Multivariable Calculus: Functions, Partial Derivatives, and Optimization Methods

1. 8 • Functions of Several Variables • Partial Derivatives • Maxima and Minima of Functions of Several Variables • The Method of Least Squares • Constrained Maxima and Minima and the Method of Lagrange Multipliers • Double Integrals Calculus of Several Variables

2. 8.1 Functions of Several Variables

3. Functions of Two Variables A real-valued function of two variablesf, consists of • A set A of ordered pairs of real numbers (x, y) called the domain of the function. • A rule that associates with each ordered pair in the domain of fone and onlyone real number, denoted by z =f(x, y).

4. Examples • Let f be the function defined by • Compute f(0, 0), f(1, 2), and f(2, 1). Solution • The domain of a function of two variablesf(x, y), is a set of ordered pairs of real numbers and may therefore be viewed as a subset of thexy-plane. Example 1, page 536

5. Examples • Find the domain of the function Solution • f(x, y) is defined for all real values of x and y, so the domain of the function f is the set of all points(x, y)in thexy-plane. Example 2, page 536

6. Examples • Find the domain of the function Solution • g(x, y) is defined for all x≠y, so the domain of the function g is the set of all points(x, y)in thexy-plane exceptthose lying on the y = x line. y y = x x Example 2, page 536

7. Examples • Find the domain of the function Solution • We require that 1 – x2 – y2 0 or x2 + y2 1 which is the set of all points(x, y) lying on and inside thecircle of radius1 with center at the origin: y 1 –1 x2 + y2= 1 x –1 1 Example 2, page 536

8. Applied Example: Revenue Functions • Acrosonic manufactures a bookshelf loudspeaker system that may be bought fully assembled or in a kit. • The demand equations that relate the unit price, p and q, to the quantities demanded weekly, x and y, of the assembled and kit versions of the loudspeaker systems are given by • What is the weekly total revenue function R(x, y)? • What is the domain of the function R? Applied Example 3, page 537

9. Applied Example: Revenue Functions Solution • The weekly revenue from selling x units assembled speaker systems at p dollars per unit isgiven byxp dollars. Similarly, the weekly revenue from selling yspeaker kits at q dollars per unit isgiven byyq dollars. Therefore, the weekly total revenuefunctionR is given by Applied Example 3, page 537

10. Applied Example: Revenue Functions Solution • To find the domain of the functionR, note that the quantities x, y, p, and q must be nonnegative, which leads to the following system of linear inequalities: Thus, the graph of the domain is: y 2000 1000 D x 1000 2000 Applied Example 3, page 537

11. Graphs of Functions of Two Variables • Consider the task of locating P(1, 2, 3) in 3-space: • One method to achieve this is to start at theoriginand measure out from there, axis by axis: z P(1, 2, 3) 3 y 1 2 x

12. Graphs of Functions of Two Variables • Consider the task of locating P(1, 2, 3) in 3-space: • Another common method is to find thexycoordinate and from there elevate to the level of the z value: z P(1, 2, 3) 3 y 2 1 (1, 2) x

13. Graphs of Functions of Two Variables • Locate the following points in 3-space: Q(–1, 2, 3),R(1, 2, –2),andS(1, –1, 0). Solution z Q(–1, 2, 3) 3 y S(1, –1, 0) –2 x R(1, 2, –2)

14. Graphs of Functions of Two Variables • The graph of a function in 3-space is a surface. • For every (x, y) in the domain of f, there is a z value on the surface. z z = f(x, y) (x, y, z) y (x, y) x

15. Level Curves • The graph of a function of two variables is often difficult to sketch. • It can therefore be useful to apply the method used to construct topographic maps. • This method is relatively easy to apply and conveys sufficient information to enable one to obtain a feel for the graph of the function.

