Chapter 17 Multivariable Calculus

Chapter Outline Partial Derivatives Chapter 17: Multivariable Calculus Chapter Outline Partial Derivatives Applications of Partial Derivatives Implicit Partial Differentiation Higher-Order Partial Derivatives Chain Rule Maxima and Minima for Functions of Two Variables 17.1) 17.2) 17.3) 17.4) 17.5) 17.6)

Chapter Outline Lagrange Multipliers Lines of Regression Chapter 17: Multivariable Calculus Chapter Outline Lagrange Multipliers Lines of Regression Multiple Integrals 17.7) 17.8) 17.9)

=> Need functions of several variables

Notation: Notation for derivatives at a given point (a,b):

17.1 HW 5 7 16 17 26 33 39

17.1 HW 5 7 16 17 26 33 39

17.2 HW 4 5 9 10 12

17.2 HW 4 5 9 10 12

17.3 HW 5 7 10 11 13

17.3 HW 5 7 10 11 13

Example: Find the second partial derivatives of f(x, y) = 3xy2 – 2y + 5x2y2, and determine the value of fxy(–1, 2). Solution: Begin by finding the first partial derivatives with respect to x and y. fx(x, y) = 3y2 + 10xy2 and fy(x, y) = 6xy – 2 + 10x2y Then, differentiate each of these with respect to x and y. fxx(x, y) = 10y2 and fyy(x, y) = 6x + 10x2 fxy(x, y) = 6y + 20xy and fyx(x, y) = 6y + 20xy -- mixed partial derivatives At (–1, 2) the value of fxy is fxy(–1, 2) = 12 – 40 = –28. For continuous mixed partial derivatives order of the differentiation does NOT matter!!!

Example – Finding Higher-Order Partial Derivatives Show that fxz = fzx and fxzz = fzxz = fzzx for the function given by f(x, y, z) = yex + x ln z. Solution: First partials: Second partials (note that the first two are equal): Third partials (note that all three are equal):

17.4 HW 18 19 21 23

17.4 HW 18 19 21 23

Chain rule for a function of one variable:

17.5 HW 3 9 10 13 17

17.5 HW 3 9 10 13 17

Economics Applications

Note that oftentimes when you get only one critical point in a practical problem, it is clear from the set-up that it is desired min/max and we do not have to do 2nd derivative test to confirm.

FYI Shows how partial derivatives are used in a theoretical argument! -- main point

17.6 HW 14 19 20 23 27 29

17.6 HW 14 19 20 23 27 29

Thus, relative min is at (-1,2) -- this is approach of the previous section! Thus, relative min is at (-1,2)

Different approach is Lagrange Multipliers:

FYI

FYI Corresponding f = xy+yz: 12 4 4 12

17.7 HW 8 9 11 14 15 18

17.7 HW 8 9 11 14 15 18

Chapter 17: Multivariable Calculus 17.8 Lines of Regression