1. Applied Calculus (MAT 121) Dr. Day Wednesday Feb 22, 2012
Using the Second Derivative
Is a function’s first derivative increasing or decreasing?
Inflection points and concavity
Increasing at an increasing rate
Increasing at a decreasing rate
Decreasing at an increasing rate
Decreasing at a decreasing rate
Wednesday, February 22, 2012
MAT 121

2. What does the second derivative tell us?
Not only can a function increase or decrease, it can do so in various ways. The second derivative helps us identify patterns in how a function increases or decreases. These patterns help us describe the concavity of a function.
The key points where concavity changes are called points of inflection. The second derivative helps us identify these key points.
Specific contextual situations are more thoroughly analyzed using the second derivative. Connections among position, velocity, and acceleration can be made using first and second derivatives.
Wednesday, February 22, 2012
MAT 121

3. Wednesday, February 22, 2012
MAT 121

4. What Possible Ways Can a Function Increase or Decrease?
Wednesday, February 22, 2012
MAT 121

5. What Possible Ways Can a Function Increase or Decrease?
Wednesday, February 22, 2012
MAT 121

6. Concavity: Patterns in Tangent Line Slopes
Wednesday, February 22, 2012
MAT 121

7. Inflection Points: Where Does Concavity Change?
Wednesday, February 22, 2012
MAT 121

8. Inflection Points: Where Does Concavity Change?
Wednesday, February 22, 2012
MAT 121

9. The Second Derivative Reveals a Function’s Concavity
Example: Determine the intervals on which the function f(x) = x3 – 3x2 – 24x is concave up and on which it is concave down. Identify all points of inflection.
Wednesday, February 22, 2012
MAT 121

10. The Second Derivative Reveals a Function’s Concavity
Example: Determine the intervals on which the function f(x) = x + 1/x is concave up and on which it is concave down. Identify all points of inflection.
Wednesday, February 22, 2012
MAT 121

11. Assignments WebAssign 4.1 due tonight. 4.2 due Thursday 2/23.
Reminder: Check results for Test #2
Wednesday, February 22, 2012
MAT 121