1. Velocity and Marginals – Calculus 10/5/15
2.3 Rates of Change
Velocity and Marginals – Calculus 10/5/15

2. Average Rates of Change

3. Average Rate of Change

5. Average Velocity
If a free falling object is dropped from a height of 100 ft, and air resistance is neglected, the height h (in feet) of the object at time t (in seconds) is given by:
ℎ=−16 𝑡
Find the average velocity of the object over each interval. a) [1,2] b) [1,1.5]

6. Instantaneous Rate of Change

7. ℎ=−16 𝑡
Find the velocity of the object when t = 1

8. Position, Velocity, and Speed
Position function ℎ=−16 𝑡 2 + 𝑣 0 𝑡+ ℎ 0
Velocity function (derivative of position)
Positive velocities indicate upward motion and negative velocities indicate downward motion
Speed – absolute value of the velocity

9. Find the Velocity
At time t= 0, a diver jumps from a diving board that is 32 feet high. Because the diver’s initial velocity is 16 feet per second, his position function is:
ℎ=−16 𝑡 2 +16𝑡+32
A) When does the diver hit the water?
B) What is the diver’s velocity at impact?

10. 1) Find the average rate of change on the interval and compare this with the instantaneous rate of change at the endpoints of the interval.
𝑓 𝑥 = 𝑥 2 −6𝑥 −1 ; −1,3
2) At 0° C, the heat loss H (in kilocalories per square meter per hour) from a person’s body can be modeled by 𝐻= 𝑣 −𝑣 , where v is the wind speed (meters per second).
Find 𝑑𝐻 𝑑𝑣 , and interpret its meaning
Find the rates of change of H when v=2 and v=5