Math 142, Chapter 8
Spring 2011
Zarestky
Chapter 8 Reading Notes
In this chapter, we will learn about functions of more than one variable and apply our calculus skills to
analyze these functions.
Section 8.1: Functions of Several Variables
By the end of this section, you should be able to:
• Evaluate a function of two or more variables.
• Plot points in the three-dimensional coordinate system.
• Sketch a graph of a function in the three-dimensional coordinate system.
• Evaluate a difference quotient for a function of two variables.
• Apply business models involving more than one independent variable.
The first few examples are to get you comfortable with multivariate functions.
Read Examples 1-5. Attempt the Matched Problems. What questions do you have?
Math 142, Chapter 8
Spring 2011
Zarestky
Graphs in three dimensions can be tougher to visualize than in two dimensions. In example 6, you will
learn about three-dimensional coordinates.
Read Example 6. Attempt the Matched Problem. What questions do you have?
Skim the remainder of the section. We will discuss graphs of three-dimensional curves as a class.
Check your understanding of this section by working a few problems from the exercises on your own.
Focus on 1-33(odd), 37, 39, 43, 45, 49, 51, 53, 57, 59, 61.
Do you have any questions?
Math 142, Chapter 8
Spring 2011
Zarestky
Section 8.2: Partial Derivatives
By the end of this section, you should be able to:
• Calculate first- and second-order partial derivatives.
• Apply business models involving more than one independent variable.
In this section, you will take derivatives of multivariate functions. Note that it is only possible to take a
derivative with respect to one variable at a time!
Work Examples 1-4 and attempt the Matched Problems. Compare your answers to the book’s
solution. What questions do you have?
Math 142, Chapter 8
Spring 2011
Zarestky
In the next example, you will calculate higher-order derivatives, analogous to the second derivative for a
function of a single variable.
Work Example 5 and attempt the Matched Problem. Compare your answers to the book’s solution.
What questions do you have?
Check your understanding of this section by working a few problems from the exercises on your own.
Focus on 1-69(odd), 79, 81, 83, 85, 93, 95.
What questions do you still have?
Math 142, Chapter 8
Spring 2011
Zarestky
Section 8.3: Maxima and Minima
By the end of this section, you should be able to:
• Determine if a critical point in three dimensions is a local maximum, local minimum, or a saddle
point by using the second derivative test.
In this section, you will identify the maximum and minimum values of multivariate functions.
Read pp. 467-469 and make notes regarding the vocabulary. How are the terms used the same or
differently than in our previous contexts?
Work Examples 1-2 and attempt the Matched Problems. Compare your answers to the book’s
solution. What questions do you have?
Math 142, Chapter 8
Spring 2011
Zarestky
Work Example 3 and attempt the Matched Problem. Compare your answers to the book’s solution.
What questions do you have?
Work Example 4 and attempt the Matched Problem. Compare your answers to the book’s solution.
What questions do you have?
Check your understanding of this section by working a few problems from the exercises on your own.
Focus on 1, 5-21(odd), 29, 31, 35.
What questions do you still have?
Math 142, Chapter 8
Spring 2011
Zarestky
Section 8.5: Method of Least Squares
By the end of this section, you should be able to:
• Apply the method of least-squares to model a data set.
Skim this section and we’ll save it for an in-class discussion.
Check your understanding of this section by working a few problems from the exercises on your own.
Focus on 1-11(odd), 15.
What questions do you still have?