1 8.3 Partial Derivatives Ex. Functions of Several Variables Chapter 8 Lecture 28

2 Partial Derivatives

3 The partial derivative of f with respect to x is the derivative of f with respect to x, when all other variables are treated as constants. Similarly, the partial derivative of f with respect to y is the derivative of f with respect to y, when all other variables are treated as constants. The partial derivatives are written

4 Ex. Partial Derivatives

5 Ex. Partial Derivatives

6 Geometric Interpretation of Partial Derivatives Plane y = b P z = f (x, y) is the slope of the tangent line at the point P(a,b, f (a,b)) along the slice through y = b. Ex.

7 Second-Order Partial Derivatives

8 Notation for Partial Derivatives

9 Marginal Cost: Linear Model  Suppose you own a company that makes two models of speakers, the Ultra Mini and the Big Stack. Your total monthly cost (in dollars) to make x Ultra Minis and y Big Stacks is given by Example: What is the significance  C/  x and  C/  y? Solution: The cost is increasing at a rate of $20 per additional Ultra Mini (if productions of Big Stacks is held constant). The cost is increasing at a rate of $40 per additional Big Stack (if productions of Ultra Mini is held constant).

10 Marginal Cost: Interaction Model  Another possibility for the cost function in the previous example is the interaction model Example: b. What is the marginal cost of manufacturing Big Stacks at a production level of 100 Ultra Minis and 50 Big Stacks per month? Solution: The marginal cost of manufacturing Ultra Minis increases by $0.1 for each Big Stack that is manufactured. a. What are the marginal costs of the two models of speakers?

11 The marginal cost of manufacturing Big Stack increases by $0.1 for each Ultra Minis that is manufactured.

12 Market Share (Cars and Light Trucks)  Based on data from , the relationship between the domestic market shares of three major U.S. manufacturers of cars and light trucks is Solution: Exercise: (Waner, Problem #43, Section 8.3) where x 1, x 2, and x 3 are, respectively, the fraction of the market held by Chrysler, Ford, and General Motors. Calculate  x 3 /  x 1 and  x 1 /  x 3. What do they signify, and how are they related to each other? General Motors’ market share decreases by 2.2% per 1% increase in Chrysler’s market share if Ford’s share is unchanged.

13 Chrysler’s market share decreases by 1% per 2.2% increase in General Motors’ market share if Ford’s share is unchanged. That is, the two partial derivatives are reciprocals of each other.