Graph Sketching - Contents 6 Graph Sketching 87 6.1...

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Unformatted text preview: Contents 6 Graph Sketching 87 6.1 Increasing Functions and Decreasing Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 87 6.2 Intervals — Monotonically Increasing or Decreasing . . . . . . . . . . . . . . . . . . . . . . . 88 6.3 Extrema — Maxima and Minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 6.4 Relative Maxima and Relative Minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 6.5 The Second Derivative Test for Relative Extrema . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.6 Concavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 6.7 Points of Inflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6.8 Asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.9 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6.10 Graph Sketching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 6.11 Graphs of Trancendental Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 6.12 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.13 Functions with Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 2 CONTENTS Chapter 6 Graph Sketching 6.1 Increasing Functions and Decreasing Functions For problems numbered 1 to 5 show whether the function is increasing or decreasing at the indicated points. 1. (a) f ( x ) = 2 x 2- 1; at x = 0, x =- 3, and x = 1 2 (b) f ( x ) = x 3- 3 x 2 + 1; at x =- 1, x = 1, x = 2, and x = 4 (c) f ( x ) = | x | - 2; at x =- 2, x =- 1, and x = 2 (d) f ( x ) = | x- 1 | ; at x = 0, x = 1 2 , and x = 2 2. (a) f ( x ) = x | x | ; at x =- 1, x = 0, and x = 3 (b) f ( x ) = x x +1 ; at x =- 1, and x = 10 (c) f ( x ) = 3 x 4 + 4 x 3 ; at x =- 2, x =- 1, x = 0, and x = 1 (d) f ( x ) = (2 x +1) ( x- 2) ; at x = 7 (e) f ( x ) = x 2 +1 x ; at x = 1 3. (a) h ( x ) = cos x 2 ; at x = π 4 (b) g ( x ) =- sin2 x ; at x = 0, x = π 4 and x = 3 π 4 (c) g ( x ) = tan x ; at x =- π 4 , x = 0, and x = π 4 (d) g ( x ) = x sin x ; at x = π 2 , and x = 0 4. (a) f ( x ) = xe x ; at x =- 10, x =- 1, x = 0, and x = 1 (b) f ( x ) = ( x + 1) e x ; at x =- 10, x =- 2, x =- 1, and x = 0 (c) f ( x ) = e x x ; at x =- 1, x = 1, and x = 10 (d) f ( x ) = e ( x- 1) 2 ; at x =- 1, x = 1, and x = 2 5. (a) h ( x ) = ln(1- x ); at x = 2, and x = 0 (b) h ( x ) = x ln x ; at x = 1, x = e- 2 , and x = e 3 (c) h ( x ) = ln 2 x ; at x = e- 1 , x = 1, and x = e (d) h ( x ) = ln(sin x ); at x = π 4 , x = π 2 , and x = 3 π 4 88 Graph Sketching 6.2 Intervals — Monotonically Increasing or Decreasing For problems numbered 6 to 11 divide the domain of the function into a finite number of intervals on each of which the function is strictly monotone. Indicate the intervals where the function is increasing and theof which the function is strictly monotone....
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This note was uploaded on 09/22/2011 for the course ECON 101 taught by Professor Mr.tull during the Spring '11 term at De La Salle University.

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Graph Sketching - Contents 6 Graph Sketching 87 6.1...

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