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Objectives: 1. Be able to apply the first derivative test to find relative extrema of a function Critical Vocabulary: Increasing, Decreasing, Constant Warm Up: Determine (using calculus) where each function is increasing, decreasing, and constant. 1. f(x) = 2x3 - 3x2 - 36x + 14 2. f(x) = (x2 - 4)2/3 x4 1 3. f ( x) x2 The previous lesson discussed the behavior of an interval (increasing, decreasing, and constant). This lesson is going to discuss the behavior of a critical number. If an interval changes from increasing to decreasing (or decreasing to increasing), then what is happening at the critical number that separates the intervals? Interval -∞ < x < 0 0<x<1 1<x<∞ Test Value x = -5 x=½ x = 25 Sign of f’(x) f’(x) = (+) Conclusion Increasing f’(x) = (-) Decreasing f’(x) = (+) Increasing Let c be a critical number of a function f that is continuous on the open interval I containing c. If f is differentiable on the interval, except possibly at c, then f(c) can be classified as follows: 1. If f’(x) changes from negative to positive at c, then f(c) is a _____________________ of f. 2. If f’(x) changes from positive to negative at c, then f(c) is a _____________________ of f. 3. If f’(x) does not change signs at c, then f(c) is neither a ______________________ of f. Summary: 1. Use increasing/decreasing test to find the intervals. 2. Use the First Derivative Test to determine if a critical number is a relative max or min. Directions: For the following exercises, find the critical numbers of f (if any). Find the open intervals on which the function is increasing or decreasing. Locate all the relative extrema. Use a graphing calculator to confirm your results. Example 1: f(x) = 2x3 - 3x2 - 36x + 14 Interval Test Value Increasing: __________ Sign f’(x) Decreasing: __________ Conclusion Constant: ___________ Relative Max: _____ Relative Min: _____ Directions: For the following exercises, find the critical numbers of f (if any). Find the open intervals on which the function is increasing or decreasing. Locate all the relative extrema. Use a graphing calculator to confirm your results. Example 2: f(x) = (x2 - 4)2/3 Interval Test Value Sign f’(x) Conclusion Increasing: __________ Decreasing: __________ Constant: ___________ Relative Max: _____ Relative Min: _____ Directions: For the following exercises, find the critical numbers of f (if any). Find the open intervals on which the function is increasing or decreasing. Locate all the relative extrema. Use a graphing calculator to confirm your results. x4 1 Example 3: f ( x) x2 Interval Test Value Sign f’(x) Conclusion Increasing: __________ Decreasing: __________ Constant: ___________ Relative Max: _____ Relative Min: _____ Page 334-335 #11-33 odd (skip 27), 55, (MUST USE CALCULUS!!!!)