Rational Definition and 616 Threads

b]1. Homework Statement [/b] lim f(x) = 2 x -> infinty lim f(x) = -2 x-> - infinty lim f(x) = - infinty x-> -4 lim f(x) = - infinty x-> 2- lim f(x) = infinty x-> 2+ relative min of 0 at x=2 relative max of -0.900466 at x=0.442818 concave down (-infinty, -4) (-4,-2)...

How can i prove that addition and mulitiplication of rational numbers follow commutative and associative law?

Homework Statement Let x_{1}, x_{2}, ... be a sequence of rational numbers in which each rational number in (0,1) occurs exactly once. Define the function, H(x) = 0 if x \leq 0, and 1 if x > 0. Next, define the function F(x)= \sum^{\infty}_{k=1} 2^{-k} H(x - x_{k}). Prove that F is...

Homework Statement Prove Proposition 1.15. Proposition 1.15. If a and b are real numbers satisfying a<b, then there are rational numbers and irrational numbers between a and b. Homework Equations Professor said to use the Archimedean property The Attempt at a Solution a < b...

Homework Statement f(x) = \frac{x}{x-1} + \frac{x+1}{3x} Homework Statement need to take the first derivative of this expression... I can do it but I am curious as too why i cannot take the derivative of \frac{x}{x-1} and then just add it to the derivative of \frac{x+1}{3x}...

Homework Statement Suppose that r is a solution of the equation: anxn + a(n−1)x(n−1) + . . . + a1x + a0 = 0 where the coefficients ak belongs to Z for k = 0, 1, . . . n, and n is greater or equal to 1. If r is a rational solution r = p/q, where p, q belong to Z and p and q are...

A long time ago I took a number theory course and really enjoyed it. At one point we were shown the proof for the theorem that a number is rational if and only if it has a periodic decimal expansion. The (<=) direction is really easy if you know some Calculus, but I remember the (=>) direction...

Prove that Cl(Q) = R in the standard topology I'm really stuck on this problem, seeing as we haven't covered limit points yet in the text and are not able to use them for this proof. Can anybody provide me with help needed for this proof? Many thanks.

Homework Statement Prove via Mathematical Induction that, The sum of n rational numbers is rational. Homework Equations The Attempt at a Solution Let N = 1 The sum of one rational number is the number it's self, which is a rational number. Assume when n = n, the sum of n...

Homework Statement 1. x-2|x| < 3 Homework Equations The Attempt at a Solution Okay, I attempted this equation below but I don't know if I'm right. x-2|x|- 3 < 0 x^2 - 2x - 3 < 0 (x - 3) (x + 1) < 0 x = 3 x = -1

Homework Statement Show that R(x) cannot be made into a complete ordered field, where R(x) is the field of rational functions. Homework Equations Definition of a complete ordered field: An ordered field O is called complete if supS exists for every non empty subset S of O that is...

I've been recently reading a book on abstract algebra and number theory, and I stumbled upon a problem that at first glance looked obvious, but I can't seem to figure out how to formally write the proof. 1.)So, let's say we have 4 integers, r,s,t,u, all greater than or equal to 1. Suppose...

how could we calculate the follwing integral ?? \int_{0}^{\infty} \frac{ K(x)}{Q(x)}dx here K(x) and Q(x) are POLYNOMIALS , of course if we had an integral over all R instead of (0 , \infty ) we could apply Cauchy's residue theorem i think there is a 'closed circuit' to perform the...

Homework Statement Show that the set of rational numbers in the interval (0, 1) cannot be expressed as the intersection of a countable collection of open sets. Homework Equations The Attempt at a Solution This sounds like something requiring proof by contradiction. There must be...

Homework Statement Show that the function f(x) = { x/2 if x is rational { x if x irrational is not differentiable at 0 Homework Equations If f is differentiable at 0 then for every e > 0 there exists some d > 0 such that when |x| < d, |(f(x)-f(0))/x - L | < e...

Are whole numbers rational numbers? If so, will any whole number divided by any other whole number result in a rational number?

Hi guys: I've got a problem I've been working on for some weeks and this might be the key to unlocking it. The question is: Given a vector in R^k, what is the measure of the set of vectors whose components are rationally dependent? Rationally dependent means for a given vector, you may...

