Rational Definition and 616 Threads

Homework Statement Prove that in any interval there exists a rational z. Homework Equations The Attempt at a Solution My professor wrote this for me when trying to explain how to prove this: a \notin Q, \epsilon rational [r, s]\in a l([r, s])<\frac{\epsilon}{2}...

Find an equation of a rational function f(x)that satisfies the conditions. We're allowed to use a calculator, by the way. (: Okay, conditions: vertical asymptote: x=-5, x=0 Horizontal asymptote: y=0 x-intercept=7; f(1)=4 On the test i had no idea how to do it, but after seeing...

Hello all, I have recently been wondering whether there is a way to determine a fraction for which the decimal expansion is a cycle of n numbers? I would like to be able to work this out myself, but I can't wait until I start my mathematics degree. So any help would be greatly appreciated...

[FONT="Trebuchet MS"]Has anyone worked out a functional Parametric chart that enables one to choose a appropriate Microscope (exclude Scanning electron MicroScopes(MS)) amongst the multitude of choices for a modern academic micro-biological & research lab, hence formulated a general guideline...

Homework Statement Solve \int_-1^1 (x^{4/3} + 4 x^{1/3}) dx Im having difficulties in algebra when solving this problem. Homework Equations The Attempt at a Solution \int_-1^1 (x^4/3 + 4 x^1/3) dx = 3/7 (1)^7/3 + 4 3/4 (1)^4/3 - 3/7 (-1)^7/3 + 4 3/4 (-1)^4/3 = 3/7 +...

Homework Statement 6x (squared) +17x+7 _________________ 2x sqaured + 7x +3 Homework Equations Simplify The Attempt at a Solution 3x+7 is the answer ____ x+3 maybe u can factor 6x by doing 2x (3x) or maybe u don't do that and take 6x (squared +17...

Homework Statement x (squared) + x - 30 _____________________ x (sqaured) -3x + 2 Find all numbers for which the rational expression is not defined. i know this is really easy stuff, but i have a hard time staying focused in math, and i have to always teach myself with the book, but sometimes...

rational function helpp needed immedietly ! 1. when astraunauts go into space, they feel lighter. this is because weight decreases as a person rises above Earth's gravitational pull according to the formula W(h) = We/(1+h/6400)^2 where We is the person's weight in Newtons, at sea level on...

Homework Statement Evaluate the limit, [SIZE="3"]WITHOUT using l'Hôpital's rule: \lim_{x \rightarrow -1} \frac{x^{1/3} + 1}{x^{1/5} + 1} Homework Equations The Attempt at a Solution I tried to use the conjugate method which does not produce a useful outcome...

Homework Statement Let S={p+q\sqrt{2} : p,q \in Q} and T={p+q\sqrt{3} : p,q \in Q}. Prove that S\capT = Q. Homework Equations See above. The Attempt at a Solution I was thinking possible using S\capT=Q S + T - S\cupT = Q But I have no idea how to combine them? I don't believe it's...

Homework Statement This problem is insanely intuitive. Define f : (0,1) \rightarrow \Re by f(x)=\begin{cases} 1/q&\text{if } x \neq 0 \text{, is rational, and }x = p/q \text{in lowest terms}\\ 0&\text{otherwise }\end{cases} Suppose \epsilon > 0. Prove that there are at most a...

Prove that if c and d are positive, then the equation \frac{c}{x-2} + \frac{d}{x-4} = 0 has at least one solution in the interval (2,4). I try to use the following theorem: If f is continuous on [a,b] and if f(a) and f(b) are nonzero and have opposite signs, then there is at least one...

Rational and irrational numbers. (semi-urgent) I need to figure this out by tomorrow =/ Homework Statement a. If a is rational and b is irrational, is a+b necessarily irrational? What if a and b are both irrational? b. If a is rational and b is irrational, is ab necessarily irrational...

The problem is: ((x^3)+x)/(x-1) And i need to break it into partial fractions... I tried long division and got: ((x^2) +x ) But the book gives me the answer of: (x^2)+x+2+(2/(x-1)) Any help would be very much appreciated, thanks.

Homework Statement Simply these rational functions: [\sqrt{(X^2)+12}-4]/(X-2) (2-\sqrt{(X^2)-5})/(X+3) (X-1)/(\sqrt{X+3}-2) Homework Equations The only example in the book used the technique of multiplying the numerator and denominator by the function p(x) if p(x) is the function...

