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

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

Wednesday, February 22, 2012 MAT 121

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

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

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

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

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

The Second Derivative Reveals a Function’s Concavity Example: Determine the intervals on which the function f(x) = x3 – 3x2 – 24x + 32 is concave up and on which it is concave down. Identify all points of inflection. Wednesday, February 22, 2012 MAT 121

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

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