1. Problem of the Day If f(x) = -x 3 + x + 1, then f '(-1) = x A) 3 B) 1 C) -1 D) -3 E) -5

2. Problem of the Day If f(x) = -x 3 + x + 1, then f '(-1) = x A) 3 B) 1 C) -1 D) -3 E) -5 f '(x) = -3x 2 + 1 - 1 x2x2

3. In the fall of 1972 President Nixon announced that the rate of increase of inflation was decreasing. This was the first time a sitting president used the third derivative to advance his case for reelection. - Hugo Rossi

4. The derivative is used to determine slope. It can also be used to determine rate of change of one variable with respect to another. population growth rates production rates water flow rates velocity and acceleration

5. 20 40 80 60 position time 1 2 100 (1, 84) (2, 36) If a billiard ball is dropped from a height of 100 feet, its height s (feet) at time t (seconds) is given by the position function s = -16t 2 + 100. Find the average velocity during the time interval [1, 2].

6. 20 40 80 60 position time 1 2 100 (1, 84) (2, 36) If a billiard ball is dropped from a height of 100 feet, its height s (feet) at time t (seconds) is given by the position function s = -16t 2 + 100. Find the average velocity during the time interval [1, 2]. Δs = s(2) - s(1) = 36 - 84 = -48 ft/sec Δt 2 - 1 1

7. 20 40 80 60 position time 1 2 100 (1, 84) (2, 36) If a billiard ball is dropped from a height of 100 feet, its height s (feet) at time t (seconds) is given by the position function s = -16t 2 + 100. What is the instantaneous velocity at t = 1?

8. 20 40 80 60 position time 1 2 100 (1, 84) (2, 36) If a billiard ball is dropped from a height of 100 feet, its height s (feet) at time t (seconds) is given by the position function s = - 16t 2 + 100. What is the instantaneous velocity at t = 1? s = -16t 2 + 100 s' = -32t s' (1) = -32(1) = -32

9. At time t = 0, a diver jumps from a diving board that is 32' above water. His position is given by s(t) = -16t 2 + 16t + 32 When does he hit the water?

10. At time t = 0, a diver jumps from a diving board that is 32' above water. His position is given by s(t) = -16t 2 + 16t + 32 When does he hit the water? What is his height when he hits the water? 0 = -16t 2 + 16t + 32 0 = -16(t 2 - t - 2) 0 = -16(t + 1)(t - 2) t = -1 or 2

11. At time t = 0, a diver jumps from a diving board that is 32' above water. His position is given by s(t) = -16t 2 + 16t + 32 What is the diver's velocity at impact?

12. At time t = 0, a diver jumps from a diving board that is 32' above water. His position is given by s(t) = -16t 2 + 16t + 32 What is the diver's velocity at impact? Velocity is rate of change or 1st derivative. s = -16t 2 + 16t + 32 s' = -32t + 16 s'(2) = -32(2) + 16 = -48 ft/sec

13. At time t = 0, a diver jumps from a diving board that is 32' above water. His position is given by s(t) = -16t 2 + 16t + 32 Using the position graph, draw the velocity graph. Then compare with the actual graph of the derivative.

14. s(t) = -16t 2 + 16t + 32 Using the position graph, draw the velocity graph. Then compare with the actual graph of the derivative. 10 30 20 1 2 (0, 32) (.5, 36) (1, 32) (1.5, 20)

15. s(t) = -16t 2 + 16t + 32 Using the position graph, draw the velocity graph. Then compare with the actual graph of the derivative. position graph 30 20 1 2 (0, 32) (.5, 36) (1, 32) (1.5, 20) 10 velocity graph 1 2 -30 -10 (.5, 0)

16. Acceleration and Higher-order derivatives Acceleration = Δv ΔtΔt s(t) = position v(t) = s'(t) = velocity a(t) = v'(t) = s''(t) = acceleration or the second derivative of position

17. f ''(x) d 2 y y'' d 2 [f(x)] D x [y] dx 2 dx 2 2 f '''(x) d 3 y y''' d 3 [f(x)] D x [y] dx 3 dx 3 3

18. An auto's velocity starting from rest is v(t) = 100t 2t + 15 Find its acceleration at 5 seconds, 10 seconds, and 20 seconds

19. An auto's velocity starting from rest is v(t) = 100t 2t + 15 Find its acceleration at 5 seconds, 10 seconds, and 20 seconds v'(t) = (2t + 15)100 - 100t(2) (2t + 15) 2 = 1500 (2t + 15) 2 at 5 seconds a(t) = 2.4 ft/sec 2 at 10 seconds a(t) = 1.22 ft/sec 2 at 20 seconds a(t) = 0.496 ft/sec 2