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

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.

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

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.

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.

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.

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

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.

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.

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.

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 = 2.50. Therefore, we get So, the real estate tax on the property with the given characteristics is $1875. Copyright © 2014, 2010, 2007 Pearson Education, Inc.

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.

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

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.