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Multivariate Calculus
Ch. 17
Multivariate Calculus
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
17.1 Functions of Several Variables
17.2 Partial Derivatives
17.1 Functions of Several Variables

If a company produces one product, x, at a cost of $10
each, then
C  x   10 x

If a company produces two products, x at a cost of $10
each and y at a cost of $15 each, then
C  x   10x  15 y

When x = 5 and y = 12, total cost is C(5, 12)
C  5, 12  10  5  15 12  230
Functions of Several Variables
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
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z = f (x, y) is a function of two independent variables if
a unique value of z is obtained from each ordered pair of
real numbers (x, y).
x and y are independent variables;
z is the dependent variable.
The set of all ordered pairs of real numbers (x, y) such
that f (x, y) is a real number is the domain of f;
the set of all values of f (x, y) is the range.
Functions of Several Variables

Example
f  x, y   2x2  xy  y 2 Find f (2, -1)
f  2,  1  2  2   2  1   1
2
2
 11

Production Function z = f (x, y)
 z = the quantity of an item produced as a function of x
and y, where x is the amount of labor and y is the
amount of capital needed to produce z units.
Functions of Several Variables
Cobb-Douglas Production Function has the form
z  P  x, y   Axa y1a
where A is a constant and 0 <  < 1
The graph of the xy-plane is an isoquant
Functions of Several Variables
Cobb-Douglas Production Function has the form
z  P  x, y   AL K 
where A is a constant and 0 <  < 1, and  = 1 - 
The graph of the xy-plane is an isoquant
Functions of Several Variables
Cobb-Douglas Production Function has the form
z  P  x, y   Axa y1a
where A is a constant and 0 <  < 1
The graph of the xy-plane is an isoquant
Find the combinations of labor and capital that will result
in an output of 100 units, given the Cobb-Douglas
production function
z  x 2 3 y1 3
Functions of Several Variables
z  x 2 3 y1 3

Let z = 100 and solve for y
100  x 2 3 y1 3
100
13

y
x2 3
Cube both sides to express y as a function of x
1003 1, 000, 000
y 2 
2
x
x
Functions of Several Variables
1, 000, 000
y
x2
How many units of capital combined with 100 workers
would result in an output of 100 units?
1, 000, 000
y
 100
2
100
How many units of capital combined with 200 workers
would result in an output of 100 units?
1, 000, 000
y
 25
2
200
Isoquant
100  x y
500
2 3 13
400
iso (equal)
quant (amount)
300
y
200
(100, 100)
100
(200, 25)
0
0
100
200
x
300
Each point (x, y) on the isoquant will result in an
output of 100 units.
Now You Try

A study of the connection between immigration and the
fiscal problems associated with the aging of the babyboom generation considered a production function of
the form
z  x 0.4 y 0.6
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where x represents the amount of labor and y the
amount of capital.
Find the equation of the isoquant at a production of 500.
17.2 Partial Derivatives
Let z  f  x, y 
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The partial derivative of f with respect to x is the
derivative of f obtained by treating x as a variable and y
as a constant.
The partial derivative of f with respect to y is the
derivative of f obtained by treating y as a variable and x
as a constant.
f x  x, y  ,
fx ,
z
,
x
f
x
are used to represent the partial derivative of z = f (x, y)
with respect to x.
Partial Derivatives
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Find fx and fy
f  x, y    x2 y  3xy  2xy 2  3 y
f x  2 xy  3 y  2 y 2
f y   x2  3x  4xy  3
f  x, y   ln  2 x  3 y 
2
fx 
2x  3 y
3
fy 
2x  3y
g ' x
Dx ln g  x   
g  x
Partial Derivatives
The notation
f x  a, b 
f
or
 a, b 
x
represents the value of a partial derivative of f with
respect to x, when x = a and y = b. (Similar symbols are
used for the partial derivative with respect to y.)
f  x, y    x2 y  3x4  8
Find fx (2, -1) and
f
 4,3
y
Partial Derivatives
f  x, y    x2 y  3x4  8
f x  2 xy  12 x
f y  x
2
3
Find fx (2, -1) and
f
 4,3
y
f x  2,  1  2  2  1  12  2   100
3
f y  4,3    4   16
2
Rate of Change
If y = f(x), then f ‘ (x) = the rate of change of y with respect to x
Likewise, if z = f(x, y), then fx = the rate of change of z with
respect to x if y is held constant.
A firm using x units of labor and y units of capital has a
production function P(x, y).
P
 the marginal productivity of labor
x
P
 the marginal productivity of capital
y
Rate of Change
A manufacturer estimates that its production function (in
hundreds of units) is given by
3
 1 1 3 2 1 3 
P  x, y    x  y 
3
3