16. Level Curves • In the 3-space graph we just saw, we can delineate the contour of the graph as it is cut by a z = c plane: z z = f(x, y) z = c y f(x, y) = c x

17. Examples • Sketch a contour map of the function f(x, y) = x2 + y2. Solution • The function f(x, y) = x2 + y2 is a revolving parabola called a paraboloid. z f(x, y) = x2 + y2 y Example 5, page 540 x

18. Examples • Sketch a contour map of the function f(x, y) = x2 + y2. Solution • A level curve is the graph of the equation x2 + y2 = c, which describes a circle with radius . • Taking different values of c we obtain: z f(x, y) = x2 + y2 y x2 + y2 = 16 4 2 –2 –4 x2 + y2 = 9 x2 + y2 = 4 x2 + y2 = 1 x2 + y2 = 16 x2 + y2 = 0 x –4 –2 2 4 x2 + y2 = 9 x2 + y2 = 4 x2 + y2 = 1 y x2 + y2 = 0 Example 5, page 540 x

19. Examples • Sketch level curves of the function f(x, y) = 2x2 – y corresponding to z = –2, –1, 0, 1, and 2. Solution • The level curves are the graphs of the equation 2x2 – y= k or for k = –2, –1, 0, 1, and 2: y 2x2 – y= –2 4 3 2 1 0 –1 –2 2x2 – y= –1 2x2 – y= 0 2x2 – y= 1 2x2 – y= 2 x –2 –1 1 2 Example 6, page 540

20. 8.2 Partial Derivatives

21. First Partial Derivatives First Partial Derivatives of f(x, y) • Suppose f(x, y) is a function of two variablesx and y. • Then, the first partial derivative of fwith respect to xatthe point(x, y) is provided the limit exists. • The first partial derivative of fwith respect to yatthe point(x, y) is provided the limit exists.

22. Geometric Interpretation of the Partial Derivative z f(x, y) y x

23. Geometric Interpretation of the Partial Derivative z f(x, y) f(x, b) y = bplane b y a (a, b) x

24. Geometric Interpretation of the Partial Derivative z f(x, y) y x

25. Geometric Interpretation of the Partial Derivative z f(x, y) f(c, y) x = c plane d y c (c, d) x

26. Examples • Find the partial derivatives∂f/∂xand ∂f/∂yof the function • Use the partials to determine the rate of change of f in the x-direction and in the y-direction at the point (1, 2) . Solution • To compute ∂f/∂x, think of the variabley as a constant and differentiate the resulting function of xwith respect tox: Example 1, page 546

27. Examples • Find the partial derivatives∂f/∂xand ∂f/∂yof the function • Use the partials to determine the rate of change of f in the x-direction and in the y-direction at the point (1, 2). Solution • To compute ∂f/∂y, think of the variablex as a constant and differentiate the resulting function of ywith respect toy: Example 1, page 546

28. Examples • Find the partial derivatives∂f/∂xand ∂f/∂yof the function • Use the partials to determine the rate of change of f in the x-direction and in the y-direction at the point (1, 2). Solution • The rate of change of f in the x-direction at the point (1, 2) is given by • The rate of change of f in the y-direction at the point (1, 2) is given by Example 1, page 546

29. Examples • Find the firstpartial derivativesof the function Solution • To compute ∂w/∂x, think of the variabley as a constant and differentiate the resulting function of xwith respect tox: Example 2, page 547

30. Examples • Find the firstpartial derivativesof the function Solution • To compute ∂w/∂y, think of the variablex as a constant and differentiate the resulting function of ywith respect toy: Example 2, page 547

31. Examples • Find the firstpartial derivativesof the function Solution • To compute ∂g/∂s, think of the variablet as a constant and differentiate the resulting function of swith respect tos: Example 2, page 547

32. Examples • Find the firstpartial derivativesof the function Solution • To compute ∂g/∂t, think of the variables as a constant and differentiate the resulting function of twith respect tot: Example 2, page 547

33. Examples • Find the firstpartial derivativesof the function Solution • To compute ∂h/∂u, think of the variablev as a constant and differentiate the resulting function of uwith respect tou: Example 2, page 547

