is the number of people who will become infected during the next 4 weeks (or the total change in
the number of infected people).EXAMPLE 13
Suppose a rumor is spreading at the rate of f (t) = 100e−0.2t new people per day. Find the number of
people who hear the rumor during the 5th and 6th days.
SOLUTION: 100 e−0.2t dt = 74 people.
If we let F ′(t) = f (t), then the integral above is the net change in F(t) from t = 4 to t = 6, or the
number of people who hear the rumor from the beginning of the 5th day to the end of the 6th.EXAMPLE 14
Economists define the marginal cost of production as the additional cost of producing one
additional unit at a specified production level. It can be shown that if C(x) is the cost at production
level x then C ′(x) is the marginal cost at that production level.
If the marginal cost, in dollars, is per unit when x units are being produced, find the change in
cost when production increases from 50 to 75 units.
SOLUTION:
We replace “cost” above by “revenue” or “profit” to find total change in these quantities.EXAMPLE 15
After t minutes, a chemical is decomposing at the rate of 10e−t grams per minute. Find the amount
that has decomposed during the first 3 minutes.
SOLUTION:EXAMPLE 16
An official of the Environmental Protection Agency estimates that t years from now the level of a
particular pollutant in the air will be increasing at the rate of (0.3 + 0.4t) parts per million per year
(ppm/yr). Based on this estimate, find the change in the pollutant level during the second year.
SOLUTION:Work†
Work is defined as force times distance: W = F × d. When a variable force F(x) moves an object
along the x-axis from a to b, we approximate an element of work done by the force over a short
distance Δx by
ΔW = F(xk) Δx,