Rational Definition and 616 Threads

Homework Statement ##f(x)## is a continuous and differentiable function. ##f(x)## takes values of the form ##^+_-\sqrt{I}## whenever x=a or b, (where ##I## denotes whole numbers) ; otherwise ##f(x)## takes real values. Also, ##|f(a)|\le |f(b)|## and ##f(c)=-1.5##. Graph of ##y=f(x)f'(x)##: The...

Homework Statement Determine a positive rational number whose square differs from 7 by less than 0.000001 (10^(-6)) Homework Equations - The Attempt at a Solution Let p/q be the required rational number. So, 7> (p/q)^(2) > 7-(0.000001) ⇒ √(7) > p/q > √(7-.000001) ⇒√(7) q> p >...

Because technically, the numerator is smaller than the denominator as −2<−1 I know it's an extremely stupid question. I mean I know that I can just multiply −1 to the numerator and the denominator and I'll get 2/1 which is greater than one. But what exactly is happening here? A number that...

$$\int\frac{\sec^2 \left({t}\right)}{\left(1+\tan\left({t}\right)\right)^3}dt$$ $$u=1+\tan\left({t}\right)\ du=\sec^2\left({t}\right) dt$$ So far ok?

$$\int\frac{x^4}{4+x^2}dx$$ this was homework for a section of expanding rational expressions $$\frac{x^4}{4+x^2}=x^2+\frac{16}{x^2 +4}-4$$ I don't see how W|F got this expansion?

I am asked: Prove that each of the following is an ideal of $\mathcal{F}(\mathcal{R})$: a. The set of all f such that f(x)=0 for every rational x b. The set of all f such that f(0)=0 My question is how do I know what the multiplicative operation is within the ring? Is multiplication the...

Homework Statement Homework Equations y = f(x) y=k(x+4)(x)(x-6) y=1/f(x) y= 1/ (k(x+4)(x)(x-6)) The Attempt at a Solution I'm more looking for clarification on how people would approach this. There is no explicit point given to deduce the value of k to determine the vertical stretch or...

Hi, 1. Homework Statement Q : A diffraction grating with 10000 lines per CM is illuminated by yellow light of wavelength 589 nm, At what angles is the 2nd order bright fringes seen ?Homework Equations From my textbook , I got this equation , d sin theta = m (λ) The Attempt at a Solution Ok so...

Homework Statement (4a/a+4)+(a+2/2a) Homework Equations Just combine and then factor out The Attempt at a Solution It's actually fairly simple, but I'm having difficulty at the end. /multiply each term by opposite denominator 4a(2a)/a+4(2a) + a+2(a+4)/2a(a+4) /combine 4a(2a)+(a+2)(a+4) /...

Homework Statement Without solving the equation 3x^2-8x-3=0 show it has 2 different rational roots.[/B]Homework Equations ax^2+bx+c=0 The Attempt at a Solution I would appreciate if someone would check my work, and advise if I have done the right or wrong thing? Thank you, Jaco [/B]...

What does it mean to take positive rational numbers to whole-number powers?

Pag 42 says: Seems inconsistent. in (9): p = x +x²+x.sin²x q = x.sin x + x.sin²x But by definition p and q are not polynomial functions. It is a mistake in the book?

Homework Statement What is Lim (1+x2)/(4-x) as x approaches 4 from the left? Prove using the definition. Homework EquationsThe Attempt at a Solution Well x≠4. Function approaches positive infinity as x approaches 4 from the left side. Let m>0 and 0<x<4. Then (1+x2)/(4-x) > x2/(4-x) > x/(4-x) >...

Is $\sin\left(10^{\circ}\right)$ rational or not? Prove it.

For example, say we have ##\frac{x^4(x - 1)}{x^2}##. The function is undefined at 0, but if we cancel the x's, we get a new function that is defined at 0. So, in this case, we have ##x^2(x - 1)##, then ##x^2(x - 1)(1)##, and since ##\frac{x^2}{x^2} = 1##, we then have ##\frac{x^4(x - 1)}{x^2}##...

How do you do it

Homework Statement A curve is given by the parametric equations ##x=t^2 +3## ##y=t(t^2+3)## Find dy/dx in terms of t and show that (dy/dx)^2 >=9 Homework Equations Parametric derivatives The Attempt at a Solution Using the chain rule, I arrived at...

