1.1. HOW DO WE MEASURE VELOCITY?
9
2 4 6 8 10 12
s
t
2 4 6 8 10 12
v
t
Figure 1.4: Axes for plotting s(t) in part (a) and v(t) in part (c) of the diver problem.
(d) Is there a connection between the two graphs that you can describe? What
can you say about the velocity graph when the height function is increasing?
decreasing? Make as many observations as you can.
3. According to the U.S. census, the population of the city of Grand Rapids, MI, was
181,843 in 1980; 189,126 in 1990; and 197,800 in 2000.
(a) Between 1980 and 2000, by how many people did the population of Grand
Rapids grow?
(b) In an average year between 1980 and 2000, by how many people did the
population of Grand Rapids grow?
(c) Just like we can find the average velocity of a moving body by computing
change in position over change in time, we can compute the average rate of
change of any function f . In particular, the average rate of change of a function
f over an interval [a, b] is the quotient
f (b) − f (a)
b − a
.
What does the quantity
f (b)− f (a)
b−a
measure on the graph of y = f (x) over the
interval [a, b]?
(d) Let P(t) represent the population of Grand Rapids at time t, where t is measured
in years from January 1, 1980. What is the average rate of change of P on the
interval t = 0 to t = 20? What are the units on this quantity?
(e) If we assume the population of Grand Rapids is growing at a rate of approximately 4% per decade, we can model the population function with the
9
2 4 6 8 10 12
s
t
2 4 6 8 10 12
v
t
Figure 1.4: Axes for plotting s(t) in part (a) and v(t) in part (c) of the diver problem.
(d) Is there a connection between the two graphs that you can describe? What
can you say about the velocity graph when the height function is increasing?
decreasing? Make as many observations as you can.
3. According to the U.S. census, the population of the city of Grand Rapids, MI, was
181,843 in 1980; 189,126 in 1990; and 197,800 in 2000.
(a) Between 1980 and 2000, by how many people did the population of Grand
Rapids grow?
(b) In an average year between 1980 and 2000, by how many people did the
population of Grand Rapids grow?
(c) Just like we can find the average velocity of a moving body by computing
change in position over change in time, we can compute the average rate of
change of any function f . In particular, the average rate of change of a function
f over an interval [a, b] is the quotient
f (b) − f (a)
b − a
.
What does the quantity
f (b)− f (a)
b−a
measure on the graph of y = f (x) over the
interval [a, b]?
(d) Let P(t) represent the population of Grand Rapids at time t, where t is measured
in years from January 1, 1980. What is the average rate of change of P on the
interval t = 0 to t = 20? What are the units on this quantity?
(e) If we assume the population of Grand Rapids is growing at a rate of approximately 4% per decade, we can model the population function with the