Higher Derivatives Concavity 2nd Derivative Test

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Presentation transcript:

Higher Derivatives Concavity 2nd 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?

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?

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? 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.

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

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

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

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

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

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

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!

Assignment Lesson 5.3 Page 345 Exercises 1 – 85 EOO