I am reading Chapter 1:"Real Numbers" of Charles Chapman Pugh's book "Real Mathematical Analysis. I need help with the proof of Theorem 7 on pages 19-20. Theorem 7 (Chapter 1) reads as follows: In the above proof, Pugh writes: " ... ... The fact that $$a \lt b$$ implies the set B \ A...

Hello, I am looking to evaluate: $$I = \int_{0}^{1} \frac{x^4(1-x)^4}{1+x^2} dx$$ I will use a rectangular contour. The image looked weird here so the upload of the image is here: http://i.stack.imgur.com/W4BfA.jpg $R$ is more like the radius of the small semi circle, we have to let $R \to...

Hello, Evaluate: $$\int_{0}^{\infty} \frac{\cos(x)}{x^2 + 1} dx$$ We know that because $f(x)$ is even:$$\int_{0}^{\infty} \frac{\cos(x)}{x^2 + 1} dx = \frac{1}{2} \cdot \int_{-\infty}^{\infty} \frac{\cos(x)}{x^2 + 1} dx$$ Consider a complex function, with $z = x + iy$ $$f(z) =...

##cosθ + isinθ = cos(θ + 2kπ) + isin(θ + 2kπ)## for ##k ∈ ℤ## ##[cosθ + isinθ]^n = [cos(θ + 2kπ) + isin(θ + 2kπ)]^n## ##cos(nθ) + isin(nθ) = cos(nθ + 2nkπ) + isin(nθ + 2nkπ)## ##cos(nθ + 2mπ) + isin(nθ + 2mπ) = cos(nθ + 2nkπ) + isin(nθ + 2nkπ)## for ##m ∈ ℤ## Now consider the special case ##n =...

What is the easiest method for adding and subtracting rational numbers

I have a question which comes from Rudin's Principles of Mathematical Analysis; specifically, from the introduction. In example 1.1, the author clearly shows that no rational numbers satisfy the equation ##p^2 = 2##. So, I am trying to imagine myself in a scenario in which I am in a time before...

When you have a rational function, such as: 3x-5/x-1 After attaining things like the x and y intercepts and asymptotes, how do you know how many "pieces" of the graph there are? With linear functions/equations, you know it's a single line. Even quadratic graphs are a single piece - albeit...

Nancy's lawn care specializes in residential lawn cutting and fertilizing. Nancy the owner has tracked her income and expenses. She has determined that her profit can be represented by the expression $5A, where A is the area of the lawn in square metres. The time in hours that it takes her to...

Hello all I have a general question. When I look for a limit of a rational function, there is this rule of dividing each term by the highest power. I wanted to ask if I should divide by the highest power, or the highest power in the denominator, and why ? I have seen different answers in...

This number is rational and normal, right? http://www.wolframalpha.com/input/?i=0.01234567890123456789... edit - You'll have to edit in the ".." because the forum doesn't recognize it as part of the link.

Homework Statement Consider the integral: \int\frac{2x^3 -4x^2 +8x +7}{(x-1)^2 (x^2 +4x +8)}{\rm{d}}x Homework EquationsThe Attempt at a Solution The degree of the denominator is 4 and the numerator's is 3, hence I thought I would try partial fractions: \frac{A}{x-1} +\frac{B}{(x-1)^2}...

Hello! (Wave)I have to check if the equation $3x^2+5y^2-7z^2=0$ has a non-trivial solution in $\mathbb{Q}$. If it has, I have to find at least one. If it doesn't have, I have to find at which p-adic fields it has no rational solution.Theorem: We suppose that $a,b,c \in \mathbb{Z}...

Hi! (Smile) I have to find for which primes $p$, the equation $x^2+y^2=3z^2$ has a rational point in $\mathbb{Q}_p$. According to my notes: Obviously, $\forall p \in \mathbb{P}, p \nmid 2 \cdot 3$, there is a rational solution in $\mathbb{Q}_p$. But,why is it obvious that the equation has a...

My https://www.amazon.com/dp/0073532320/?tag=pfamazon01-20 gives a rule of thumb to divide by the highest power in the denominator for the following problem to demonstrate a slant (oblique) asymptote: \lim_{x\to\infty} \frac{4x^3+5}{-6x^2-7x} = \lim_{x\to\infty}...

