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

2024
Views:

291

Pages:

2
Author:

Luciano Kirby

Question 1 Do all rational functions have vertical asymptotes? Why or why not? If not, give an example of a rational function that does not have a vertical asymptote. Answer All rational functions MUST have at least 1 vertical asymptote. The reason for this is that the polynomial in the denominator of the rational function when equated with 0, yields at least 1 value of x on which the denominator becomes 0 and the function approaches + and -∞ Question 2 Graph the rational functions. Include the graphs and equations of the asymptotes and dominant terms. (x+3) / (x+2) Answer (x+3) / (x+2) = (x+2) / (x+2) + 1/(x+2) = 1 + 1/(x+2) It has a vertical asymptote at x=-2. 1/(x+2) is dominant near the asymptote, and 1 is dominant as x goes to ∞ or -∞. (It has the same horizontal asymptote, y=1.) Question 3 Write two different rational functions whose graphs have the same end behavior as the graph of y = 3x² Answer y = 3x² Pick some factor to multiply top and bottom y = 3x². { (x+2)/(x+2) = (3x³ + 6x²) / (x+2) } This adds a "hole" at x = -2 since that would be undefined, but otherwise behaves exactly like y = 3x² because of course the factors cancel out. Pick another factor to multiply top and bottom y = 3x². { (x-5)/(x-5) = (3x³ - 15x²) / (x-5) } "Hole" at x=5, but otherwise the same as y = 3x². A simpler option could be to just add some constant to 3x² Question 4 (a) How do you find vertical asymptotes of rational functions? (b) Let s be the rational function s(x) = (anxⁿ + an-₁xⁿ⁻¹ + ... + a₁x + a₀) / (bmxᵐ + bm-₁xᵐ⁻¹ + ... + b₁x + b₀) How do you find the horizontal asymptote of s? (c) Find the vertical and horizontal asymptotes of f(x) = (5x² + 3) / (x² - 4) Answer a) We find a vertical asymptote of a rational function such that we find a rational zeros of the denominator i.e the line x = a is a vertical asymptote of a rational function, where a is zero of the denominator. b) Let s(x) be a rational function s(x) = (anxⁿ + an-₁xⁿ⁻¹ + ... + a₁x + a₀) / (bmxᵐ + bm-₁xᵐ⁻¹ + ... + b₁x + b₀) then If n < m, then s(x) has horizontal asymptote y = 0, If n = m, then s has horizontal asymptote y = an / bm If n > m, then s has no horizontal asymptote. c) We want to find the denominator in the factored form f(x) = (5x² + 3) / (x² - 4) Since, we can express the denominator in the factored form x² - 4 = (x - 2)(x+2) we can see that zeros 2 and -2, therefore from the part a) the lines x=2 and x=-2 are vertical asymptotes. We can see that n=2 and m=2 i.e n=m. So, from the part b) we obtain that the horizontal asymptote is y=5. Question 5 The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x: Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits. lim (x-3) / √(4x² + 25) x->∞ Answer Highest power of x in denominator is x², but since it is beneath square root, we are going to divide both numenator and denominator with x. lim (x-3) / √(4x² + 25) x->∞ = lim (1 - (3/x)) / √(4 + (25/x²)) = (1-0) / √4+0 = 1 / 2 x->∞