Math for Bus. and Eco. Chapter 3
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Chapter Three Function of Two Variables
description
function of two variables, finding relative extrema by derivative, constraint optimization
Transcript of Math for Bus. and Eco. Chapter 3
Chapter Three
Function of Two Variables
2
A function f of the two variables x and y is a rule that assigns to each ordered pair (x, y) of real numbers in some set one and only one real number denoted by f(x,y).
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The Domain of a Function of Two Variables
The domain of the function f(x, y) is the set of all ordered pairs (x, y) of real numbers for which f(x, y) can be evaluated.
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Example 1
For f(x, y)3xy2 find a) f(2,3)b) f(2, 2
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22,3 3 2 3 15f
2
2, 2 3 2 2 8f
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Example 2
For . Find the domain of f and f(0,1)
The domain
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, lnxf x y e y
( , ) / 0 and 0x y x y
0 0(0,1) ln1 0 1f e e
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Example 3
a) Find the domain of f.b) Compute f(1, 2)
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23 5,
x yf x y
x y
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Example 4
A pharmacy sells two brands of aspirin. Brand A sells for $1.25 per bottle and Brand B sells for $1.50 per bottle.a) What is the revenue function for aspirin?b) What is the revenue for aspirin if 100
bottles of Brand A and 150 bottles B are sold?
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Example 4 (Continued)
a) Let x the number of bottles of Brand A sold y the number of bottles of Brand B sold. Then, the revenue function is
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, 1.25 1.50R x y x y
b) 100,150 1.25 100 1.50 150
125 225
350
R
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Cobb-Douglas Production Functions
Economists use a formula called the Cobb-Douglas Production Functions to model the production levels of a company (or a country). Output Q at a factory is often regarded as a function of the amount K of capital investment and the size L of the labor force. Output functions of the form
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1,Q K L AK L
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Cobb-Douglas Production Functions (Cont.)
where A and are positive constants and 01 have proved to be especially useful in economic analysis. Such functions are known as Cobb-Douglas production function.
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Example 5
Suppose that the function Q500x0.3y0.7 represents the number of units produced by a company with x units of labor and y units of capital.a) How many units of a product will be
manufactured if 300 units of labor and 50 units of capital are used?
b) How many units will be produced if twice the number of units of labor and capital are used?
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Example 5 (Cont.)a)
b) If number of units of labor and capital are both doubled, then x2300600 and y250100
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0.3 0.7300,50 500 300 50
500 5.535 15.462
42,791 units
Q
0.3 0.7600,100 500 600 100
500 6.815 25.119
85,592 units
Q
DefinitionLet z f x, y)a) The first partial derivative of f with
respect to x is
b) The first partial derivative of f with respect to y is:
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Par
tial
Der
ivat
ives
0
, ,, limx x
f x x y f x yzf x y
x x
0
, ,, limy y
f x y y f x yzf x y
y y
Computation of Partial Derivatives
oThe function or fx is obtained by differentiating f with respect to x, treating y as a constant.
oThe function or fy is obtained by differentiating f with respect to y, treating x as a constant.
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Par
tial
Der
ivat
ives /z x
/z y
Example 1For the function
Find and
Treating y as a constant, we obtain
Treating x as a constant, we obtain
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Par
tial
Der
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ives
/z x /z y 2 2, 4 3 5f x y x xy y
8 3z
x yx
3 10z
x yy
Example 2, 3Find the partial derivatives fx and fy if
Find fx (1,2) and fy (1,2) if
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Par
tial
Der
ivat
ives
2 2 2, 2
3
yf x y x xy
x
2, xyf x y xe y
Example 4Suppose that the production function Q(x,y) 2000 x 0.5y0.5 is known. Determine the marginal productivity of labor and the marginal productivity of capital when 16 units of labor and 144 units of capital are used.
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Par
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Example 4 (Cont.)
substituting x16 and y144, we obtain
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0.50.5 0.5
0.5
0.50.5 0.5
0.5
10002000 0.5
10002000 0.5
Q yx y
x x
Q xx y
y y
0.5
0.5(16,144)
1000(144) 1000 123000 units
(16) 4
Q
x
0.5
0.5(16,144)
1000(16) 1000 4333.33 units
(144) 12
Q
y
Example 4 (Cont.)
Thus we see that adding one unit of labor will increase production by about 3000 units and adding one unit of capital will increase production by about 333 units.
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Par
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Der
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Example 5It is estimated that the weekly output at a certain plant is given by the function
Q(x,y) 1200x500yx2 yx3 y2 units, where x is the number of skilled workers and y the number of unskilled workers employed at the plant. Currently the work force consists of 30 skilled workers, and 60 unskilled workers. Use marginal analysis to estimate the change in the weekly output that will result from the addition of 1 more skilled worker if the number of unskilled workers is not changed.
