1. An Application of the fundamental theorem of calculus: Rate graphs
Section 4-5

2. The Fundamental Theorem of Calculus: The interpretation of a rate graph
The function 𝑓(𝑡) is the rate of change of the amount of “something” with units of “something”/unit of time.
The FTC tells us that the definite integral of this rate of change is the total accumulation in the amount of “something” over the given time interval

3. The graph of a function 𝑓(𝑡) consists of a
quarter circle and line segments.
Let g be the function given by .
a) Find
Graph of f

4. Find all values of x on the open interval
at which g has a relative maximum .
Find the absolute minimum value of g on
And the value of x at which it occurs
Graph of f

5. d) Find the x-coordinate of each point of inflection of the graph of g on.
Graph of f

6. 2)

7. 3) The graph of the velocity , in ft/sec, of a car traveling
on a straight road, for is shown in the figure.
a) Find the average acceleration of the car,
over the interval

8. b) Find an approximation for the acceleration of the car at t = 20.

9. Approximate the distance with a Riemann sum, using the midpoints of three subintervals of equal length. Explain the meaning of this integral.

10. (modification of 2006 BC 4)
Rocket A has positive velocity v(t) after being launched upward from an initial height of 0 feet at time t = 0 seconds. The velocity of the rocket is recorded for selected values of t over the interval seconds as shown in the table below
t (seconds) 10 20 30 40 50 60 70 80
(ft per sec) 5 14 22 29 35 44 47 49

11. 4) Sketch a graph of the data
t (seconds) 10 20 30 40 50 60 70 80
(ft per sec) 5 14 22 29 35 44 47 49

12. (modification of 2006 BC)
a) Use a midpoint Riemann sum with 3 subintervals of equal length to approximate
Then explain the meaning of in terms of the rocket’s flight.

13. (modification of 2006 BC)
Rocket B is launched upward with an acceleration of feet per second per second.
At time t=0 seconds, the initial height of the rocket is 0 feet, and the initial velocity is 2 feet per second. Which of the two rockets is traveling faster at t = 80 seconds?

14. Home Work
Use a section header for each of the topics, so there is a clear transition to the audience.
Worksheet 4-5