(9/30/08)
Math 10A. Lecture Examples.
Sections 2.2 and 2.3. The derivative at a point and as a function†
Example 1 The function A=A(t)whose graph is shown in Figure 1 gives the percent-
age of alcohol in a person’s blood thours after he has consumed three fluid
ounces of alcohol.(1) The tangent line at t=1in Figure 2 passes through
the points P= (1,0.18)and Q= (2,0.28). What is the (instantaneous) rate
of change of the person’s blood alcohol level with respect to time one hour
after consuming the alcohol?
t2 4 6 8 10 12
A(percent of alcohol)
0.2
0.1
0.3
(hours)
A=A(t)
t2 4 6 8 10 121
A(percent of alcohol)
0.2
0.1
0.3
(hours)
A=A(t)
P
Q
FIGURE 1 FIGURE 2
Answer: A0(1) = 0.1 percent per hour
Example 2 Figure 3 shows the graph of the rate of oxygen consumption r=r(v)(liters
per minute) of a swimmer as a function of his velocity v(meters per second)
and the tangent line to the graph at v=1.(2) (a) Why is r(1.4)greater than
r(1)? (b) What is the rate of change of his oxygen consumption with respect
to velocity at v=1? (Give the units.)
v0.8 1 1.2 1.4
r(liters per minute)
1
2
3
4
(meters per second)
(1.2,2.66)
(1,2.1)
r=r(v)
FIGURE 3
Answer: (a) r(1.4) is greater than r(1) because the swimmer uses more oxygen when he swims faster.
(b) r0(1) = [Slope of the tangent line] = 2.66 −2.1
1.2−1= 2.8 liters per minute per meters per second.
†Lecture notes t o accompany Sections 2.2 and 2.3 of Calculus by Hughes-Hallett et al.
(1)Data adapted from Encyclopædia Britannica, Vol. 1, Chicago: Encyclopædia Britannica, Inc., 1965, p. 548.
(2)Data adapted from The Human Machine by R. Alexander, New York, NY: Columbia University Press, 1992, p. 117.
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