1. Rates of Change and Tangent Lines
2.1
Rates of Change and Tangent Lines
AP CALCULUS
MS. OLIFER

2. What you’ll learn about
Displacement
Average and Instantaneous Rate of Change
Instantaneous Velocity as Limit

3. Change in Position Change in position = velocity x change in time
But what if velocity isn’t constant?

4. Tangent Lines and Instantaneous Rate of Change

5. Average and Instantaneous Speed

6. Instantaneous Velocity
Average velocity over a very small time interval is very close to instantaneous velocity.

7. Instantaneous Velocity
Galileo’s Formula

8. Example 1, pg. 61
Conclusion: average velocity converges to instantaneous velocity or that instantaneous velocity is the limit of average velocity as the length of time interval shrinks to zero.
Average velocity = slope of a secant line

9. Example on pg. 62 Conclusion:
As the time interval shrinks to zero, the slope of the secant line approaches the slope of the tangent line.
Instantaneous rate of change = slope of the tangent line at a given point.

10. Instantaneous Rate of Change
To estimate the instantaneous rate of change at a point, compute the average rate of change over several intervals.