1. Tangent Lines and Derivatives
AP Calculus
September 19 & 20, 2016
Mrs. Agnew

2. Essential Question Essential Vocabulary
What is the derivative and how do you calculate it?
Essential Vocabulary
Slope of a curve
Instantaneous rate of change
Tangent line
Derivative
Numerical Derivative

3. Tangent Lines and Velocities
When computing the slope of a tangent
line, why is it necessary to evaluate a limit?
The formula above gives us the slope of the tangent line to the curve at x = a.
Also gives the instantaneous velocity of an object at time t = a.
Instantaneous Rate of Change

4. The Derivative THE DERIVATIVE
The special limits below occur so widely, they are given a special name…
THE DERIVATIVE
Examples…

5. What is the Derivative? The derivative represents…
the slope of the tangent line
the instantaneous rate of change
the instantaneous velocity
If the derivative is slope of tangent line, then equation of tangent line at x = a is:
Examples…

6. What is the Derivative?
The instantaneous rate of change is the limit of the average rates of change over smaller and smaller intervals.
The derivative, f ’(a), is the instantaneous rate of change in y = f(x) with respect to x at x = a.
Examples…

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

8. Homework: 9/20/2016 Page 155 – 156 (Stewart)
#3 – 7, 10, 13, 14, 15, 18, 25, 28, 30, 33