1. Chapter 17
Multivariable Calculus

2. INTRODUCTORY MATHEMATICAL ANALYSIS
0. Review of Algebra
Applications and More Algebra
Functions and Graphs
Lines, Parabolas, and Systems
Exponential and Logarithmic Functions
Mathematics of Finance
Matrix Algebra
Linear Programming
Introduction to Probability and Statistics

3. INTRODUCTORY MATHEMATICAL ANALYSIS
Additional Topics in Probability
Limits and Continuity
Differentiation
Additional Differentiation Topics
Curve Sketching
Integration
Methods and Applications of Integration
Continuous Random Variables
Multivariable Calculus

4. Chapter 17: Multivariable Calculus
Chapter Objectives
To discuss functions of several variables and to compute function values.
To compute partial derivatives.
To develop the notions of partial marginal cost, marginal productivity, and competitive and complementary products.
To find partial derivatives of a function defined implicitly.
To compute higher-order partial derivatives.
To find the partial derivatives of a function by using the chain rule.

5. Chapter 17: Multivariable Calculus
Chapter Objectives
To apply the second-derivative test for a function of two variables.
To find critical points for a function.
To develop the method of least squares.
To compute double and triple integrals.

6. Chapter Outline Functions of Several Variables Partial Derivatives
Chapter 17: Multivariable Calculus
Chapter Outline
17.1)
Functions of Several Variables
Partial Derivatives
Applications of Partial Derivatives
Implicit Partial Differentiation
Higher-Order Partial Derivatives
Chain Rule
Maxima and Minima for Functions of Two Variables
17.2)
17.3)
17.4)
17.5)
17.6)
17.7)

7. Chapter Outline Lagrange Multipliers Lines of Regression
Chapter 17: Multivariable Calculus
Chapter Outline
Lagrange Multipliers
Lines of Regression
Multiple Integrals
17.8)
17.9)
17.10)

8. 17.1 Functions of Several Variables
Chapter 17: Multivariable Calculus
17.1 Functions of Several Variables
Example 1 – Functions of Two Variables
A function can involve 2 or more variables, e.g.
a is a function of two variables. Because the denominator is zero when y = 2, the domain of f is all
(x, y) such that y  2.
b. h(x, y) = 4x defines h as a function of x and y. The domain is all ordered pairs of real numbers.
c. z2 = x2 + y2 does not define z as a function of x and y.

9. The plane intersects the x-axis when y = 0 and z = 0.
Chapter 17: Multivariable Calculus
17.1 Functions of Several Variables
Example 3 – Graphing a Plane
Sketch the plane
Solution:
The plane intersects the x-axis when y = 0 and z = 0.

10. Sketch the surface z = x2 is a parabola.
Chapter 17: Multivariable Calculus
17.1 Functions of Several Variables
Example 5 – Sketching a Surface
Sketch the surface
Solution:
z = x2 is a parabola.

11. Chapter 17: Multivariable Calculus
17.2 Partial Derivatives
Partial derivative of f with respect to x is given by
Partial derivative of f with respect to y is given by

12. Solution: The partial derivatives are
Chapter 17: Multivariable Calculus
17.2 Partial Derivatives
Example 1 – Finding Partial Derivatives
If f(x, y) = xy2 + x2y, find fx(x, y) and fy(x, y). Also, find fx(3, 4) and fy(3, 4).
Solution: The partial derivatives are
Thus, the solutions are

13. By partial differentiating, we get
Chapter 17: Multivariable Calculus
17.2 Partial Derivatives
Example 3 – Partial Derivatives of a Function of
Three Variables
If find
Solution:
By partial differentiating, we get

14. By partial differentiating, we get
Chapter 17: Multivariable Calculus
17.2 Partial Derivatives
Example 4 – Finding Partial Derivatives
If find
Solution:
By partial differentiating, we get

15. 17.3 Applications of Partial Derivatives
Chapter 17: Multivariable Calculus
17.3 Applications of Partial Derivatives
Example 1 – Marginal Costs
Interpretation of rate of change:
A company manufactures two types of skis, the Lightning and the Alpine models. Suppose the joint-cost function for producing x pairs of the Lightning model and y pairs of the Alpine model per week is
where c is expressed in dollars. Determine the marginal costs ∂c/∂x and ∂c/∂y when x = 100 and y = 50, and interpret the results.

