Engineering Calculus Notes 262

Engineering Calculus Notes 262 - x = y = . 2 and z can vary...

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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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