4.4. THE FUNDAMENTAL THEOREM OF CALCULUS
261
want to compute the average value of r on [4, 10]. Thus,
r AVG[4,10] =
1
10 − 4
10
4
r(t) dt ≈
44.282
6
= 7.380,
which has its units measured in gallons per day.
Activity 4.12.
During a 40-minute workout, a person riding an exercise machine burns calories at a
rate of c calories per minute, where the function y = c(t) is given in Figure 4.36. On the
interval 0 ≤ t ≤ 10, the formula for c is c(t) = −0.05t 2 + t + 10, while on 30 ≤ t ≤ 40,
its formula is c(t) = −0.05t 2 + 3t − 30.
10
20
30
40
5
10
15
cal/min
min
y = c(t)
Figure 4.36: The rate c(t) at which a person exercising burns calories, measured in calories
per minute.
(a) What is the exact total number of calories the person burns during the first 10
minutes of her workout?
(b) Let C(t) be an antiderivative of c(t). What is the meaning of C(40) − C(0) in
the context of the person exercising? Include units on your answer.
(c) Determine the exact average rate at which the person burned calories during
the 40-minute workout.
(d) At what time(s), if any, is the instantaneous rate at which the person is burning
calories equal to the average rate at which she burns calories, on the time
interval 0 ≤ t ≤ 40?
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