University:
University of Michigan
Course:
MATH 115 | Calculus I
Academic year:

2024
Views:

306

Pages:

2
Author:

Luciano Kirby

Question 1 "A rational function is defined as the quotient of polynomials in which the denominator has a degree of at least 1" If we are talking merely about x, then I get the concept. A rational function f(x) could be written as „P(x) , where q(x) ≠ 0." q(x) The issue that I'm having is that of talking about rational functions of n variables. For instance, what would be the meaning of 'f(x,y) is a rational function of x & y'? Answer 1 The definition applies to n variables as well. For any two polynomials p,q, the function f(x₁, x₂,... xₙ) = p(x₁, x₂,... xₙ) / q(x₁, x₂,... xₙ) is a rational function (if q is not constant). For example, f(x, y) = x/y is a rational function of 2 variables. Answer 2 For instance, f(x, y) = p(x,y) / q(x,y) is a rational function of x and y if p and q are polynomials in x and y... So p and q are finite sums of terms of the form axⁿyᵐ, where a is a numerical coefficient and n and m are whole numbers... and, of course, the degree of q is 1 or more... Question 2 What is rational function and how do you find domain, vertical and horizontal asymptotes. Also what is "holes" with all limits and continuity and discontinuity? Answer A rational function is where there are xs Question 3 How do you write a rational function that has the following properties: a zero at x= 4, a hole at x= 7, a vertical asymptote at x= -3, a horizontal asymptote at y = 2/5 ? Answer A zero at x=4 means we have (x-4) as a factor in numerator; a hole at x=7 means, we have x-7 a factor both in numerator as well as denominator; a vertical asymptote at x=-3 means x+3 a factor in denominator only a horizontal asymptote at y = 2/5 means highest degrees in both numerator and denominator are equal and their coefficients are in ratio of 2:5 Hence desired rational function is (2(x-4)(x-7)) / (5(x-7)(x+3)) i.e. (2x² - 22x + 56) / (5x² - 10x - 105) See its graph down below. Observe vertical asymptote x=-3 and horizontal asymptote y = 2/5. We have a zero at x=4 as function passes through (4,0). Hole is not seen as x-7 cancels out, but we know, the function is not defined at this point. graph {(2x² - 22x + 56) / (5x² - 10x - 105), [-10.67, 9.33, -4.4, 5.6]} Question 4 Help me with regarding this rational function: f(x) = (x-3) / (-x+1) Answer (x-3) / (-x+1) = (-2) / (-x+1) + (-1) The first term can be any real but 0. Question 5 Help with figuring out the end behavior of a Rational Function. What is the end behavior of: f(x) = (2x² + 2) / (x² + 9) Any help would be good Answer f(x) = (2x² + 2) / (x² + 9) = (2(x² + 9) + 2 - 2(9)) / (x² + 9) = (2(x² + 9) - 16) / (x² + 9) = 2 - 16/(x² + 9) As x is large, the term in blue will vanish, as the denominator have far bigger magnitude compared to the numerator. Hence the limit would be 2. Of course, not every rational number will have their numerator and denominator to have the same degree. Try to write p(x) / q(x) = t(x) + p₁(x) / q₁(x) where degree of p₁(x) is smaller than degree of q₁(x). As x is huge, p₁(x) / q₁(x) becomes close to 0 and it will behave like t(x).