1. Second Derivatives and Graphs of Derivatives
AP Calculus
Mrs. Agnew
September 26-27, 2016

2. Essential Stuff Essential Question Essential Vocabulary
How do you find second derivatives?
How do you graph derivatives?
Essential Vocabulary
Second derivative
Instantaneous Rate of Change
Numerical Derivatives

3. Review: Tangent Lines How do you find the equation of a tangent line?
What is the slope of the tangent line?
What is the Calculus equation for the tangent line?

4. Review: Rates of Change
How do you find the instantaneous velocity?
What is the instantaneous rate of change?
The derivative gives the slope of the tangent line, the instantaneous velocity and the instantaneous rate of change.

5. The Derivative Is… Slope of the curve Instantaneous Velocity
Rate of Change
Slope of the tangent line

6. Numerical Derivatives
The numerical derivative is the value of the derivative at a given x value.
Interpret the meaning of a numerical deriv.
We can use the calculator to find the numerical derivative
NDERIV
dy/dx on graph screen
Example: Use calc to find tangent line to y = xlnx at e and 1/e.

7. Second Derivatives
The derivative of a function is a function itself… so it also has a derivative.
The derivative of a derivative is known as the SECOND DERIVATIVE.
Notations:
How would you find the second derivative?

8. Second Derivatives Interpretation of the Second Derivative:
“The rate of change of the rate of change”
Rate of change of the slope of curve
Acceleration
In other words, it is how quickly something is increasing or decreasing.
It is possible to find third, fourth, etc. derivatives as well.
Guided Practice

9. Graphing Derivatives Key components of graph of derivative:
When f is increasing, f '(x) > 0.
When f is decreasing, f '(x) < 0.
When f is at an extreme value, f '(x) = 0.
Positive slope: derivative above x-axis. Negative slope: derivative below x-axis. Slope of zero: derivative on x-axis.

10. Graphing Derivatives
Magnitude of slope tells you how far above and below the x-axis to go.
Steeper slope means further away from x- axis.
Stewart Page 178 #1, 3, 5, 6, 12, & Examples…
Graphing Derivatives on Calculator

11. Homework: September 27
Stewart - page 167 – 170 #4, 5, 7, 10, 35, 37