Applied Mathematics for Business and Economics

Lecture Note Differentiation

Example 1
Suppose the total cost in dollars of manufacturing q units of a certain commodity is

if the current level of production is 40 units, estimate how the

total cost will change if 40.5 units are produced.

()

Cq=++ 3510 q^2 q

Solution
In this problem, the current value of the variable isq= 40 and the change in the

variable isΔ=q 0.5, by the approximation formula, the corresponding change in cost is

Δ=CC()40.5 −C( (^40) )C′′( (^40) )Δ= ×qC( (^40) ) 0.5
since
Cq′′( ) 6=+q5 and C( 40 )=×+=6 40 5 245
it follows that
Δ×=×=CC ′() 40 0.5 245 0.5 $122.50
Example 2
The daily output at a certain factory is QL( )= 900 L^13 unitswhere Ldenotes the size
of the labor force measured in worker-hours. Currently, 1,000 worker-hours of labor
are used each day. Use calculus to estimate the number of additional worker-hours of
labor that will be needed to increase daily output by 15 units.
Solution
Solve forΔLusing approximating formula
ΔQQL L ′( )×Δ
with
Δ= =QL15, 1, 000 and QL L′( )= 300 −^23
to get
()
23
15 300 1, 000 L
−
≈ Δ
or
()
(^1523152)
1, 000 10 5 worker-hours
300 300

Δ≈L = × =

4.1 Approximation of Percentage change .....................................................

The percentage change of a quantity expresses the change in that quantity as a
percentage of its size prior to the change. In particular,
change in quantity
Percentage change 100
size of quantity

=

This formula can be combined with the approximation formula and written in
functional notation as follows.

Approximation Formula for Percentage Change

If Δxis a (small) change in x, the corresponding percentage change in the

function f(x)is

()

( )

()

Percentage change in 100 100

f f xx

f

fx fx

Δ ′ ×Δ

=×  ×