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Sec 5: Vertical Asymptotes & the Intermediate Value Theorem

Sec 5: Vertical Asymptotes & the Intermediate Value Theorem. 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.

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Sec 5: Vertical Asymptotes & the Intermediate Value Theorem

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  1. Sec 5: Vertical Asymptotes & the Intermediate Value Theorem

  2. 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. Definition of a Vertical Asymptote

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

  4. 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. INTERMEDIATE VALUE THEOREM

  5. 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 1

  6. 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. Example 2

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

  8. 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 Start Unit 2 Test Thursday

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