1. Higher Derivatives Concavity 2nd Derivative Test
Lesson 5.3

2. 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?

3. 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?

4. 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

5. 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.

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

7. 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

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

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

10. 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

11. 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

12. 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!

13. Assignment
Lesson 5.3
Page 345
Exercises 1 – 85 EOO