Applied Mathematics for Business and Economics

Lecture Note Function of Two Variables

That is, 4 months from now the monthly demand for Brand A will be decreasing at
the rate of 3.65 bottles per month.

3.2 The Total differential
Recall from chapter 2 that if y is a function of x,
dy
yx
dx

Δ  Δ

whereΔxis a small change in the variable x and Δyis the corresponding change in the

function y.The expression

dy

dy x

dx

=Δthat was used to approximate Δywas called the

differential of y. Here is the analogous approximation formula for functions of two
variables.

Approximation Formula

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 change in z is

zz

zxy

x y

∂ ∂

ΔΔ+

∂∂

 Δ

Remark 2
z
x
x

∂

Δ

∂

change in z due to the change in x for fixed y.

z

y

y

∂

Δ

∂

change in z due to the change in y for fixed x.

The Total Differential

If z is a function of x and y, the total differential of z is

zz

dz x y

x y

∂ ∂

= Δ+ Δ

∂∂

Example 3

At a certain factory, the daily output isQKL= 60 12 13units, where K denotes the

capital investment measured in units of $1,000 and L the size of the labor force
measured in worker-hours. The current capital investment is $ 900,000 and 1,000 and
labor are used each day. Estimate the change in output that will result if capital
investment is increased by $1,000 and labor is increased by 2 worker-hours.

Solution
Apply the approximation formula with K=900, L=1000, ΔKL=Δ=1, a n d 2 to get^

30 1/ 2 1/ 3 20 1/ 2 2/ 3

11

30 10 1 20 30 2

30 100

22 units

QQ

QKL

KL

KL K KL L− −

∂∂

ΔΔ+Δ

∂∂

=Δ+ Δ

⎛⎞ ⎛ ⎞

=× ××+×× ×⎜⎟ ⎜ ⎟

⎝⎠ ⎝ ⎠

=



That is, output will increase by approximately 22 units.