Multivariate Calculus

1
Multivariate Calculus

• Ch. 14

2
Multivariate Calculus

• 14.1 Functions of Several Variables
• 14.2 Partial Derivatives

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14.1 Functions of Several Variables

• If a company produces one product, x, at a cost
  of 10 each, then
• If a company produces two products, x at a cost
  of 10 each and y at a cost of 15 each, then
• When x 5 and y 12, total cost is C(5, 12)

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Functions of Several Variables

• 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.

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Functions of Several Variables

• Example
• 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.

Find f (2, -1)
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Functions of Several Variables

• Cobb-Douglas Production Function has the form
• where A is a constant and 0 lt a lt 1

The graph of the xy-plane is an isoquant
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Functions of Several Variables

• Cobb-Douglas Production Function has the form
• where A is a constant and 0 lt a lt 1

The graph of the xy-plane is an isoquant
8
Functions of Several Variables

• Cobb-Douglas Production Function has the form
• where A is a constant and 0 lt a lt 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
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Functions of Several Variables

• Let z 100 and solve for y

Cube both sides to express y as a function of x
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Functions of Several Variables
How many units of capital combined with 100
workers would result in an output of 100 units?
How many units of capital combined with 200
workers would result in an output of 100 units?
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Isoquant
iso (equal) quant (amount)
(100, 100)
(200, 25)
Each point (x, y) on the isoquant will result in
an output of 100 units.
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Now You Try pg. 879, 20
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Now You Try pg. 879, 20
14
14.2 Partial Derivatives

• 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.

are used to represent the partial derivative of z
f (x, y) with respect to x.
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Partial Derivatives

• Find fx and fy

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Partial Derivatives

• The notation
• 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.)

Find fx (2, -1) and
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Partial Derivatives
Find fx (2, -1) and
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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).
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Rate of Change
A manufacturer estimates that its production
function (in hundreds of units) is given
by where x is units of labor and y is units of
capital.

• Find the number of units produced when 27 units
  of labor and 64 units of capital are utilized.
• Find and interpret fx (27, 64) (marginal
  productivity of labor).
• What would be the approximate effect on
  production of increasing labor by 1 unit?

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Rate of Change

• Find the number of units produced when 27 units
  of labor and 64 units of capital are utilized.

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Rate of Change

• Find and interpret fx (27, 64) (marginal
  productivity of labor).

22
Rate of Change

• Find and interpret fx (27, 64) (marginal
  productivity of labor).

• Production will increase by 688.4 units if 1 unit
  of labor is added while capital is held constant.

23
Now You Try pg. 889, 43
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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

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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,
For substitute commodities,
For complementary commodities,
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Example

• Suppose the demand function for flour in a
  certain community is given by
• and the demand for bread is given by
• where pf is the dollar price of a pound of flour,
  and pb is the dollar price of a loaf
  of bread

and
Determine whether flour and bread are substitute
or complementary commodities or neither.
Flour and bread are complementary commodities
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Now You Try

• Given the following pair of demand functions, use
  partial derivatives to determine whether the
  commodities are substitute, complementary, or
  neither.

D1 Demand for product 1 p1 Price of product
1 D2 Demand for product 2 p2 Price of product
2
28
?
Chapter 14
End