Hey guys I'm wondering if someone could hep me solve this integral. I've been working at it for a few days now (as part of a project I'm doing over the summer) and have gotten stuck. I think I need to make some substitution but I can't see what it is to make. -\int\frac{dI}{I(R+BI+CI^2)} I...

Homework Statement Y1=(x3+3x2-4)/(x2)Homework Equations Slant Asymptote at y=x+3. (X-1) and (X+2) appear to be intercepts in the back of the book. How do I factor the numerator to get that?The Attempt at a Solution I know this is simple and there is a method to find the zeros of the numerator...

Hi all. I have always wondered: If we e.g. look at functions given by f(x) = \frac{\cos x}{x^2}, \quad g(x) = \frac{\sin x}{x^2}, \quad h(x) = \frac{\exp x}{x^2}, then does the degree of the denominator exceed the degree of the nominator by 1 or by 2?

Homework Statement Hi all. My question has to do with integrating rational functions over the unit circle. My example is taken from here (page 2-3): http://www.maths.mq.edu.au/%7Ewchen/lnicafolder/ica11.pdf We wish to integrate the following \int_0^{2\pi } {\frac{{d\theta }}{{a + \cos...

Homework Statement a) 2x^4 - 15x^3 + 23x^2 + 15x - 25 = 0 b) 12x^3 - 20x^2 + 23x - 10 = 0 Homework Equations The Attempt at a Solution I was wondering if you guys could check my answers. For A: Possibles: +/- {1, 5, 25} / {1, 2} Actual: 5, 5/2

I was wondering what Q/Z, or the rational numbers modulo the integers. I am struggling to visualize what the cosets may be. Thank you for your time.

what are the x and y intercepts y= x+2/x-7 rational expression is y=-2/7 and x= -2

Hi all This is my first post so please be gentle with me! Limit of this rational function as x approaches infinity? f(x) = (x^3 - 2x)/(2x^2 - 10) I was under the impression that if the degree of the polynomial of the numerator exceed that of the denominator then there could be no...

Homework Statement Use linear approximation to approximate sqrt((3.2)^2 + 2(2.1) + (4.3)^2) with a rational number (a ratio of integers). Homework Equations f(x,y) = sqrt(x^2 + 2y + z^2) f(x,y) = (x^2 + 2y + z^2)^1/2 The Attempt at a Solution x = 3 ∆x = 2/10 y = 2 ∆y =...

I've done a calculus problem. The LHS is my answer and the RHS is the answer in the book. \frac{2(x-1)^2}{(x^2-1)^2} = \frac{2}{(x+1)^2} What was done to simplify this equation?

Homework Statement Find an equation of a rational function f that satisfies the given conditions: vertical asymptotes: x=-3, x=1 horizontal asymptote: y=0 x-intercept: -1; f(0)= -2 hole at x=2Homework Equations The Attempt at a Solution Vertical Asymptotes are (x+3) and (x-1). Also (x-2)...

Suppose a,b,c,d are integers and a DOES NOT equal c. Suppose that x is a real number that satisfies the equation: (ax+b)/(cx+d)=1 Must x be rational? If so, express x as a ratio of two integers. I have no idea how to begin this problem.

Homework Statement 3x+1 2x-4 > 0 The Attempt at a Solution My answer came to be (x< 1/3) U (x>2)

I wanted help on solving Solving Rational Inequalities. I watched this video and i was wondering if that was the best or easiest way. Our teacher taught us a much harder and weirder way using multiple number lines instead of one. If you do watch the video the man shows examples with open dots...

Homework Statement Homework Equations Idk what to do after I multiply the LCD I get confused please help me. The Attempt at a Solution All I know is that the LCD is (x+5)(x+1)(x-2) and when I multiply into every fraction I get this. (X+5)(X-2)(X-2) - (X+5)(X+1)(X+1)...

Homework Statement [(4x-4x2)/x2+2x-3] * [(x2+x-6)/4x] Homework Equations The Attempt at a Solution I'm so lazy, though, I do all this stuff mentally, so... Oby-kaby. I individualized all groups into smaller units(I don't know how to say that in algebrish), by dividing by least...

Homework Statement Let n/m be a positive rational number in lowest terms. By examining the long division algorithm, show that the decimal expansion of n/m is eventually periodic, and that the period divides phi(m). For simplicity,you may assume that (m, 10) = 1. Homework Equations...

Hi, I was wondering whether two rational functions f,g whch coincide on the unit circle actually coincide on all of C. I would say yes. Let D be the set of all complex numbers with the poles of both f and g removed (let's assume there are no poles on the unit circle). This is then open...

