found ...
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1answer
78 views

Why does the graph of $x^3/x^3$ not have a horizontal asymptote?

I am a graduate student studying math, and am actually teaching College Algebra right now. But every once in a while, I come upon something new in a subject that I have supposedly mastered. Why does ...
2
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35 views

Truncation error in Padè approximants

Suppose only the following data are known about a rational function $R(x)=P(x)/Q(x)$ (for $P,Q$ polynomials): (a) the degree of $P$ is $\leq m$ and the degree of $Q$ is $\leq n$; (b) the first $k$ ...
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28 views

Variation of the argument of a rational function along a circle

Crossposted here on MO. Let $f:\bar{\mathbb C}\to \bar{\mathbb C}$ be a rational function, and take a circle $C$ not crossing the zero- and polar-locus of $f$. The argument principle tells us the ...
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3answers
40 views

Help solve rational expression

I need help solving this rational expression. Divide $$\frac{4x^4 + 6x^3 + 3x - 1}{2x^2 + 1}$$ How do you solve this problem? Where do I start?
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Setting up word problem for finding length and width

Word Problem: The length of a rectangular sign is $3$ feet longer than the width. If the sign has space for $54$ square feet of advertising, find its length and width. I have not idea where to start. ...
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43 views

Rationals over an interval

Suppose $I$ is an interval $[a,b]$. It is noted that $a$ and $b$ are real integers. Divide the interval into $n$ parts with step size $h=(b-a)/n$. Clearly all the points $a$, $a+h$, ...
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1answer
45 views

Solving the polynominal: $s(t) = -16t^2 + 48t + 160$

The height of a ball is thrown directly upward from an initial height of $160$ ft with an initial velocity of $48$ ft per second is given by the function: $s(t) = -16t^2 + 48t + 160$, where $s(t)$ ...
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110 views

How to solve the equation $(3x+4)/[4x(x-4] - 1/(x-4) = 1/4x$? [closed]

I'm having a difficult time with prove the next equation: $$\frac{3x+4}{4x\left(x-4\right)}-\frac{1}{x-4}\:=\:-\frac{1}{4x}$$ Can somebody help me with the process ?
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4answers
76 views

Solving for $x$: $\;\frac8{x-2}-\frac{13}2=\frac3{2x-4}.$

I have solved this sort of problem before, but frustratingly I have forgotten to. The problem is: Solve for $x$: $$\dfrac8{x-2}-\dfrac{13}2=\dfrac3{2x-4}.$$ So hence the title, How should I ...
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41 views

The value of $a/b+b/a-ab$ under the condition $ab=a-b$ (AMC 10 problem)

I am confused about the problem below. I answered correctly but used substitution to solve instead of changing everything to the same denominator as shown in the solution. The problem: Two non-zero ...
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34 views

Derivative of rational function help.

consider $$f(x)=\frac{1}{2x-4}$$ The derivative should be $\displaystyle -\frac{1}{2(2x-4)^2}$ However I get $\displaystyle -\frac{2}{(2x-4)^2}$ my workflow: $$\begin{array}{} f'(x)&= ...
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1answer
37 views

Supremum of a rational function

Let $f(z)$ be a rational function in the complex plane such that $f$ does not have any poles in $\{z:\Im z\ge0\}$. Prove that $\sup\{|f(z)|:\Im z\ge0\}=\sup\{|f(z)|:\Im z=0\}$. Let $\Gamma_r$ be a ...
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1answer
111 views

There cannot exist a rational function $f: \mathbb{R} \to \mathbb{R}$ injective, not surjective

I was looking for a rational function $f: \mathbb{R} \to \mathbb{R}$ that looks like $\arctan$, in that it is injective not surjective well-defined on all $x\in \mathbb{R}$ (no vertical ...
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47 views

Where rational functions are undefined

I have another question/comment I'd like a fresh pair of eyes on The question is "A rational function can have infinitely many x-values at which it is not continuous" I know since Q(x) in the ...
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21 views

min max of a rational function

The variable is a vector $x \in \mathbb{R}^n \times\mathbb{R}^m_+ \times \mathcal{E}$ where $\mathcal{E}$ is an ellipsoid of dimension $e$. I would like to find the min and max of the following ...
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35 views

Rational function on a noetherian scheme

Let $X$ be a noetherian scheme and $f$ a rational function on $X$, so by definition the domain of $X$ includes all associated points of $X$. I think the following is true: $f$ is regular on $X$ if and ...
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356 views

The solutions for the equation $\frac{a}{c-b+1}+\frac{b}{a-c+1}+\frac{c}{b-a+1}=0.$

How can I find the solution for the following equation in $a,b \mbox{ and } c$. $$\frac{a}{c-b+1}+\frac{b}{a-c+1}+\frac{c}{b-a+1}=0.$$ Also $b-c \neq 1$, $c-a \neq 1$ and $a-b \neq 1$. Thanks!
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50 views

Branch of mathematics that studies groups / rings or rational functions

I'm not really a mathematician, and looking for some literature which could potentially help me in research. Im interested in algebra of rational functions (ratios of polynomials) of finite order. ...
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1answer
19 views

Termwise differentiation of sequence of rational functions when the uniform limit is analytic

Given a sequence $\{f_n(x)\}$ of rational functions which converges uniformly to the analytic function $\{g(x)\}$ on $[a, b]$ ($f_n(x)$ are defined on $[a, b]$ and hence are analytic), what can we say ...
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43 views

Inverse functions problem

There are two functions $f\colon\mathbb Q \to \mathbb Q \setminus \{-1\}$ and $g\colon\mathbb Q \to \mathbb Q \setminus \{1\}$. $$g(x) = \frac{f(x)}{f(x)+1}.$$ Prove that if there is a inverse ...
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Values of $m$ for which $y^2 + 2xy + 2x -my -3$ can be factorised

