Higher Derivatives Concavity 2 nd Derivative Test Lesson 5.3

Think About It Just because the price of a stock is increasing … does that make it a good buy? When might it be a good buy? When might it be a bad buy? What might that have to do with derivatives? 2

Think About It It is important to know the rate of the rate of increase! The faster the rate of increase, the better. Suppose a stock price is modeled by What is the rate of increase for several months in the future? 3

Think About It Plot the derivative for 36 months The stock is increasing at a decreasing rate Is that a good deal? What happens really long term? 4 Consider the derivative of this function … it can tell us things about the original function

Higher Derivatives The derivative of the first derivative is called the second derivative Other notations Third derivative f '''(x), etc. Fourth derivative f (4) (x), etc. 5

Find Some Derivatives Find the second and third derivatives of the following functions 6

Velocity and Acceleration Consider a function which gives a car's distance from a starting point as a function of time The first derivative is the velocity function The rate of change of distance The second derivative is the acceleration The rate of change of velocity 7

Concavity of a Graph Concave down Opens down Concave up Opens up 8 Point of Inflection where function changes from concave down to concave up

Concavity of a Graph Concave down Decreasing slope Second derivative is negative Concave up Increasing slope Second derivative is positive 9

Test for Concavity Let f be function with derivatives f ' and f '' Derivatives exist for all points in (a, b) If f ''(x) > 0 for all x in (a, b) Then f(x) concave up If f ''(x) < 0 for all x in (a, b) Then f(x) concave down 10

Test for Concavity Strategy Find c where f ''(c) = 0 This is the test point Check left and right of test point, c Where f ''(x) < 0, f(x) concave down Where f ''(x) > 0, f(x) concave up Try it 11

Determining Max or Min Use second derivative test at critical points When f '(c) = 0 … If f ''(c) > 0 This is a minimum If f ''(c) < 0 This is a maximum If f ''(c) = 0 You cannot tell one way or the other! 12

Assignment Lesson 5.3 Page 345 Exercises 1 – 85 EOO 13