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= (, )