34. Examples • Find the firstpartial derivativesof the function Solution • To compute ∂h/∂v, think of the variableu as a constant and differentiate the resulting function of vwith respect tov: Example 2, page 547

35. Examples • Find the firstpartial derivativesof the function Solution • Here we have a function ofthree variables, x,y, and z, and we are required to compute • For short, we can label these first partial derivatives respectively fx, fy, and fz. Example 3, page 549

36. Examples • Find the firstpartial derivativesof the function Solution • To find fx, think of the variablesy and z as a constant and differentiate the resulting function of xwith respect tox: Example 3, page 549

37. Examples • Find the firstpartial derivativesof the function Solution • To find fy, think of the variablesx and z as a constant and differentiate the resulting function of ywith respect toy: Example 3, page 549

38. Examples • Find the firstpartial derivativesof the function Solution • To find fz, think of the variablesx and y as a constant and differentiate the resulting function of zwith respect toz: Example 3, page 549

39. The Cobb-Douglas Production Function • The Cobb-Douglass Production Function is of the form f(x, y) = axby1–b (0 < b < 1) where a and b are positive constants, x stands for the cost of labor, y stands for the cost of capital equipment, and f measures the output of the finished product.

40. The Cobb-Douglas Production Function • The Cobb-Douglass Production Function is of the form f(x, y) = axby1–b (0 < b < 1) • The first partial derivativefxis called the marginal productivity of labor. • It measures the rate of change of production with respect to the amount of money spent on labor, with the level of capital kept constant. • The first partial derivativefyis called the marginal productivity of capital. • It measures the rate of change of production with respect to the amount of money spent on capital, with the level of labor kept constant.

41. Applied Example: Marginal Productivity • A certain country’s production in the early years following World War II is described by the function f(x, y) =30x2/3y1/3 when x units of labor and y units of capital were used. • Compute fx and fy. • Find the marginal productivity of labor and the marginal productivity of capital when the amount expended on labor and capital was 125 units and 27 units, respectively. • Should the government have encouraged capital investment rather than increase expenditure on labor to increase the country’s productivity? Applied Example 4, page 550

42. Applied Example: Marginal Productivity f(x, y) =30x2/3y1/3 Solution • The first partial derivatives are Applied Example 4, page 550

43. Applied Example: Marginal Productivity f(x, y) =30x2/3y1/3 Solution • The required marginal productivity of labor is given by or 12 units of output per unit increase inlabor expenditure (keeping capital constant). • The required marginal productivity of capital is given by or 27 7/9 units of output per unit increase incapital expenditure (keeping labor constant). Applied Example 4, page 550

44. Applied Example: Marginal Productivity f(x, y) =30x2/3y1/3 Solution • The government should definitely have encouraged capital investment. • A unitincrease in capital expenditure resulted in a much faster increase in productivity than a unitincrease in labor: 27 7/9 versus 12 per unit of investment, respectively. Applied Example 4, page 550

45. Second Order Partial Derivatives • The first partial derivativesfx(x, y) andfy(x, y) of a function f(x, y) of two variables x and y are also functions of x and y. • As such, we may differentiate each of the functions fx and fy to obtain the second-order partial derivatives of f.

46. Second Order Partial Derivatives • Differentiating the function fxwith respect tox leads to the second partial derivative • But the function fx can also be differentiatedwith respect toy leading to a different second partial derivative

47. Second Order Partial Derivatives • Similarly, differentiating the function fywith respect toy leads to the second partial derivative • Finally, the function fy can also be differentiatedwith respect tox leading to the second partial derivative

48. Second Order Partial Derivatives • Thus, foursecond-order partial derivatives can be obtained of a function of two variables: When both are continuous

49. Examples • Find the second-order partial derivatives of the function Solution • First, calculate fx and use it to find fxx and fxy: Example 6, page 552

50. Examples • Find the second-order partial derivatives of the function Solution • Then, calculate fy and use it to find fyx and fyy: Example 6, page 552