1. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Higher Order Derivatives & Concavity OBJECTIVES  Find derivatives of higher order.  Determine Concavity of a function  Find the relative extrema of a function using the Second-Derivative Test.  Given a formula for distance, find velocity and acceleration. 5.3

2. Slide 1.8- 2 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Higher-Order Derivatives: Consider the function given by Its derivative f is given by The derivative function f can also be differentiated. We can think of the derivative f as the rate of change of the slope of the tangent lines of f. It can also be regarded as the rate at which is changing.

3. Slide 1.8- 3 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Higher-Order Derivatives (continued): We use the notation f  for the derivative. That is, We call f  the second derivative of f. For the second derivative is given by

4. Slide 1.8- 4 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Higher-Order Derivatives (continued): Continuing in this manner, we have When notation like gets lengthy, we abbreviate it using a symbol in parentheses. Thus is the n th derivative.

5. Slide 1.8- 5 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Higher-Order Derivatives (continued): For we have

6. Slide 1.8- 6 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Higher-Order Derivatives (continued): notation for the second derivative of a Leibniz’s notation for the second derivative of a function given by y = f(x) is read “the second derivative of y with respect to x.” The 2’s in this notation are NOT exponents.

7. Slide 1.8- 7 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Higher-Order Derivatives (concluded): If then

8. Slide 1.8- 8 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 1: For find 1.8 Higher Order Derivatives

9. Slide 1.8- 9 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 1 (cont.)

10. Slide 1.8- 10 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 2: For find and.

11. Slide 1.8- 11 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley By the Extended Chain Rule, Using the Product Rule and Extended Chain Rule,

12. Slide 1.8- 12 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley DEFINITION: The velocity of an object that is s(t) units from a starting point at time t is given by

13. Slide 1.8- 13 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley DEFINITION:

14. Slide 1.8- 14 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 4: For s(t) = 10t 2 find v(t) and a(t), where s is the distance from the starting point, in miles, and t is in hours. Then, find the distance, velocity, and acceleration when t = 4 hr.

15. Slide 2.2 - 15 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley DEFINITION: Suppose that f is a function whose derivative f exists at every point in an open interval I. Then f is concave up on I if f is concave down on I if f is increasing over I. f is decreasing over I. f  is positive over I. f  is negative over I.

16. Slide 2.2 - 16 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley THEOREM 4: A Test for Concavity 1. If f  (x) > 0 on an interval I, then the graph of f is concave up. ( f is increasing, so f is turning up on I.) 2. If f  (x) < 0 on an interval I, then the graph of f is concave down. ( f is decreasing, so f is turning down on I.)

17. Slide 1.8- 17 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Theorem 6: Finding Points of Inflection. If a function f has a point of inflection, it must occur at a point (x, y) where f ”(x) = 0 or f “ (x) does not exist. x would have to be in the domain of f, in order for there to be a point of inflection at (x, y).

18. Slide 1.8- 18 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley To find points of inflection 1.Find f‘(x) 2.Find f”(x). 3.Find any values of x that are in the domain of f, that make f”(x)=0 or where f”(x) does not exist. These are Possible Points of Inflection. 4.Draw a f” number line and divide it into intervals with the possible points of inflection. 5.Determine the concavity (+ concave up, - concave down) for each interval. 6.If the concavity changes at any of the x values, then a point of inflection occurs at that x. 7.Find f(x) for any of these x values. The point (x, y) will be a point of inflection.

19. Slide 2.2 - 19 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley THEOREM 5: The Second Derivative Test for Relative Extrema Suppose that f is differentiable for every x in an open interval (a, b) and that there is a critical value c in (a, b) for which f (c) = 0. Then: 1. f (c) is a relative minimum if f  (c) > 0. 2. f (c) is a relative maximum if f  (c) < 0. For f  (c) = 0, the First-Derivative Test can be used to determine whether f (c) is a relative extremum.