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

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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
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2 CONTENTS
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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 ) = - sin 2 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
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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 the intervals where it is decreasing. Example : f ( x ) = 6 x 4 - 20 x 3 - 6 x 2 + 72 x + 12 Solution : f 0 ( x ) = 12(2 x 3 - 5 x 2 - x + 6) = 12( x + 1)(2 x - 3)( x - 2) (see Chapter 0 page 5 Theorem I) f 0 ( x ) = 0 at x = - 1 , 3 2 or 2. Because f is a continuous function we can conclude f ( x ) is strictly monotone on each of the intervals ( -∞ , - 1), ( - 1 , 3 2 ), ( 3 2 , 2), (2 , + ). Compute a value f 0 ( - 2) = - 336 < 0 to conclude that f is decreasing on ( -∞ , - 1) due to the fact f is continuous on ( -∞ , - 1) and - 2 ( -∞ , - 1). f 0 (0) > 0 increasing on ( - 1 , 3 2 ) f 0 ( 7 4 ) < 0 decreasing on ( 3 2 , 2 ) f 0 (3) > 0 increasing on (2 , + ). NOTE: To use this method you must check that f is continuous on the given interval and state this as part of your solution. 6. (a) f ( x ) = ( x - 1) 2 + 1 (b) f ( x ) = x 3 + 2 (c) f ( x ) = | x - 2 | (d) f ( x ) = x | x | 7. (a) f ( x ) = 1 ( x - 1)( x - 2) (b) f ( x ) = 4 x 3 - 3 x (c) f ( x ) = ( x 2 +1) x 2 (d) f ( x ) = x - 3 8. (a) f ( x ) = ( x 2 + 2 x + 1) 1 2 (b) f ( x ) = 1 ( x 2 +4) 1 2 (c) f ( x ) = | ( x - 1)( x - 2)( x - 3) | (d) f ( x ) = p | x - 2 | 9. (a) f ( x ) = e x - x (b) f ( x ) = xe 2 x (c) f ( x ) = e x x +1 (d) f ( x ) = e x 2 - x - 2 10. (a) f ( x ) = x 2 ln x (b) f ( x ) = ln x x (c) f ( x ) = ln | x | (d) f ( x ) = ln( x 2 + 1) 11. (a) s ( x ) = sin ( x - π 2 ) (b) f ( x ) = | sin x | (c) f ( x ) = sin | x | (d) f ( x ) = sin x + cos x
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6.3 Extrema — Maxima and Minima 89 6.3 Extrema — Maxima and Minima Examples maximum relative maximum relative minimum minimum a b relative minimum relative maximum minimum maximum The maximum (global maximum) is the highest value a function attains on the given domain. The minimum (global minimum) is the lowest value a function attains on the given domain. Some functions do not have a maximum or a minimum.
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