Page 222 - How To Solve Word Problems In Calculus

Solution
Exact solution
The cost of producing 101 baseballs is

2
C(101) = 0.05(101) + 0.5(101) + 50 = $610.55
The cost of producing 100 baseballs is
2
C(100) = 0.05(100) + 0.5(100) + 50 = $600.00
st
The cost of producing the 101 baseball is
C(101) − C(100) = 610.55 − 600.00 = $10.55

Approximation using marginal cost
MC(x) = C (x) = 0.1x + 0.5

MC(100) = C (100) = 0.1(100) + 0.5 = $10.50

The derivative offers an excellent approximation.

EXAMPLE 7
A manufacturer estimates that the cost of producing x units of
2
a product is C(x) = 0.1x + 3x + 100 dollars and that he will
be able to sell x units when the price is
p(x) = 25 − 0.05x dollars per unit.

(a) Determine the marginal cost function and use it to ap-
proximate the cost of producing the 21st unit. Compare
this with the actual cost.
(b) Determine the marginal revenue function for this product.
Use it to approximate the additional revenue from the sale
of the 21st unit and compare with the actual value.

Solution
(a) MC(x) = C (x) = 0.2x + 3

MC(20) = $7.00

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