Homework Statement Find the rational number representation of the repeating decimal. 1.0.\overline{36}Homework Equations The Attempt at a Solution I know it has something to do with infinite geometric sequences but I'm not sure what. what would your ratio be for a repeating decimal, I've...

Homework Statement The definite integral of (t^3 + t -1)/(sin(t)) from 2 to x^2 Homework Equations The Attempt at a Solution First off, I don't have the solution anywhere, my teacher just gave this to us to work on for the final exam review. I can think of a few things. I...

Homework Statement \int \frac{4x^5-1}{(x^5+x+1)^2} dx = ? Homework Equations The solution is - \frac{x}{x^5 + x + 1} The Attempt at a Solution Other than getting lucky and noticing immediately that this could be the derivative of a fraction, I do not see an easy way to solve this. The...

Homework Statement find \int\frac{x^{2}-2x-1}{(x-1)^{2}(x^{2}+1)}dx Homework Equations None The Attempt at a Solution \frac{x^{2}-2x-1}{(x-1)^{2}(x^{2}+1)} = \frac{A}{x-1} + \frac{B}{(x-1)^{2}} + \frac{Cx + D}{x^{2}+1} x^{2}-2x-1 = A(x-1)(x^{2}+1) + B(x^{2}+1) + (Cx + D)(x-1)^{2}...

Homework Statement "Prove that the sum of two rational numbers is a rational number." I just started on proof writing, so I'll just like to verify if I'm not missing anything here, and get some comments about the style. The attempt at a solution Theorem. If a,b \in \mathbb{Q} then a+b \in...

Ok so, This summer I will be taking a Pre-calc/trig course intensive, to get ready to take calculus in the fall, to start up my track for physics. I got a Pre Calculus Workbook For Dummies and I have to say so far I'm not too pleased. I have already found a bunch of typos, and when there...

Is there a way ( a theorem ) to find a rational number for a given irrational number such that it is an approximation to it to the required decimal places of accuracy. For example 22/7 is an approximate for pi for 2 decimal places.

Hi there I have a Rational function y = 1 / x^2-1 . I have a good idea what the graph looks it, it will have vertical asymptotes at -1 and 1 and I can work out the y intercept (-1 concave down). However I'm not sure about the other parts to the question. Homework Statement dy/dx...

Homework Statement Find the critical numbers of the function: F(x) = x^(4/5) (x - 4)^(2) Homework Equations None. The Attempt at a Solution I differentiated and got to (1 / 5th root of x) (x - 4)(2x + 4/5(x-4)) but I don't know how I can simplify the expression to be able...

Homework Statement Solve for x. Homework Equations 3/2 + 2/2x-4 = 1/x-2 The Attempt at a Solution LCD = 2(x-2) 3/2 + 2/2x-4 = 1/x-2 Mult. all terms by 2(x-2) 3x - 6 + 2 = 2 3x = 6 x = 2 What am I missing here?

Homework Statement For all a in the set of real numbers, if a is rational, a + \sqrt{2} is irrational. You may use that \sqrt{2} is irrational and the sum and difference of rational numbers is rational. Homework Equations The Attempt at a Solution My proof seems way too simple, I don't trust...

Homework Statement Hi, I have this function: f(x ) = 0 (x is irrational) or f(x) = 1/q for rational p/q in lowest terms. show that this function is not differentiable anywhere The Attempt at a Solution This is the answer from the solutions book: consider [(f(a+h) - f(a ) ) /...

Hey all, I'm new here so I'm a little noobish at the formatting capabilities of PF. Trying my best though! :P Homework Statement Let a, b, c, d \in Q, where \sqrt{b} and \sqrt{d} exist and are irrational. If a + \sqrt{b} = c + \sqrt{d}, prove that a = c and b = d. Homework...

How to state the equations of a rational functions with the following asymptotes? (1)x=2, y=-3 (2)y=0, x=4 (3)y=0

How do you multiply an equation that has 2 or more numerators. such as : 8x + 8 x - 1 ______ X _______ X2 - 2x + 1 2x + 2 And don't say anything like, "I'm not doing your homework for you" or anything stupid like...

Homework Statement Write the partial fraction decomposition of the rational expression. Check your result algebraically. (x2 – 7x + 16)/[(x + 2)(x2 – 4x + 5)] The Attempt at a Solution [A/(x+2)] + [(Bx+C)/(x2-4x+5)] x2-7x+16= A(X2-4x+5)+(Bx+C)(x+2)...

