Answer
(a) $C'(x) = 2-0.04~x+0.00021~x^2$
(b) $C'(100) = 0.1$
When the production level is 100 units, the additional cost of producing each item is $\$0.10$
(c) The cost of producing the 101st item is $\$0.10$
The value of $C'(100)$ is a very good approximation of the cost of producing the 101st item.
Work Step by Step
(a) $C(x) = 920+2x-0.02~x^2+0.00007~x^3$
We can find the marginal cost function:
$C'(x) = 2-0.04~x+0.00021~x^2$
(b) We can find $C'(100)$:
$C'(100) = 2-0.04~(100)+0.00021~(100)^2$
$C'(100) = 2-4+2.1$
$C'(100) = 0.1$
When the production level is 100 units, the additional cost of producing each item is $\$0.10$
(c) We can find the total cost of producing 101 items:
$C(101) = 920+2(101)-0.02~(101)^2+0.00007~(101)^3$
$C(101) = \$990.10$
We can find the total cost of producing 100 items:
$C(100) = 920+2(100)-0.02~(100)^2+0.00007~(100)^3$
$C(100) = \$990.00$
We can find the cost of producing the 101st item:
$C(101)- C(100) = \$990.10 - \$990 = \$0.10$
The cost of producing the 101st item is $\$0.10$
The value of $C'(100)$ is a very good approximation of the cost of producing the 101st item.
