Engineering Calculus Notes 262

Engineering Calculus Notes 262 - △ x = △ y = ± 2 and z...

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250 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION This is an easy calculation, but the answer is only an estimate; by comparison, a calculator “calculation” of f (2 . 9 , 1 . 2) gives 27 . 25 5 . 220. As a second example, we consider the accuracy of the result of the calculation of a quantity whose inputs are only known approximately. Suppose, for example, that we have measured the height of a rectangular box as 2 feet, with an accuracy of ± 0 . 1 ft , and its a base as 5 × 10 feet, with an accuracy in each dimension of ± 0 . 2 ft . We calculate the volume as 100 ft 3 ; how accurate is this? Here we are interested in how far the actual value of f ( x,y,z ) = xyz can vary from f (5 , 10 , 2) = 100 when x and y can vary by at most
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Unformatted text preview: △ x = △ y = ± . 2 and z can vary by at most △ z = ± . 1. The best estimate of this is the diFerential: f ( x,y,z ) = xyz ∂f ∂x ( x,y,z ) = yz ∂f ∂y ( x,y,z ) = xz ∂f ∂z ( x,y,z ) = xy and at our point ∂f ∂x (5 , 10 , 2) = 20 ∂f ∂y (5 , 10 , 2) = 10 ∂f ∂z (5 , 10 , 2) = 50 so the diFerential is d (5 , 10 , 2) f ( △ x, △ y, △ z ) = 20 △ x + 10 △ y + 50 △ z which is at most (20)(0 . 2) + (10)(0 . 2) + (50)(0 . 1) = 4 + 2 + 5 = 11 . We conclude that the ±gure of 100 cubic feet is correct to within ± 11 cubic feet. Exercises for § 3.3 Practice problems:...
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