16. Solution: The marginal costs are
Chapter 17: Multivariable Calculus
17.3 Applications of Partial Derivatives
Example 1 – Marginal Costs
Solution: The marginal costs are
Thus,
and

17. Chapter 17: Multivariable Calculus
17.3 Applications of Partial Derivatives
Example 3 – Marginal Productivity
A manufacturer of a popular toy has determined that the production function is P = √(lk), where l is the number of labor-hours per week and k is the capital (expressed in hundreds of dollars per week) required for a weekly production of P gross of the toy. (One gross is 144 units.) Determine the marginal productivity functions, and evaluate them when l = 400 and k = 16. Interpret the results.

18. Solution: Since , Thus, Chapter 17: Multivariable Calculus
17.3 Applications of Partial Derivatives
Example 3 – Marginal Productivity
Solution: Since ,
Thus,

19. 17.4 Implicit Partial Differentiation
Chapter 17: Multivariable Calculus
17.4 Implicit Partial Differentiation
Example 1 – Implicit Partial Differentiation
We will look into how to find partial derivatives of a function defined implicitly.
If , evaluate when x = −1, y = 2, and
z = 2.
Solution: Using partial differentiation, we get

20. 17.5 Higher-Order Partial Derivatives
Chapter 17: Multivariable Calculus
17.5 Higher-Order Partial Derivatives
Example 1 – Second-Order Partial Derivatives
We obtain second-order partial derivatives of f as
Find the four second-order partial derivatives of
Solution:

21. Solution: By implicit differentiation,
Chapter 17: Multivariable Calculus
17.5 Higher-Order Partial Derivatives
Example 3 – Second-Order Partial Derivative of an Implicit Function
Determine if
Solution: By implicit differentiation,
Differentiating both sides with respect to x, we obtain
Substituting ,

22. Chapter 17: Multivariable Calculus
17.6 Chain Rule
If f, x, and y have continuous partial derivatives, then z is a function of r and s, and

23. The demand functions for the cameras and film are given by
Chapter 17: Multivariable Calculus
17.6 Chain Rule
Example 1 – Rate of Change of Cost
For a manufacturer of cameras and film, the total cost c of producing qC cameras and qF units of film is given by
The demand functions for the cameras and film are given by
where pC is the price per camera and pF is the price per unit of film. Find the rate of change of total cost with respect to the price of the camera when pC = 50 and pF = 2.

24. Solution: By the chain rule,
Chapter 17: Multivariable Calculus
17.6 Chain Rule
Example 1 – Chain Rule
Example 3 – Chain Rule
Solution: By the chain rule,
Thus,
a. Determine ∂y/∂r if
Solution: By the chain rule,

25. Solution: By the chain rule,
Chapter 17: Multivariable Calculus
17.6 Chain Rule
Example 3 – Chain Rule
b. Given that z = exy, x = r − 4s, and y = r − s, find ∂z/∂r in terms of r and s.
Solution: By the chain rule,
Since x = r − 4s and y = r − s,

26. 17.7 Maxima and Minima for Functions
Chapter 17: Multivariable Calculus
17.7 Maxima and Minima for Functions
of Two Variables
Relative maximum at the point (a, b) is shown as
RULE 1
Find relative maximum or minimum when
RULE 2 Second-Derivative Test for Functions of Two Variables
Let D be the function defined by
If D(a, b) > 0 and fxx(a, b) < 0, relative maximum at (a, b);
If D(a, b) > 0 and fxx(a, b) > 0, relative minimum at (a, b);
If D(a, b) < 0, then f has a saddle point at (a, b);
If D(a, b) = 0, no conclusion.

