1. 2.1: Rates of Change & Limits

2. Suppose you drive 200 miles, and it takes you 4 hours. Then your average speed is: If you look at your speedometer during this trip, it might read 65 mph. This is your instantaneous speed.

3. Suppose a ball is dropped from the upper observation deck of the CN tower, 450m above the ground. How fast is the ball falling after 3s? The following table shows the results over successively smaller periods Time interval Instantaneous Speed 3 < t < 434.3 3 < t < 3.129.89 3 < t < 3.0529.645 3 < t < 3.0129.449 3 < t < 3.00129.4049 Recall s = 4.9t 2 It appears that it approaches 29.4m/s Instantaneous speed =

4. Recall s = 4.9t 2 Instantaneous speed = When x = 3, then the instantaneous speed = 9.8(3) = 29.4

5. A rock falls from a high cliff. The position of the rock is given by: After 2 seconds: average speed over the first 2s: What is the instantaneous speed at 2 seconds?

6. for some very small change in t where h = some very small change in t

7. The limit as h approaches zero: 0 = = =

8. A spherical balloon is being inflated. Find the rate of change of the volume with respect to the radius when the radius is 10 cm. Recall V(r) = The rate of change of V w.r.t. r is about 1260 cm 3 /cm.

9. Review: average slope: slope at a point: average velocity: instantaneous velocity: If is the position function: These are often mixed up by Calculus students! So are these! velocity = slope 