5.3 :Higher Order Derivatives, Concavity and the 2 nd Derivative Test

1. 5.3:Higher Order Derivatives, Concavity and the 2nd Derivative Test Objectives: To find Higher Order Derivatives To use the second derivative to test for concavity To use the 2nd Derivative Test to find relative extrema

2. If a function’s derivative is f’, the derivative of f’, if it exists, is the second derivative, f’’. You can take 3rd, 4th,5th, etc. derivative Notations Second Derivative: Third Derivative: For n> 4, the nth derivative is written f(n)(x)

3. 1. Find f(4)(x). 2. Find f’’(0).

4. Find f’’(x). 1. 2.

5. Find f’’’(x).

6. If a function describes the position of an object along a straight line at time t: s(t) = position s’(t) = v(t) = velocity (can be + or - ) s’’(t) = v’(t) = a(t) = acceleration If v(t) and a(t) are the same sign, object is speeding up If v(t) and a(t) are opposite signs, object is slowing down

7. Suppose a car is moving in a straight line, with its position from a starting point (in ft) at time t (in sec) is given by s(t)=t3-2t2-7t+9 a.) Find where the car is moving forwards and backwards. b.) When is the car speeding up and slowing down?

8. Concavity of a Graph • How the curve is turning, shape of the graph • Determined by finding the 2nd derivative • Rate of change of the first derivative • Concave Up: y’ is increasing, graph is “smiling”, cup or bowl • Concave Down: y’ is decreasing, graph is “frowning”, arch • Inflection point: where a function changes concavity • f’’ = 0 or f’’ does not exist here

9. Precise Definition of Concave Up and Down A graph is Concave Up on an interval (a,b) if the graph lies above its tangent line at each point in (a,b) A graph is Concave Down on an interval (a,b) if graph lies below its tangent line at each point in (a,b) At inflection points, the graph crosses the tangent line

10. Test for Concavity • f’ and f’’ need to exist at all point in an interval (a,b) • Graph is concave up where f’’(x) > 0for all points in (a,b) • Graph is concave down where f’’(x) < 0 for all points in (a,b) Find inflection points and test on a number line. Pick x-values on either side of inflection points to tell whether f’’ is > 0 or < 0

11. Find the open intervals where the functions are concave up or concave down. Find any inflection points. 1.

14. Second Derivative Test for Relative Extrema Let f’’(x) exist on some open interval containing c, and let f’(c) = 0. • If f’’(c) > 0, then f(c) is a relative minimum • If f’’(c) < 0, then f(c) is a relative maximum • If f’’(c) = 0 or f’’(c) does not exist, use 1st derivative test

15. Find all relative extrema using the 2nd Derivative Test. 1. 2.