Sec 5: Vertical Asymptotes & the Intermediate Value Theorem

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Presentation transcript:

Sec 5: Vertical Asymptotes & the Intermediate Value Theorem

Definition of a Vertical Asymptote If f(x) approaches ±∞ as x approaches c from the left or right, then the line x = c is a vertical asymptote. Vertical Asymptotes can be determined by finding where there is non-removable discontinuity in a rational function.

Ex 1: Determine all Vertical Asymptotes A. C. B. D.

INTERMEDIATE VALUE THEOREM If f is continuous on [a, b] and k is any number between f(a) and f(b), then there is a c in [a, b] such that f(c) = k.

Example 1 Use the IVT to show that f(x) = x³ + 2x – 1 has a zero (x-intercept so y = 0) in the interval [0, 1].

Example 2 Verify that the IVT applies to the indicated interval and find the value of c that is guaranteed by the theorem. f(x) = x² - 6x + 8 on the interval [-1, 3] where f(c) = 0.

HOMEWORK Pg 85 #9-15 odds Pg 78 #83, 84, 95 *check answers in the solution manual

Start Unit 2 Test Thursday Calculating limits with a table (numerically) Finding limits with a graph Finding limits analytically: substitution, rationalization, factoring Properties of Limits Two Special Trig Limits Continuity & Discontinuity: Removable & Non-Removable One-Sided Limits Existence Theorem Intermediate Value Theorem Infinite Limits & their Properties