1. The problem statement, all variables and given/known dat If f and g are differentiable functions with f(O) = g(0) = 0 and g'(O) not equal 0, show that lim f(x) = f'(0) x->0 g(x) g'(0) The Attempt at a Solution I know that lim as x→a f(a) = f(a) if function is continuous. since its...

Dear All, Please help me understand how ## \sqrt{2} ## divide by 0 is rational as stated in the excerpt from alan F beardon's book?

Im trying to find the integral of ( sec(t)^2 ) / ( (tan(t)^3) + (tan(t)^2) ). I've managed to get the integral into the form 1 / (u^3 + u^2) where u = tan(t), however I am having difficulty proceeeding from there. Could someone take a look at the working out I have attached and let me...

prove that $\sqrt[3]{45+29\sqrt{2}}+ \sqrt[3]{45-29\sqrt{2}} $ is rational

Homework Statement mtan for f(x) = 5/√ 3x ... at x=1 Homework Equations msec = y2-y1 / x2-x1 The Attempt at a Solution The two points I got from the equation: (1, 5/√ 3) and (1+h, 5/√ 3+h) msec = f(1+h) - f(1) / h = (5/√ 3+h - 5/√ 3) / h ... multiply top and bottom by denominators (√ 3+h)...

Is there a way to distinguish between rational functions that have the same limit at both ends and those that don't? I think I might have answered my own question, but let's say I evaluate a rational function, and it turns out to be a coefficient ratio with no variables (3/2). Does that mean...

List all values of x for which each rational expression is undefined: $$\frac{x^2-9}{x^2-3x-10}$$ Answer is 3, -3 q1) Are these -, and positive answer interchangeable ones because it is a rational expression so when I see rational that's going to signal it's going to need 2 answers (either...

I have a question in math I need help with please: Show that the product of a rational number and it's inverse is equal to 1, with one exception. what is the exception? can anyone help please?

is ln(√eπ)/π a rational number? where π =3.14...

Hello, I am having difficulty solving my math problems. Simplify the expression: (12+r-r^2)/(r^3 +3r^2) The answer is (4-r)/r^2 I know that i can expand 12+r-r^2 as (-r+4)(r+3) But i cannot figure out the rest. Please help me. Thanks

$$g(x)=\begin{cases}x^2, & \text{ if x is rational} \\[3pt] 0, & \text{ if x is irrational} \\ \end{cases}$$ a) Prove that $\lim_{{x}\to{0}}g(x)=0$ b) Prove also that $\lim_{{x}\to{1}}g(x) \text{ D.N.E}$ I've never seen a piecewise function defined that way...hints?

Homework Statement Find all integer roots that satisfy (3x + 1)/(x - 4) < 1. The Attempt at a Solution I would do this: Make it an equation and find x such that (3x + 1)/(x - 4) = 1. 3x + 1 = x - 4 2x = -5 x = -5/2 Then check if the inequality is valid for values smaller than x and for...

If its not problem for you to check these last two. I got no more of these. If you have some kind a book with tasks like this on internet I would love to print it out so I can have some fun. 1ST answer 20/3 2ND answer -20/3

My first question is: is this formula (at the bottom) a known formula? In this subject i haven't explained how i build up the formula. So far i think it is equal to the gamma function of Euler with \Gamma\left(\frac{m_1}{m_2}+1\right)= \frac{m_1}{m_2}\ ! with m_1 , m_2 \in...

Show that for $p,q,r$ if $pq+qr+rp=1$ then $(p^2+1)(q^2+1)(r^2+1)$ is square of rational number

Hello All, Padmanabhan's discussion of dynamics mentions that in general the two dimensional harmonic oscillator fills the surface of a two torus. He further notes that there will be an extra isolating integral of motion provided that the ratio of frequencies is a rational number. This last...

The Problem Let x and y be real numbers such that y<x, using the Dedekind cut construction of reals prove that there is always a rational q such that y<q<x What I've done Since I can associate a cut to every real number, let x^∗ be the cut associated to x and y^∗ the one associated...

Is there a rational function,not series, that approximates e^x ? for example (x+1)/(x+3)

can you explain what it means when they are not "Integral" and not "Rational"? $\frac{4y}{x} = 4yx^{-1 }$ is not integral in x $3x\sqrt{y}z^3$ not rational in y