11DIFFERENTIATION - Department of Mathematics

726 11 DIFFERENTIATION
SOLUTION ✔
a. Using the general power rule, we obtain
N(t) d
dt [100(64 4t)2/3 ]
100 d
(64 4t)2/3
dt
1002 1/3 d
(64 4t) (64 4t)
3 dt
200
3 (64 4t)1/3 (4)
800
3(64 4t) 1/3
b. The rate at which the membership was increasing when the center first
opened is given by
N(0) 800
66.7
1/3 3(64)
or approximately 67 people per week.
c. The rate at which the membership was increasing at the beginning of the
40th week is given by
N(40)
800
43.9
1/3 3(64 160)
or approximately 44 people per week.
d. The membership when the center first opened is given by
N(0) 100(64) 2/3 100(16)
or approximately 1600 people. The membership at the beginning of the 40th
week is given by
N(40) 100(64 160) 2/3 3688.3
or approximately 3688 people.
Group Discussion
The profit P of a one-product software manufacturer depends on
the number of units of its products sold. The manufacturer estimates that
it will sell x units of its product per week. Suppose P g(x) and x f(t),
where g and f are differentiable functions.
1. Write an expression giving the rate of change of the profit with respect
to the number of units sold.
2. Write an expression giving the rate of change of the number of units
sold per week.
3. Write an expression giving the rate of change of the profit per week.