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Par
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Second-Order Partial Derivatives
If z f(x, y) the partial derivative of fx with respect to x is
the partial derivative of fx with respect to y is
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Par
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Der
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2
2orxx x x
z zf f
x x x
2
orxy x y
z zf f
y x y x
Second-Order Partial Derivatives (Cont.)
If z f(x, y) the partial derivative of fy with respect to x is
the partial derivative of fy with respect to y is
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Par
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Der
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2
oryx y x
z zf f
x y x y
2
2oryy y y
z zf f
y y y
Example 6
Compute the four second-order partial derivatives of the function
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Par
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3 2, 5 2 1f x y xy xy x
Example 7
Find all four second partial derivatives of
then find
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Par
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2, ln 4f x y x y
2,1 2xxf
Remark
The two partial derivatives fxy and fyx are sometimes called the mixed second-order partial derivatives of f and fxyfyx .
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Chain Rule for Partial Derivatives
Suppose z is a function of x and y, each of which is a function of t then z can be regarded as a function of t and
26The
Cha
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App
roxi
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ion
by th
e To
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iffe
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ial
dz z dx z dy
dt x dt y dt
Chain Rule for Partial Derivatives
Remark 1
rate of change of z with respect
to t for fixed y. rate of change of z with respect to t for fixed x.
27The
Cha
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ule;
App
roxi
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ion
by th
e To
tal D
iffe
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ial
z dx
x dt
z dy
y dt
Example 1
Find
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Cha
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App
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by th
e To
tal D
iffe
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ial 2 2if 3 , 2 1,and
dzz x xy x t y t
dt
Example 1
By the chain rule,
Which you can rewrite in terms of t
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2 3 2 3 2dz z dx z dy
x y x tdt x dt y dt
2 24(2 1) 6 3(2 1)(2 ) 18 14 4dz
t t t t t tdt
The
Cha
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ule;
App
roxi
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ion
by th
e To
tal D
iffe
rent
ial
Example 1
By the chain rule,
Which you can rewrite in terms of t
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2 3 2 3 2dz z dx z dy
x y x tdt x dt y dt
2 24(2 1) 6 3(2 1)(2 ) 18 14 4dz
t t t t t tdt
The
Cha
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ule;
App
roxi
mat
ion
by
the
Tota
l Dif
fere
ntia
l
Example 2A health store carries two kinds of multiple vitamins, Brand A and Brand B. Sales figures indicate that if Brand A is sold for x dollars per bottle and Brand B for y dollars per bottle, the demand for Brand A will be
It is estimated that t months from now the price of Brand A will be
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The
Cha
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App
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by
the
Tota
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2, 300 20 30 bottles/month Q x y x y
2 0.05 dollars per bottlex t
Example 2 (Cont.)It is estimated that t months from now the price of Brand A will be
and the price of Brand B will be
At what rate will the demand for Brand A be changing with respect to time 4 months from now?
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The
Cha
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App
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by
the
Tota
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2 0.05 dollars per bottlex t
2 0.1 dollars per bottley t
Example 2 (Cont.)Our goal is to find dQ/dt when t 4. Using chain rule, we obtain
whenand hence,
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The
Cha
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App
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ion
by
the
Tota
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fere
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1 240 0.05 30 0.05
dQ Q dx Q dy
dt x dt y dt
x t
4, 2 0.05 4 2.2t x
40 2.2 0.05 30 0.05 0.5 3.65dQ
dt
Approximation Formula
Suppose z is a function of x and y. If ∆x denotes a small change in x and ∆y a small change in y, the corresponding change in z is
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The
Tot
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iffe
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ial
z zz x y
x y
Remark 2
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The
Tot
al d
iffe
rent
ial change in z due to the
change in x for fixed y.
change in z due to the change in y for fixed x.
z
xx
z
yy
The Total Differential
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The
Tot
al d
iffe
rent
ial If z is a function of x and y, the total
differential of z is z z
dz x yx y
Example 3
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The
Tot
al d
iffe
rent
ial
At a certain factory, the daily output is Q=60 K1/2 L2/3 units, 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.
Example 3
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The
Tot
al d
iffe
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ial
Apply the approximation formula with K900, L1000, ∆K1 and ∆L2 to get
That is, output will increase by approximately 22 units.
1/2 1/3 1/2 2/330 20
1 130 10 1 20 30 2
30 100
22 units
Q QQ K L
K L
K L K K L L
percentage change
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App
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of P
erce
ntag
e C
hang
e
The percentage change of a quantity expresses the change in the quantity as a percentage of its size prior to the change. In particular,
change in quantityPercentage change 100
size of quantity
Approximation of Percentage Change
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App
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mat
ion
of P
erce
ntag
e C
hang
e
Suppose z is a function of x and y. If ∆x denotes a small change in x and ∆x a small change in y , the corresponding percentage change in z is
% 100 100
z zx y
z x yz
z z