where x is units of labor and y is units of capital.
1. Find the number of units produced when 27 units of labor
and 64 units of capital are utilized.
2. Find and interpret Px (27, 64) (marginal productivity of
labor).
3. What would be the approximate effect on production of
increasing labor by 1 unit?
Rate of Change
 1 1 3 2 1 3 
P  x, y    x  y 
3
3

3
1. Find the number of units produced when 27 units of labor
and 64 units of capital are utilized.
2
1 3
1 3 
1
P  27, 64     27    64  
3
3

3
 46.656  4, 665.6 units
Rate of Change
3
 1 1 3 2 1 3 
P  x, y    x  y 
3
3

2. Find and interpret Px (27, 64) (marginal productivity of
labor).
1 1 3 2 1 3
3
Let u  x  y and let P  u
3
3
1 4 3 
P P u
4 
 3u   x 
Then


x u x
 9

4
 1 1 3 2 1 3   1  4 3 
 3  x  y    x 
3
3
  9

1  4 3  1 1 3 2 1 3 
 x  x  y 
3
3
3

4
Rate of Change
3
 1 1 3 2 1 3 
P  x, y    x  y 
3
3

2. Find and interpret Px (27, 64) (marginal productivity of
labor).
4
1  4 3  1 1 3 2 1 3 
Px  x  x  y 
3
3
3

4
1
2
4 3  1
1 3
1 3 
Px  27, 64    27    27    64  
3
3
3

 .0041.1111  .1667 
4
 .0041167.91  .6884
3. Production will increase by 688.4 units if 1 unit of
labor is added while capital is held constant.
Now You Try
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A car dealership estimates that the total weekly sales of
its most popular model are a function of the car’s list
price p and the interest rate i in percent offered by the
manufacturer. The approximate weekly sales are given
by
f  p, i   99 p  .5 pi  .0025 p2
a. Find the weekly sales if the average list price is $19,400
and the manufacturer is offering an 8% interest rate.
b. Find and interpret fp(p, i) and fi(p, i)
c. What would be the effect on weekly sales if the price is
$19,400 and the interest rate rises from 8% to 9%?
Substitute and Complementary
Commodities
• Two commodities are said to be substitute
commodities if an increase in the quantity demanded
for either results in a decrease in the quantity
demanded for the other.
• Butter and margarine
• Two commodities are said to be complementary
commodities if a decrease in the quantity demanded
for either results in an decrease in the quantity
demanded for the other.
• 35 mm cameras and film
Substitute and Complementary
Commodities
Given:
p1 = the price of product 1, p2 = the price of product 2
D1 = demand for product 1, D2 = demand for product 2
According to the law of demand,
D1
D2
 0 and
0
p1
p2
For substitute commodities,
D1
D2
 0 and
0
p2
p1
D1
D2
 0 and
0
For complementary commodities,
p2
p1
Example
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Suppose the demand function for flour in a certain
community is given by
10
D f  p f , pb   500 
 5 pb
pf  2

pb
 5  0
and
and the demand for bread is given by
Db
7
 2  0
Db p f , pb  400  2 p f 
p f
pb  3
where pf is the dollar price of a pound of flour,
and pb is the dollar price of a loaf of bread

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D f

Determine whether flour and bread are substitute or
complementary commodities or neither.
Flour and bread are complementary commodities
Now You Try
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Given the following pair of demand functions, use
partial derivatives to determine whether the
commodities are substitute, complementary, or neither.
D1  500  6 p1  5 p2
D2  200  2 p1  5 p2
D1 = Demand for product 1
p1 = Price of product 1
D2 = Demand for product 2
p2 = Price of product 2
Chapter 17
End