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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 1 of 62 Chapter 7 Functions of Several Variables

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 2 of 62 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 Chapter Outline

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 3 of 62 § 7.1 Examples of Functions of Several Variables

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 4 of 62 Functions of More Than One Variable Cost of Material Tax and Homeowner Exemption Level Curves Section Outline

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 5 of 62 Functions of More Than One Variable DefinitionExample Function of Several Variables: A function that has more than one independent variable

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 6 of 62 Functions of More Than One VariableEXAMPLE SOLUTION Let. Compute g(1, 1) and g(0, -1).

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 7 of 62 Cost of MaterialEXAMPLE SOLUTION (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. TOPLEFT SIDERIGHT SIDEBACK 3555 xyyz xz Area (sq ft) Cost (per sq ft)

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 8 of 62 Cost of Material The total cost is the sum of the amount of cost for each side of the enclosure, CONTINUED Similarly, the cost of the top is 3xy. Continuing in this way, we see that the total cost is

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 9 of 62 Tax & Homeowner ExemptionEXAMPLE (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 homeowners 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 homeowners exemption of $5000, assuming a tax rate of $2.50 per hundred dollars of net assessed value.

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 10 of 62 Tax & Homeowner ExemptionSOLUTION We are looking for T. We know that v = 200,000, x = 5000 and r = Therefore, we get CONTINUED So, the real estate tax on the property with the given characteristics is $1875.

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 11 of 62 Level Curves DefinitionExample 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.

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 12 of 62 Level CurvesEXAMPLE SOLUTION Find a function f (x, y) that has the curve y = 2/x 2 as a level curve. Since level curves occur where f (x, y) = c, then we must rewrite y = 2/x 2 in that form. This is the given equation of the level curve. Subtract 2/x 2 from both sides so that the left side resembles a function of the form f (x, y). Therefore, we can say that y – 2/x 2 = 0 is of the form f (x, y) = c, where c = 0. So, f (x, y) = y – 2/x 2.

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