I'm trying to compose f(x)= x + \frac{1}{x} with itself. e.g. f \circle f I have x + \frac{1}{x} + \frac{1}{x+(1/x)} Now I multiplied \frac{1}{x + (1/x)} * \frac{x}{x} and I got: x + \frac{1}{x} + \frac{x}{x^2+1} This is not the correct answer according to the book. Books is...

Hello, I found one excercise - convert a rational number 63/64(base - 10) to a number system with a base of 4 using Radix conversion. Searching throught the internet i found this formula (i hope it`s the correct one :) ) - http://img255.imageshack.us/img255/903/races3.jpg Unlike...

Homework Statement Prove by contradiction: If a and b are rational numbers and b != 0, and r is an irrational number, then a+br is irrational. In addition, I am to use only properties of integers, the definitions of rational and irrational numbers, and algebra. You guys should also know that...

Hey all! It's been a while since I've done this, how do you integrate a rational function, where the denominator cannot be factored, again? For example, \int \frac{x}{x^{4}-1} dx Thanks, in advance!

Homework Statement i've tried many partial fraction methods but none of y answers are correct in the end, please help me evaluate the integral for f(x)= (10x+2)/(x-5)(x^2 + 1) Homework Equations there are no relevant equations given The Attempt at a Solution A/x-5 + Bx+C/x^2 +1

Evaluate the integral: int ((x+1)/(x-1)) dx is this possible? i tried searching for this but there was no answer

Hi all, i found this problem in a topology book, but it seems to be of an analysis flavour. I'm stumped. Show that the collection of all points in R^2 such that at least one of the coordinated is rational is connected. My gut says that it should be path-connected too (thus connected), but...

So one of my problems is the square root of 768/square root of 384. I tried multiplying it by the square root of 384/square root of 384, but i ended up with a crazy answer. Should I simply the radicals first and then try it?...never mind, I divide radical 768/radical 384 to get radical 2, right...

Homework Statement Here's the question: int(1/(x(x^2+3)*sqrt(1-x^2)))dx Homework Equations According to the textbook, the answer should be: (1/3)*ln((1-sqrt(1-x^2))/x)+(1/12)*ln((2+sqrt(1-x^2))/(x^2+3))+C The Attempt at a Solution 1) let t = sqrt(1-x^2), so dt = (-x)/sqrt(1-x^2)...

Homework Statement I have a picture of a rational function and I need help finding out the equation of the function. Homework Equations Vertical Asymptote/ (Bx+C) The Attempt at a Solution I know I have to find X and Y intercepts, which is what I did. But how do I find the...

prove that the square root of 2 plus the square root of 3 is not rational? does always the sum of two not rational numbers is a not rational number? i know the proof 2 = a^2/b^2 i separately proved that square root of 2 and square root of 3 are irrational how two prove that the sum...

My textbook says the following: For a closed interval J_n = [a_n, b_n] "A nested sequence of rational intervals give rise to a separation of all rational numbers into three classes (A so-called Dedekind Cut). The first class consists of the rational numbers r lying to the left of the...

Consider the set of rational numbers, under the usual metric d(x,y)=|x-y| I am pretty sure that this space is totally disconnected, but I can't convince myself that the set {x} U {y} is a disconnected set. It seems obvious, but I can't find two non-empty disjoint open sets U,V such that U...

Hello. Between a rational and a irrational is there a rational? and a irrational? and vice-versa? I know that between 2 rationals there is a rational and a irrational and that between 2 irrationals there is a rational and a irrational, but i cannot figure this out... please help. Thanks.

Hello! I've been solving a few of these problems but I'm stuck on this one, trying to simplify one of the steps. Homework Statement Find dy/dx of: y = x(x^2 +1)^1/2 Attempt at a solution y1 = x (1/2)(x^2 + 1)^-1/2 * (2x) + (x^2 +1)^1/2 * 1 I get the the equation above but I have...

Homework Statement prove that it is possible that an irrational number raised to another irrational, can be rational. you are given root2 to root2 to root2 Homework Equations The Attempt at a Solution i have shown that root2 to root2 to root2 is rational, but would appreciate a...

Cross multiply ... and then multiply by the common denominator... how do you do it do you do it in your head..by cross multiplying or the other way It seems like cross multiplying is less formal ...and I don't know if its 'right' to take the shortcut --since you have to do this...