Multivariate Calculus Ch. 17

Multivariate Calculus 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 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)

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.

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)

Functions of Several Variables Cobb-Douglas Production Function has the form 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 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 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

Functions of Several Variables Let z = 100 and solve for y Cube both sides to express y as a function of x

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?

Isoquant (100, 100) (200, 25) iso (equal) quant (amount) 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 baby- boom generation considered a production function of the form 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 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.

Partial Derivatives Find f x and f y

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 f x (2, -1) and

Partial Derivatives Find f x (2, -1) and

Rate of Change Likewise, if z = f ( x, y ), then f x = 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 ). If y = f ( x ), then f ‘ ( x ) = the rate of change of y with respect to x

Rate of Change 1.Find the number of units produced when 27 units of labor and 64 units of capital are utilized. 2.Find and interpret P x (27, 64) (marginal productivity of labor). 3.What would be the approximate effect on production of increasing labor by 1 unit? 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.

Rate of Change 1.Find the number of units produced when 27 units of labor and 64 units of capital are utilized.

Rate of Change 2.Find and interpret P x (27, 64) (marginal productivity of labor).

Rate of Change 2.Find and interpret P x (27, 64) (marginal productivity of labor). 3.Production will increase by units if 1 unit of labor is added while capital is held constant.

Now You Try 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 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 f p ( p, i ) and f i ( 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: p 1 = the price of product 1, p 2 = the price of product 2 D 1 = demand for product 1, D 2 = demand for product 2 According to the law of demand, For substitute commodities, For complementary commodities,

Example Suppose the demand function for flour in a certain community is given by and the demand for bread is given by where p f is the dollar price of a pound of flour, and p b is the dollar price of a loaf of bread Determine whether flour and bread are substitute or complementary commodities or neither. and Flour and bread are complementary commodities

Now You Try Given the following pair of demand functions, use partial derivatives to determine whether the commodities are substitute, complementary, or neither. D 1 = Demand for product 1 p 1 = Price of product 1 D 2 = Demand for product 2 p 2 = Price of product 2

Chapter 17 End 