AB Calc Sect Notes Monday, November 28, 2011

Transcription

1 Assignments & Opportunities: I will TRY to have Sketchpad projects back to you next Monday or Tuesday. Tomorrow: p268; 5,22,27,45 & p280; 9 AB Calc Sect Notes Monday, November 28, 2011 Today's Topics & Class Plan: Do warm-up immediately w/ your partner In large group, we will complete the Notes & Sample Problem handout for Sections 4.2 & Extrema and Derivative Tests Study and be able to use AND identify each idea and theorem from these notes. AB Calc Assignment Page Warm-up: Warming Up to Derivatives Let f(t) be the temperature at time t where you live and suppose that at time t = 3 you feel uncomfortably hot. How do you feel about the given data in each case below? 1. f (3) = 2, f (3) = 4 2. f (3) = 2, f (3) = 4 3. f (3) = 2, f (3) = 4 4. f (3) = 2, f (3) = 4. Notes 4.3: Deriv. Tests AB Calculus - Hardtke Name 1. Rolle's Thrm: If f is differentiable over [a, b] and if f( ) = f( ), then f'(c ) = for at least one number c in. Chapter 4 Page 1

2 Notes 4.3: Deriv. Tests Name 1. Rolle's Thrm: If f is differentiable over [a, b] and if f( ) = f( ), then f'(c ) = for at least one number c in. 2. Mean Value Theorem (MVT): If f is differentiable on [a, b], then there exists at least one c in (a, b) such that f'(c ) = or equivalently, f(b) - f(a) = f'(c )(b - a). 3. Increasing/Decreasing Test: i. If f'(x) > 0 for every x in (a, b), then f is (increasing or decreasing) on [a, b]. ii. If f'(x) < 0 for every x in (a, b), then f is (increasing or decreasing) on [a, b 4. First Derivative Test: Let c be a critical number of a continuous function f. i. If f' changes from positive to negative at c, then f(c) is a of f. ii. If f' changes from negative to positive at c, then f(c) is a of f. iii. If f' does not change signs at c, then f(c) is not a of f. 5. Definition of Concavity: The graph of f is i. Concave (Upward or Downward) if f' is increasing on I (i.e., interval I) ii. Concave (Upward or Downward) if f' is decreasing on I 6. Test for Concavity: The graph of f is i. Concave (Upward or Downward) if f"(x) > 0 on I ii. Concave (Upward or Downward) if f"(x) < 0 on I 7. Definition of Pt of Inflection: A point (c, f(c) ) on a graph of f is a point of inflection if: i. f is continuous at c AND ii. There is an open interval (a, b) containing c such that the graph is concave upward on (a, c) and concave on (, ), or vice versa. *You can shorten this by saying "the concavity changes at P(c, f(c) )." 8. Second Derivative Test: Suppose that f is continuous near c. i. If f (c) = 0 and f"(c) < 0, then f has a local at c. ii. If f (c) = 0 and f"(c) > 0, then f has a local at c. Warning: If f"(c) = 0, the second derivative test is not applicable. In such cases, use first deriv. test. 9. Closed Interval Test: When a problem involves a closed interval, don't forget that global extrema can occur at or at either, so you must compare these -coordinates. 10. IVT: Given f is cont. over [a, b], for any N such that f(a) N f(b), there exists a c in (a,b) such that f(c) = N. 11. Extreme Value Theorem: Given f is cont. over [a, b], then f must have an absolute max & absolute min in [a, b]. Chapter 4 Page 2

3 12. In which graph below are the slopes of the tangents increasing? Does this coordinate with CU or CD?. 13. Visualize the Second Derivative Test using graphs A and B above. If f (c) = 0 in a CD interval, could (c, f(c)) be a local min? 14. Use the second deriv. test to find local extrema of f(x) = x 2 - x 4, inflection pts and concavity. Step 1: f'(x) = Thus, crit #s are: Step 2: f"(x) = Now evaluate f" at the crit numbers: Step 3: Use signs of f" & plug into f to determine the appropriate y-coordinates for local maxima: local minima: Step 4: To locate pts of inflection & concavity, solve f" for zero and examine sign of f"(x) in each interval: Chapter 4 Page 3

4 15. Use the First Derivative Test to find the extrema of f(x) = x3 6x2 + 9x + 2 over [, ]. Step 1: f'(x) = Thus, crit #s are: Step 2: Check where f changes signs on each interval determined by the crit #s. 16. Find the extrema of f(x) = x3 6x2 + 9x + 2 over [ 1, 4]. (Same function as in question 15). Chapter 4 Page 4

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