Applied Mathematics for Business and Economics

Lecture Note Differentiation

Example 3
The GNP of a certain country wasNt t()=++^25 t 200 billion dollars t years after

1990. Use calculus to estimate the percentage change in the GNP during the first
      quarter of 1998.

Solution
Use the formula

()

()

Percentage change in 100

Nt t

N

Nt

′ Δ



Witht= 8 , Δ=t 0.25 and Nt′()= 25 t+ to get

( )

()

()

2

80.25

Percentage change in 100

8

28 50.25

100

858200

1.73 percent

N

N

N

′ ×

×+

+×+







Example 4

At a certain factory, the daily outputis QK( )=4, 000K^12 units, where K denotes the

firm’s capital investment. Use calculus to estimate the percentage increase in output
that will result from a 1 percent increase in capital investment.

Solution

The derivative isQK′()=2, 000K−^12. The fact that K increases by 1 percent means

thatΔ=K 0.01K. Hence,

( )

()

(^12) ()
12
Percentage change in 100

2, 000 0.01

100

4, 000

0.5 percent

QK K

Q

QK

kK

K

−

′ Δ

=

=



4.2 Marginal Analysis in Economics ............................................................

In economics, the use of the derivative to approximate the change in a function
produced by a 1-unit change in its variable is called marginal analysis. For example,

if Cx( )is the total production cost incurred by a manufacturer when x units are

produced andR(x)is the total revenue derived from the sale of x units, then is

called the marginal cost and

Cx′()

R′(x) is called the marginal revenue. If production (or

sales) is increased by 1 unit, then Δx= 1 and the approximation formula:

Δ= +Δ− ≈ ΔCCx x Cx Cx x( ) ( ) ′( )

becomes
Δ= +− ≈CCx() 1 Cx Cx( ) ′( )

while

ΔR=+Δ− ≈ ΔRx x Rx R x x()( ) ′( )

becomes