Calculus Multivariable McCallum, Hughes-Hallett, Gleason 4th Ed ch14 solutions

Calculus: Multivariable

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14.1 SOLUTIONS 951 CHAPTER FOURTEEN Solutions for Section 14.1 Exercises 1. If h is small, then f x (3 , 2) f (3 + h, 2) - f (3 , 2) h . With h = 0 . 01 , we Fnd f x (3 , 2) f (3 . 01 , 2) - f (3 , 2) 0 . 01 = 3 . 01 2 (2+1) - 3 2 (2+1) 0 . 01 = 2 . 00333 . With h = 0 . 0001 , we get f x (3 , 2) f (3 . 0001 , 2) - f (3 , 2) 0 . 0001 = 3 . 0001 2 (2+1) - 3 2 (2+1) 0 . 0001 = 2 . 0000333 . Since the difference quotient seems to be approaching 2 as h gets smaller, we conclude f x (3 , 2) 2 . To estimate f y (3 , 2) , we use f y (3 , 2) f (3 , 2 + h ) - f (3 , 2) h . With h = 0 . 01 , we get f y (3 , 2) f (3 , 2 . 01) - f (3 , 2) 0 . 01 = 3 2 (2 . 01+1) - 3 2 (2+1) 0 . 01 = - 0 . 99668 . With h = 0 . 0001 , we get f y (3 , 2) f (3 , 2 . 0001) - f (3 , 2) 0 . 0001 = 3 2 (2 . 0001+1) - 3 2 (2+1) 0 . 0001 = - 0 . 9999667 . Thus, it seems that the difference quotient is approaching - 1 , so we estimate f y (3 , 2) ≈ - 1 . 2. Using Frst Δ x = 0 . 1 and Δ y = 0 . 1 , we have the estimates: f x (1 , 3) f (1 . 1 , 3) - f (1 , 3) 0 . 1 = 0 . 0470 - 0 . 0519 0 . 1 = - 0 . 0493 , and f y (1 , 3) f (1 , 3 . 1) - f (1 , 3) 0 . 1 = 0 . 0153 - 0 . 0519 0 . 1 = - 0 . 3660 . Now, using Δ x = 0 . 01 and Δ y = 0 . 01 , we have the estimates: f x (1 , 3) f (1 . 01 , 3) - f (1 , 3) 0 . 01 = 0 . 0514 - 0 . 0519 0 . 01 = - 0 . 0501 , and f y (1 , 3) f (1 , 3 . 01) - f (1 , 3) 0 . 01 = 0 . 0483 - 0 . 0519 0 . 01 = - 0 . 3629 .
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952 Chapter Fourteen /SOLUTIONS 3. (a) Negative. As the price of beef goes up, we expect people to buy less beef and so the quantity of beef sold goes down. If b increases, we expect Q to decrease. (b) Positive. As the price of chicken goes up, we expect people to buy more beef. As c increases, we expect Q to increase. (c) We estimate that Δ Q Δ b - 213 1 kg/dollar . If the price of beef rose by one dollar, the store would sell approximately 213 fewer kilograms of beef. 4. (a) We expect f p to be negative because if the price of the product increases, the sales usually decrease. (b) If the price of the product is $8 per unit and if $ 12000 has been spent on advertising, sales increase by approximately 150 units if an additional $ 1000 is spent on advertising. 5. (a) The units of ∂c/∂x are units of concentration/distance. (For example, (gm/cm 3 )/cm.) The practical interpretation of ∂c/∂x is the rate of change of concentration with distance as you move down the blood vessel at a ±xed time. We expect ∂c/∂x < 0 because the further away you get from the point of injection, the less of the drug you would expect to ±nd (at a ±xed time). (b) The units of ∂c/∂t are units of concentration/time. (For example, (gm/cm 3 )/sec.) The practical interpretation of ∂c/∂t is the rate of change of concentration with time, as you look at a particular point in the blood vessel. We would expect the concentration to ±rst increase (as the drug reaches the point) and then decrease as the drug dies away.
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Calculus Multivariable McCallum, Hughes-Hallett, Gleason 4th Ed ch14 solutions

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