CWA CH 5 FINAL_97_14 - Chapter 5 GRAPHS AND THE DERIVATIVE...

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Chapter 5 GRAPHS AND THE DERIVATIVE 5.1 Increasing and Decreasing Functions 1. By reading the graph, f is (a) increasing on (1 ; 1 ) and (b) decreasing on ( ¡1 ; 1) : 2. By reading the graph, f is (a) increasing on ( ¡1 ; 4) and (b) decreasing on (4 ; 1 ) : 3. By reading the graph, g is (a) increasing on ( ¡1 ; ¡ 2) and (b) decreasing on ( ¡ 2 ; 1 ) : 4. By reading the graph, g is (a) increasing on (3 ; 1 ) and (b) decreasing on ( ¡1 ; 3) : 5. By reading the graph, h is (a) increasing on ( ¡1 ; ¡ 4) and ( ¡ 2 ; 1 ) and (b) decreasing on ( ¡ 4 ; ¡ 2) : 6. By reading the graph, h is (a) increasing on (1 ; 5) and (b) decreasing on ( ¡1 ; 1) and (5 ; 1 ) : 7. By reading the graph, f is (a) increasing on ( ¡ 7 ; ¡ 4) and ( ¡ 2 ; 1 ) and (b) decreasing on ( ¡1 ; ¡ 7) and ( ¡ 4 ; ¡ 2) : 8. By reading the graph, f is (a) increasing on ( ¡ 3 ; 0) and (3 ; 1 ) and (b) decreasing on ( ¡1 ; ¡ 3) and (0 ; 3) : 9. (a) Since the graph of the function is positive for x < ¡ 1 and x > 3 , the intervals where f ( x ) is increasing are ( ¡1 ; ¡ 1) and (3 ; 1 ) : (b) Since the graph of the function is negative for ¡ 1 < x < 3 , the interval where f ( x ) is decreasing is ( ¡ 1 ; 3) . 10. (a) Since the graph of the function is positive for 3 < x < 5 ; the intervals where f ( x ) is increasing is (3 ; 5) : (b) Since the graph of the function is negative for x < 3 and x > 5 , the interval where f ( x ) is decreasing are ( ¡1 ; 3) and (5 ; 1 ) . 11. (a) Since the graph of the function is positive for x < ¡ 8 ; ¡ 6 < x < ¡ 2 : 5 and x > ¡ 1 : 5 ; the intervals where f ( x ) is increasing are ( ¡1 ; ¡ 8) ; ( ¡ 6 ; ¡ 2 : 5) ; and ( ¡ 1 : 5 ; 1 ) : (b) Since the graph of the function is negative for ¡ 8 < x < ¡ 6 and ¡ 2 : 5 < x < ¡ 1 : 5 , the in- tervals where f ( x ) is decreasing are ( ¡ 8 ; ¡ 6) and ( ¡ 2 : 5 ; ¡ 1 : 5) : 12. (a) Since the graph of the function is positive for x < ¡ 3 ; ¡ 3 < x < 3 , and x > 3 , the intervals where f ( x ) is increasing are ( ¡1 ; ¡ 3) ; ( ¡ 3 ; 3) , and (3 ; 1 ) : (b) Since the graph of the function is never negative, there are no intervals where f ( x ) is decreasing. 13. y = 2 : 3 + 3 : 4 x ¡ 1 : 2 x 2 (a) y 0 = 3 : 4 ¡ 2 : 4 x y 0 is zero when 3 : 4 ¡ 2 : 4 x = 0 x = 3 : 4 2 : 4 = 17 12 and there are no values of x where y 0 does not exist, so the only critical number is x = 17 12 : 297
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298 Chapter 5 GRAPHS AND THE DERIVATIVE Test a point in each interval. When x = 0 ; y 0 = 3 : 4 ¡ 2 : 4(0) = 3 : 4 > 0 : When x = 2 ; y 0 = 3 : 4 ¡ 2 : 4(2) = ¡ 1 : 4 < 0 : (b) The function is increasing on ¡ ¡1 ; 17 12 ¢ : (c) The function is decreasing on ¡ 17 12 ; 1 ¢ : 14. y = 1 : 1 ¡ 0 : 3 x ¡ 0 : 3 x 2 (a) y 0 = ¡ 0 : 3 ¡ 0 : 6 x y 0 is zero when ¡ 0 : 3 ¡ 0 : 6 x = 0 x = ¡ 0 : 3 0 : 6 = ¡ 1 2 and there are no values of x where y 0 does not exist, so the only critical number is x = ¡ 1 2 : Test a point in each interval. When x = ¡ 1 ; y 0 = ¡ 0 : 3 ¡ 0 : 6( ¡ 1) = 0 : 3 > 0 : When x = 0 ; y 0 = ¡ 0 : 3 ¡ 0 : 6(0) = ¡ 0 : 3 < 0 : (b) The function is increasing on ¡ ¡1 ; ¡ 1 2 ¢ : (c) The function is decreasing on ¡ ¡ 1 2 ; 1 ¢ : 15. f ( x ) = 2 3 x 3 ¡ x 2 ¡ 24 x ¡ 4 (a) f 0 ( x ) = 2 x 2 ¡ 2 x ¡ 24 = 2( x 2 ¡ x ¡ 12) = 2( x + 3)( x ¡ 4) f 0 ( x ) is zero when x = ¡ 3 or x = 4 ; so the critical numbers are ¡ 3 and 4 : Test a point in each interval.
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