Chapter 8 Multivariable Calculus Section 2 Partial Derivatives (Part 1)

2 Barnett/Ziegler/Byleen Business Calculus 12e Learning Objectives for Section 8.2 Partial Derivatives The student will be able to evaluate partial derivatives.

3 Barnett/Ziegler/Byleen Business Calculus 12e Introduction to Partial Derivatives

4 Barnett/Ziegler/Byleen Business Calculus 12e Example For a company producing only one type of surfboard, the cost function is C(x) = x, where x is the number of boards produced. Differentiating with respect to x, we obtain the marginal cost function C (x) = 70. This means that there is an increase in cost by $70 for a one-unit increase in the number of boards at any production level.

5 Barnett/Ziegler/Byleen Business Calculus 12e Example (continued)

6 Barnett/Ziegler/Byleen Business Calculus 12e Example (continued)

7 Barnett/Ziegler/Byleen Business Calculus 12e Partial Derivatives Notation & Definition If z = f (x, y), then the partial derivative of f with respect to x is defined by and is denoted by The partial derivative of f with respect to y is defined by and is denoted by

8 Barnett/Ziegler/Byleen Business Calculus 12e Example 1 Let z = f (x, y) = 3x 2 + 2xy – y 3. Find the partial derivative of z with respect to x. Find the partial derivative of z with respect to y.

9 Graphical Interpretation Barnett/Ziegler/Byleen Business Calculus 12e

10 Example 2 To find the slope of the line tangent to the function at P(1, 1, 3) that is parallel to the xz-plane, the y variable is treated as constant. Barnett/Ziegler/Byleen Business Calculus 12e The slope of the line tangent to the function at P(1, 1, 3) is 3.

11 Example 2 (continued) Barnett/Ziegler/Byleen Business Calculus 12e A slice of the graph showing the function in the xz-plane at y= 1 The slope of the tangent line is 3. A graph of z = x 2 + xy + y 2 is shown. For the partial derivative at (1, 1, 3), that leaves y constant. The corresponding tangent line is parallel to the xz-plane.

12 Barnett/Ziegler/Byleen Business Calculus 12e Example 3

13 Barnett/Ziegler/Byleen Business Calculus 12e Example 4 Using the Chain Rule Let f (x, y) = (5 + 2xy 2 ) 3 Hint: Think of the problem as z = u 3 and u = 5 + 2xy 2 a. Find f x (x, y) f x (x, y) = 3 (5 + 2xy 2 ) 2 · 2y 2 = 6y 2 (5 + 2xy 2 ) 2 b. Find f y (x, y) f y (x, y) = 3 (5 + 2xy 2 ) 2 · 4xy = 12xy(5 + 2xy 2 ) 2 The inside derivative. The outside derivative.

14 Example 5 Barnett/Ziegler/Byleen Business Calculus 12e Quotient Rule!

15 Homework Barnett/Ziegler/Byleen Business Calculus 12e

Chapter 8 Multivariable Calculus Section 2 Partial Derivatives (Part 2)

17 Barnett/Ziegler/Byleen Business Calculus 12e Learning Objectives for Section 8.2 Partial Derivatives The student will be able to evaluate second-order partial derivatives.

18 Review Barnett/Ziegler/Byleen Business Calculus 12e

19 Barnett/Ziegler/Byleen Business Calculus 12e Second-Order Partial Derivatives Taking a second-order partial derivative means taking a partial derivative of the first partial derivative. If z = f (x, y), then 1 st 2 nd 2 nd order partial derivatives are used to determine concavity.

20 Barnett/Ziegler/Byleen Business Calculus 12e Example 1 Let f (x, y) = x 3 y 3 + x + y 2 a. Find f xx (x, y) b. Find f xy (x, y) c. Find f yx (x, y)

21 Barnett/Ziegler/Byleen Business Calculus 12e Example 2

22 Barnett/Ziegler/Byleen Business Calculus 12e Example 3

23 Example 4 Barnett/Ziegler/Byleen Business Calculus 12e

24 Homework Barnett/Ziegler/Byleen Business Calculus 12e

Chapter 8 Multivariable Calculus Section 2 Partial Derivatives (Part 3)

26 Barnett/Ziegler/Byleen Business Calculus 12e Learning Objectives for Section 8.2 Partial Derivatives The student will be able to solve applications involving partial derivatives.

27 Example 1 Barnett/Ziegler/Byleen Business Calculus 12e

28 Example 1 (continued) Barnett/Ziegler/Byleen Business Calculus 12e At the current level of 4000 units of labor and 2500 units of capital, if we increase labor by 1-unit and keep capital fixed at 2500 units, this will increase production by approximately 4.53 units. Marginal productivity of labor

29 Example 1 (continued) Barnett/Ziegler/Byleen Business Calculus 12e At the current level of 4000 units of labor and 2500 units of capital, if we increase capital by 1-unit and keep labor fixed at 4000 units, this will increase production by approximately units. Marginal productivity of capital

30 Example 1 (continued) Barnett/Ziegler/Byleen Business Calculus 12e

31 Example 2 Barnett/Ziegler/Byleen Business Calculus 12e

32 Example 2 (continued) Barnett/Ziegler/Byleen Business Calculus 12e With a tire pressure of 22 psi and a speed of 40 mph, the car gets 42.6 mpg. The mpg increases by 0.3 mpg for each 1 psi increase in tire pressure and a fixed speed of 40 mph.

33 Example 2 (continued) Barnett/Ziegler/Byleen Business Calculus 12e The mpg decreases by 0.8 mpg for each 1 mph increase in speed and a fixed tire pressure of 22 psi.

34 Example 3  The Cycle Shop makes a high performance bike and a standard bike. The weekly demand and cost functions are: Barnett/Ziegler/Byleen Business Calculus 12e x = weekly demand for the high performance bike p = price of a high performance bike y = weekly demand for the standard bike q = price of a standard bike

35 Example 3 (continued) Barnett/Ziegler/Byleen Business Calculus 12e

36 Example 3 (cont.) Barnett/Ziegler/Byleen Business Calculus 12e If the company currently makes 10 high performance and 5 standard bikes: A.What is the change in profit if we increase the number high performance bikes by 1 and hold the number of standard bikes at 5? B.What is the change in profit if we increase the number standard bikes by 1 and hold the number of high performance bikes at 10? C.What would you recommend to the company?

37 Example 3 (continued) Barnett/Ziegler/Byleen Business Calculus 12e The company should increase the number of standard bikes by 1 and keep the number of high performance bikes at 5 because the profit increases by $80.

38 Homework Barnett/Ziegler/Byleen Business Calculus 12e