Chapter7

# Chapter7 - Chapter 7 Functions of Several Variables...

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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
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
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS , 11e – Slide 5 of 62 Functions of More Than One Variable Definition Example Function of Several Variables : A function that has more than one independent variable ( 29 w z xy w z y x f - = 5 , , , 2

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Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS , 11e – Slide 6 of 62 Functions of More Than One Variable EXAMPLE EXAMPLE SOLUTION SOLUTION Let . Compute g (1, 1) and g (0, -1). ( 29 2 2 2 , y x y x g + = ( 29 3 1 2 1 1 2 1 1 , 1 2 2 = + = + = g ( 29 ( 29 2 1 2 0 1 2 0 1 , 0 2 2 = + = - + = - g
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS , 11e – Slide 7 of 62 Cost of Material EXAMPLE EXAMPLE SOLUTION 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. TOP LEFT SIDE RIGHT SIDE BACK 3 5 5 5 xy yz yz 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 CONTINUED [ ] [ ] . 5 feet square in side back of area back of foot square per cost xz = Similarly, the cost of the top is 3 xy . Continuing in this way, we see that the total cost is ( 29 . 5 10 3 5 5 5 3 , , xz yz xy xz yz yz xy z y x C + + = + + + =
Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS , 11e – Slide 9 of 62 EXAMPLE 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

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Chapter7 - Chapter 7 Functions of Several Variables...

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