1. Functions of Several Variables
Chapter 7
Functions of Several Variables
Copyright © 2014, 2010, 2007 Pearson Education, Inc.

2. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Chapter Outline
Examples of Functions of Several Variables
Partial Derivatives
Maxima and Minima of Functions of Several Variables
Lagrange Multipliers and Constrained Optimization
The Method of Least Squares
Double Integrals
Copyright © 2014, 2010, 2007 Pearson Education, Inc.

3. Examples of Functions of Several Variables
7.1
Examples of Functions of Several Variables
Copyright © 2014, 2010, 2007 Pearson Education, Inc.

4. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Section Outline
Functions of More Than One Variable
Cost of Material
Tax and Homeowner Exemption
Level Curves
Copyright © 2014, 2010, 2007 Pearson Education, Inc.

5. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Functions of More Than One Variable
Definition
Example
Function of Several Variables: A function that has more than one independent variable
Copyright © 2014, 2010, 2007 Pearson Education, Inc.

6. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Functions of More Than One Variable
EXAMPLE
Let Compute g(1, 1) and g(0, -1).
SOLUTION
Copyright © 2014, 2010, 2007 Pearson Education, Inc.

7. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Graphs of Functions of More Than One Variable
Copyright © 2014, 2010, 2007 Pearson Education, Inc.

8. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Cost of Material
EXAMPLE
(Cost) Find a formula C(x, y, z) that gives the cost of material for the rectangular enclose in the figure, with dimensions in feet, assuming that the material for the top costs $3 per square foot and the material for the back and two sides costs $5 per square foot.
SOLUTION
TOP
LEFT SIDE
RIGHT SIDE
BACK 3 5 xy yz xz
Cost (per sq ft)
Area (sq ft)
Copyright © 2014, 2010, 2007 Pearson Education, Inc.

9. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Cost of Material
CONTINUED
The total cost is the sum of the amount of cost for each side of the enclosure,
Similarly, the cost of the top is 3xy. Continuing in this way, we see that the total cost is
Copyright © 2014, 2010, 2007 Pearson Education, Inc.

10. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Tax & Homeowner Exemption
EXAMPLE
(Tax and Homeowner Exemption) The value of residential property for tax purposes is usually much lower than its actual market value. If v is the market value, then the assessed value for real estate taxes might be only 40% of v. Suppose the property tax, T, in a community is given by the function
where v is the estimated market value of a property (in dollars), x is a homeowner’s exemption (a number of dollars depending on the type of property), and r is the tax rate (stated in dollars per hundred dollars) of net assessed value.
Determine the real estate tax on a property valued at $200,000 with a homeowner’s exemption of $5000, assuming a tax rate of $2.50 per hundred dollars of net assessed value.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.

11. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Tax & Homeowner Exemption
CONTINUED
SOLUTION
We are looking for T. We know that v = 200,000, x = 5000 and r = Therefore, we get
So, the real estate tax on the property with the given characteristics is $1875.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.

12. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Level Curves
Definition
Example
Level Curves: For a function f (x, y), a family of curves with equations f (x, y) = c where c is any constant
An example immediately follows.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.

13. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Level Curves
Copyright © 2014, 2010, 2007 Pearson Education, Inc.

14. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Level Curves
EXAMPLE
Find a function f (x, y) that has the curve y = 2/x2 as a level curve.
SOLUTION
Since level curves occur where f (x, y) = c, then we must rewrite y = 2/x2 in that form.
This is the given equation of the level curve.
Subtract 2/x2 from both sides so that the left side resembles a function of the form f (x, y).
Therefore, we can say that y – 2/x2 = 0 is of the form f (x, y) = c, where c = 0. So, f (x, y) = y – 2/x2.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.