27. Find the critical points of the following functions. a. Solution:
Chapter 17: Multivariable Calculus
17.7 Maxima and Minima for Functions of Two Variables
Example 1 – Finding Critical Points
Find the critical points of the following functions. a.
Solution:
Since
we solve the system and get b.
we solve the system and get and

28. we solve the system and get
Chapter 17: Multivariable Calculus
17.7 Maxima and Minima for Functions of Two Variables
Example 1 – Finding Critical Points c.
Since
we solve the system and get

29. Chapter 17: Multivariable Calculus
17.7 Maxima and Minima for Functions of Two Variables
Example 3 – Applying the Second-Derivative Test
Examine f(x,y) = x3 + y3 − xy for relative maxima or minima by using the second derivative test.
Solution: We find critical points,
which gives (0, 0) and (1/3, 1/3).
Now,
Thus,
D(0, 0) < 0  no relative extremum at (0, 0).
D(1/3,1/3)>0 and fxx(1/3,1/3)>0 relative minimum at (1/3,1/3)
Value of the function is

30. Examine f(x, y) = x4 + (x − y)4 for relative extrema.
Chapter 17: Multivariable Calculus
17.7 Maxima and Minima for Functions of Two Variables
Example 5 – Finding Relative Extrema
Examine f(x, y) = x4 + (x − y)4 for relative extrema.
Solution: We find critical points at (0,0) through
D(0, 0) = 0  no information.
f has a relative (and absolute) minimum at (0, 0).

31. Chapter 17: Multivariable Calculus
17.7 Maxima and Minima for Functions of Two Variables
Example 7 – Profit Maximization
A candy company produces two types of candy, A and B, for which the average costs of production are constant at $2 and $3 per pound, respectively. The quantities qA, qB (in pounds) of A and B that can be sold each week are given by the joint-demand functions
where pA and pB are the selling prices (in dollars per pound) of A and B, respectively. Determine the selling prices that will maximize the company’s profit P.

32. Since ∂2P/∂p2A< 0, we indeed have a maximum.
Chapter 17: Multivariable Calculus
17.7 Maxima and Minima for Functions of Two Variables
Example 7 – Profit Maximization
Solution:
The total profit is given by
The profits per pound are pA − 2 and pB − 3,
The solution is pA = 5.5 and pB = 6.
Since ∂2P/∂p2A< 0, we indeed have a maximum.

33. Chapter 17: Multivariable Calculus
17.8 Lagrange Multipliers
Example 1 – Method of Lagrange Multipliers
Lagrange multipliers allow us to obtain critical points.
The number λ0 is called a Lagrange multiplier.
Find the critical points for z = f(x,y) = 3x − y + 6, subject to the constraint x2 + y2 = 4.
Solution: Constraint
Construct the function
Setting , we solve the equations to be

34. How should the output be distributed in order to minimize costs?
Chapter 17: Multivariable Calculus
17.8 Lagrange Multipliers
Example 3 – Minimizing Costs
Suppose a firm has an order for 200 units of its product and wishes to distribute its manufacture between two of its plants, plant 1 and plant 2. Let q1 and q2 denote the outputs of plants 1 and 2, respectively, and suppose the total-cost function is given by
How should the output be distributed in order to minimize costs?

35. Solution: We minimize c = f(q1, q2), given the constraint
Chapter 17: Multivariable Calculus
17.8 Lagrange Multipliers
Example 3 – Minimizing Costs
Solution: We minimize c = f(q1, q2), given the constraint
q1 + q2 = 200.

36. Chapter 17: Multivariable Calculus
17.8 Lagrange Multipliers
Example 5 – Least-Cost Input Combination
Find critical points for f(x, y, z) = xy+ yz, subject to the constraints x2 + y2 = 8 and yz = 8.
Solution:
Set
We obtain 4 critical points:
(2, 2, 4) (2,−2,−4) (−2, 2, 4) (−2,−2,−4)

37. Chapter 17: Multivariable Calculus
17.9 Lines of Regression

38. Chapter 17: Multivariable Calculus
17.9 Lines of Regression
Example 1 – Determining a Linear-Regression Line
By means of the linear-regression line, use the following table to represent the trend for the index of total U.S. government revenue from 1995 to 2000 (1995 = 100).

39. Solution: Perform the arithmetic
Chapter 17: Multivariable Calculus
17.9 Lines of Regression
Example 1 – Determining a Linear-Regression Line
Solution: Perform the arithmetic

40. Chapter 17: Multivariable Calculus
17.10 Multiple Integrals
Example 1 – Evaluating a Double Integral
Definite integrals of functions of two variables are called (definite) double integrals, which involve integration over a region in the plane.
Find
Solution:

41. Find Solution: Example 3 – Evaluating a Triple Integral
Chapter 17: Multivariable Calculus
17.10 Lines of Regression
Example 3 – Evaluating a Triple Integral
Find
Solution: