Handouts6 - Differentiation2

Handouts6 - Differentiation2 - MATH0201 BASIC CALCULUS...

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Unformatted text preview: MATH0201 BASIC CALCULUS MATH0201 BASIC CALCULUS Differentiation II Dr. WONG Chi Wing Department of Mathematics, HKU MATH0201 BASIC CALCULUS First Derivative and Monotonicity Partition Numbers Monotonicity of Function Local Extrema Second Derivative and Concavity Concavity of Graphs of Functions Criterion on Concavity Point of Inflection Local Extrema Again Curve Sketching Graphs of Polynomial Functions Graphs of Rational Functions Optimization Over Closed and Bounded Intervals Unique Critical Value Applied Optimization Reference § 3.1–5 of the textbook. MATH0201 BASIC CALCULUS First Derivative and Monotonicity Partition Numbers Definition 1 Suppose that f is defined on I n S where I is an interval and S is a finite subset of I , maybe empty. x 2 I is said to be a partition number of f if 1. x 2 S , or 2. either f is discontinuous at x or f ( x ) = 0. Example 2 (Example 26, Handout–Limit) Partition numbers of f ( x ) = x 3 + 3 x 2 4 x 12 are 3, 2, and 2. Example 3 (Example 27, Handout–Limit) Partition numbers of f ( x ) = x + 1 x 2 are 1 and 2. MATH0201 BASIC CALCULUS First Derivative and Monotonicity Partition Numbers Observe that I Two open intervals are determined by a partition number: ( 1 ; c ) and ( c ; 1 ) : I k + 1 open intervals are obtained by k partition numbers: ( 1 ; c 1 ) ; ( c 1 ; c 2 ) ; ( c 2 ; c 3 ) ; ::: ( c k 1 ; c k ) ; ( c k ; 1 ) : Let f be continuous (on I n S ). In view of the Intermediate value theorem, if a and b are consecutive partition numbers, then f must be of constant sign on ( a ; b ) . MATH0201 BASIC CALCULUS First Derivative and Monotonicity Monotonicity of Function Theorem 4 Let f be differentiable over ( a ; b ) . 1. f is increasing over ( a ; b ) if f ( x ) > for any x 2 ( a ; b ) : 2. f is decreasing over ( a ; b ) if f ( x ) < for any x 2 ( a ; b ) : MATH0201 BASIC CALCULUS First Derivative and Monotonicity Monotonicity of Function Procedure Determine the intervals on which the function f ( x ) is Increasing/Decreasing. 1. Determine partition numbers of f ( x ) . 2. Apply the sign test on each interval I determined by the partition numbers. 3. If f ( x ) > 0/ f ( x ) < 0 on I , then f ( x ) is increasing/decreasing there. MATH0201 BASIC CALCULUS First Derivative and Monotonicity Monotonicity of Function Example 5 Determine the intervals on which f ( x ) = 27 x x 3 is increasing. Solution. I f ( x ) = 27 3 x 2 = 3 ( 9 x 2 ) = 3 ( 3 x )( 3 + x ) . I The partition numbers of f are x = 3 and x = 3. I The intervals determined are ( 1 ; 3 ) , ( 3 ; 3 ) and ( 3 ; 1 ) . Use the sign test I f ( x ) is increasing on ( 3 ; 3 ) . MATH0201 BASIC CALCULUS First Derivative and Monotonicity Monotonicity of Function Theorem 6 Let f ( x ) be increasing on ( a ; c ) and on ( c ; b ) . If f ( x ) is continuous at x = c, then f ( x ) is increasing on ( a ; b ) ....
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This note was uploaded on 04/22/2011 for the course MATH 201 taught by Professor C.wong during the Spring '11 term at HKU.

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Handouts6 - Differentiation2 - MATH0201 BASIC CALCULUS...

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