calculus_l'hospital - MA103 Week 5 Lecture Notes How...

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MA103 Week 5 Lecture Notes How Derivatives Affect the Shape of a Graph (Text Reference: §4.3) Monotonicity (Increasing/Decreasing) S Definition: A function f is called (strictly) increasing on an interval I if f Ÿ x 1   f Ÿ x 2   whenever x 1 , x 2 I and x 1 x 2 . (That is, for all x 1 x 2 in I , f Ÿ x 1   f Ÿ x 2   .) Similarly, f is (strictly) decreasing on I if x 1 , x 2 I and x 1 x 2 implies f Ÿ x 1   f Ÿ x 2   . S f is said to be monotonic on I if f is either increasing on I or decreasing on I Ž note that this term is not used in your text S Test for Monotonic Functions (Increasing/Decreasing Test): 1. f U Ÿ x   0 for all x Ÿ a , b   implies f is increasing on Ÿ a , b   ; 2. f U Ÿ x   0 for all x Ÿ a , b   implies f is decreasing on Ÿ a , b   ; Ž proof may be discussed in lecture S example: Examine the monotonicity (i.e., where the function is increasing/decreasing) of f Ÿ x   1 4 x 4 " x 3 x 2 1 sol’n: f U Ÿ x   x 3 " 3 x 2 2 x x Ÿ x " 1  Ÿ x " 2   The critical numbers are x 0,1,2. Analyze the sign of f U Ÿ x   using intervals determined by the critical numbers. f U " " ". 01 2 . f |{ | { Thus f is decreasing on Ÿ ". ,0   : Ÿ 1,2   and increasing on Ÿ 0,1   : Ÿ 2, .   . S First Derivative Test: Suppose c is a critical number of a continuous function f . 1. If the sign of f U Ÿ x   is positive to the left of c and negative to the right of c , then f has a local maximum at c . 2. If the sign of f U Ÿ x   is negative to the left of c and positive to the right of c , then f has a local minimum at c . 3. If the sign of f U Ÿ x   does not change at c , then f has no local extremum at c . S example: In the above example, f would have a local maximum value at x 1 and local minimum values at x 0 and x 2 by the First Derivative Test. S Note that if a function f has just one critical number c in an interval I , then if f has a local minimum at c , f Ÿ c   is the minimum value of f on I , and if f has a local maximum at c , f Ÿ c   is the maximum value of
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This note was uploaded on 04/30/2011 for the course MATH 205 taught by Professor Tseng during the Spring '08 term at American College of Computer & Information Sciences.

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calculus_l'hospital - MA103 Week 5 Lecture Notes How...

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