For what values of $m$, will the expression $y^2 + 2xy + 2x -my -3$ be capable of resolution into two rational linear factor? This is how I did it: $$y^2 + 2xy + 2x -my -3 = y^2+(2x-m)y+2x-3$$ ...
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1answer
47 views

rational parameterization of quartic

With the curve $x^4 - 6x^2 - y^2 + 1 = 0$ in the range of $x$ inside of $(-1,1)$, I can only identify two rational points $(0,1)$ and $(0,-1)$. Is it possible to determine if there are others?
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1answer
40 views

Inverse LaPlace Transform of the square root of Rational, Monic 1st Degree Polynomials

I tried to find this in Churchill's Operational Mathematics which has a good variety of transform pairs, but no matches for what appears a simple expression. Does anyone have a solution for the ...
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1answer
39 views

Rational parametrization of points on sphere of irrational radius

I am trying to figure out if a closed form parametrization can exist for finding all of the rational points on a sphere of radius $\sqrt{2}$. ie, x=f(u,v), y=g(u,v), and z=h(u,v) where f,g,and h are ...
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integrality of certain rational numbers

Let $P,Q\in\mathbb Q[X]$ be relatively prime polynomials ($X$ being an indeterminate). Assume that $Q(0)=1$ and that $P/Q$ is in $\mathbb Z[[X]]$. Does this imply that $P$ and $Q$ are in $\mathbb ...
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17 views

recursive relation in rational expression form

I am looking for a closed form expression for the variables $n_i$ that are stationary solutions of the recursive relation: $ n_i(t+1)=n_i(t)\sum_mf_{i,m}\frac{K_m}{\sum_j f_{j,m}n_j(t)}$ i.e. the ...
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79 views

expand a rational function in a power series

$$\frac{4-x}{(2-x)(1-x)^2}$$ Expand in ascending powers of x, stating when the expansion is valid; also write down the coefficient of $x^n $
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Solving $L= \frac{a^2}{2a^2+bc}+\frac{b^2}{2b^2+ac}+\frac{c^2}{2c^2+ab}$ priveded $a+b+c=0$

Let $a,b,c$ be such that $a+b+c=0$ and suppose that $$L= \frac{a^2}{2a^2+bc}+\frac{b^2}{2b^2+ac}+\frac{c^2}{2c^2+ab}.$$ Find the value of $L$. I can only see the symmetry of these function ...
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1answer
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Find the sum of the maximum and minimum

For a real number $x$ find the sum of the maximum and minimum. $$y=\frac{x^2-2x-3}{2x^2+2x+1}$$ This is a sample question for a math competition. It seems like calculus would be used to solve this, ...
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Geometry aspect of a extreme value problem

In a plain with orthogonal coordinate $XOY$, set point $A(a,a)$, and $P$ is a point in function $y=\frac{1}{x}$,where $x>0$. If the distance between $P$ and $A$ is $2\sqrt{2}$.Find all $a$ ...
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1answer
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How to prove $\frac{1}{x}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+2\sqrt{\frac{1}{ab}+\frac{1}{ac}+\frac{1}{bc}}$

Question: Let $a,b,c>0$ are give numbers and $x>0$, such that $$ \sqrt{\dfrac{a+b+c}{x}}=\sqrt{\dfrac{b+c+x}{a}}+\sqrt{\dfrac{c+a+x}{b}}+\sqrt{\dfrac{a+b+x}{c}} $$ show that $$ ...
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1answer
47 views

Solve this simple polynomial

$$\text{Problem: }{x^3-x^2-x-2 \over x^2 + x - 6}$$ My textbook was able to come up with $(x-2)(x^2+x+1)$ $$\text{Textbook: }{(x-2)(x^2+x+1) \over (x-2)(x+3)}$$ I've tried grouping and using the ...
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20 views

Conditions under which asymptotes can be intersected/crossed?

What are the conditions under which some horizontal asymptotes can be crossed, and why is this acceptable? I'm speaking specifically to rational functions, such as in the case of the function $$ f(x) ...
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19 views

Notational issue

Let $K = F(t)$. If $r \in K: (\nexists c \in F: r(t) = c \forall t)$ is a rational function and $L = F(r(t))$, then what form does $f \in L$ have? Is it a rational function where the coefficients are ...
2
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1answer
156 views

How to evaluate $\int_0^1\frac{1+x^4}{1+x^6}\,dx$

$$\int_0^1\frac{1+x^4}{1+x^6}\,dx$$ Can anyone help me solve the question? I am struggling with this.
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26 views

Solve the inequality $(1008-99K-0.75K^2)/(42-0.5K)>0$

How do you solve $$\frac{1008-99K-0.75K^2}{42-0.5K} > 0$$ for $K$? I don't think you can just get rid of the denominator by multiplying to the other side right?
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2answers
55 views

How do i define 'complex rational function'?

http://en.wikipedia.org/wiki/Rational_function I don't get the definition in wikipedia. It would be great to define "complex rational function" with the domain $\overline{\mathbb{C}}$, namely ...
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30 views

I am not sure if the answer for the dividing rational expressions problem (attached) should be simplified

$$ \frac{x}{x+2} \div \frac{1}{x^2 - 4} $$ Original image: http://i.stack.imgur.com/EEqd4.jpg -I am not sure if the answer for the problem (which is attached) would be $\frac{x^3-4x}{x+2}$ or if it ...
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0answers
16 views

Looking for methods/results for explicitly bounding iterations of rational functions

This is a cross-post of http://mathoverflow.net/questions/155775/looking-for-methods-results-for-explicitly-bounding-iterations-of-rational-funct But I received no answer there to the actual ...