Applied Mathematics for Business and Economics

Lecture Note Function of Two Variables

3.3 Approximation of Percentage Change
The percentage change of a quantity expresses the change in the quantity as a
percentage of its size prior to the change. In perticular,
change in quantity
Percentage change 100
size of quantity

=

Approximation of Percentage Change

Suppose z is a function of x and y. If Δx denotes a small change in x and Δy

a small change in y , the corresponding percentage change in z is

Percentage change in 100 100

zz

x y

z xy

z

zz

∂ ∂

Δ +Δ

Δ ∂∂

= 

Example 4
Use calculus to approximate the percentage by which the volume of a cylinder
increases if the radius increases by 1 percent and the height increases by 2 percent.

Solution

The volume of a cylinder is given by the function Vrh( ,)=πrh^2

hh0.02

, where r is the

radius and h the height. The fact that r increases by 1 percent means that
and the fact that h increases by 2 percent means that

Δ=rr0.01
Δ =. By the
approximation formula for percentage change

() ()

2

2

22

2

2

2

Percentage change in 100

2 0.01 0.02

100

0.02 0.02

100

0.04

100 4 percent

VV

rh

V rh

V

rh r r h

rh

rh rh

rh

rh

rh

ππ

π

ππ

π

π

π

∂ ∂

Δ+ Δ

∂∂

+

=

+

=

==



Example 5
At a certain factory, output is given by the Cobb-Douglas production function

QKL AKL(), = α^1 −α, where A and αare positive constants with 01 <α< , and where

K denotes the capital investment and L the size of the labor force. Use calculus to
estiamate the percentage by which output will change if both capital and labor are
increased by 1 percent. (Answer: 1%)

4 Relative Maxima and Minima

In geometric terms, a relative maximimum of a functionf(xy, )is a peak, a point on

the surface that is higher than any nearby point on the surface. A relative

minimum is the bottom of a valley, a point that is lower than any nearby point on the
surface.

zfxy= (, )