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
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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 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
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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 + + = + + + =
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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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