1. Chapter 17
Multivariable Calculus

2. 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)

3. 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)

4. => Need functions of several variables

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

8. 17.1 HW

9. 17.1 HW

17. 17.2 HW

18. 17.2 HW

22. 17.3 HW

23. 17.3 HW

24. 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 – x2y
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!!!

25. 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):

27. 17.4 HW

28. 17.4 HW

29. Chain rule for a function of one variable:

34. 17.5 HW

35. 17.5 HW

42. Economics Applications

43. 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.

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

45. 17.6 HW

46. 17.6 HW

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

48. Different approach is Lagrange Multipliers:

52. FYI

53. FYI
Corresponding f = xy+yz:

54. 17.7 HW

55. 17.7 HW

56. Chapter 17: Multivariable Calculus
17.8 Lines of Regression