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
QKL
KL
KL K KL L− −
∂∂
ΔΔ+Δ
∂∂
=Δ+ Δ
⎛⎞ ⎛ ⎞
=× ××+×× ×⎜⎟ ⎜ ⎟
⎝⎠ ⎝ ⎠
=
That is, output will increase by approximately 22 units.