I have been playing around with numbers in different bases and then I thought, what if they were in fractional bases. I found a way to convert numbers to fractional bases and have been searching on the internet and not found a similar way to do this. Anyway, here is an example of how I would do...

Hello, I am trying to prove the following... lim (x+3) \left|x+5\right|/x+5 x\rightarrow-5 from the left, L=+2 from the right, L=-2 I used delta-epsilon on the right hand limit and got \delta = \epsilon However, I'm not sure how to proceed when I get to this step while trying to prove the...

In the field of formal rational functions, construct a nest of closed, bounded intervals whose intersection is empty. (That is, show that the Nested Intervals property fails in this field) I know it has to involve radical 2 but because that is the only number we know is irrational but other...

Hello, Just wondering if this is a correct way to say the set of RATIONAL numbers is countable: Rationals (Q) is countable , because for every Q = p/q , such that p & q are positive INTS (Z) and since the set of positive INTs (Z) is countable ( a 1:1 correspondence) Q is countable because it...

Homework Statement A security system costs $9/month in costs (fixed cost). If a new security system costs $1500 to purchase and last for 8 years: a)Determine the rational function that gives the annual cost of a security system as a function of the number of years you own the system. Using...

Homework Statement A scientist predicted that the population of fish in a lake could be modeled by the function f(t)= 40t/(t^2+1), where t is given in days. The function that actually models the fish population is g(t)=45t/(t^2+8t+7). Determine where g(t)>f(t). Homework Equations...

how can i find the range of a rational function for ex. y=1/x+1

Homework Statement A home security system costs $9/month in electricity costs (you may assume this is a fixed cost over the duration of the time involved in this question). A new security system costs $1500 to purchase and lasts for 8 years. Determine a rational function that gives the...

Homework Statement Evaluate the limit: lim_{(x)\rightarrow(2)}\frac{x^2+x-6}{sqrt(x+4)-sqrt(6)} Homework Equations N/A The Attempt at a Solution I've drawn the graph which indicates that the at x=2, y=0, so 0 would seem to be the limit. I could not, however, get the limit...

Homework Statement Decompose the rational expression into a sum of partial fractions: (x+1)/(3(x-2)2) I am familiar with the method of decomposing fractions into a sum of partials fractions (solving for A, B, C, etc.). What is confusing me is the coefficient 3 in the denominator. Do I...

Dear Forum, I've been trying to find a proof for the following: \int \frac{x}{a^2+x^2}dx = \frac{1}{2}\ln|a^2+x^2|+c After many hours I've resorted to asking for help - any ideas anyone? cheers, mazzo

Homework Statement Find the taylor expansion of the following formula in the case where r > > d to the first order in \epsilon = \frac{d}{r} \frac{1}{r_{+}} = \frac{1}{\sqrt{r^{2} + (\frac{d}{2})^{2} - rdcos\theta}} Homework Equations (1 + \epsilon)^{m} = 1+m\epsilon, where...

Homework Statement Let x and y be real numbers with x<y and write an inequality involving a rational number p/q capturing what we need to prove. Multiply everything in your inequality by q, then explain why this means you want q to be large enough so that q(y-x)>1 . Explain how you...

Homework Statement Prove: For every rational number z, there exists irrational numbers x and y such that x + y = z. Homework Equations by definition, a rational number can be represented by ratio of two integers, p/q. The Attempt at a Solution Is there a way to do this by...

There was a part c and d from a question I couldn't answer. Let R = \{ a/b : a, b \in \mathbb{Z}, b \equiv 1 (\mod 2) \}. a) was find the units, b) was show that R\setminus U(R) is a maximal ideal. Both I was successful. But c) is find all primes, which I believe i only found one...the...

Homework Statement Not so much a homework problem, but a problem that is annoying me because of its simplicity. Not all cubic polynomials with rational coefficients can be factorized by the rational root theorem (or is this false?). What I am finding hard to comprehend is how a cubic with...

Express \frac{1+\sqrt{2}}{3-\sqrt{2}} as a+b\sqrt{2} where a and b are rational numbers. I started by \frac{1+\sqrt{2}}{3-\sqrt{2}} * \frac{3+\sqrt{2}}{3+\sqrt{2}} But, I obtain \frac{5}{7}-\frac{4}{7}\sqrt{2} I believe that, here, a and b are rational, but is there a more tidy version? I...