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Engineering Calculus Notes 263

Engineering Calculus Notes 263 - 01 inches thick use the...

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3.3. DERIVATIVES 251 1. Find all the partial derivatives of each function below: (a) f ( x,y ) = x 2 y 2 xy 2 (b) f ( x,y ) = x cos y + y sin x (c) f ( x,y ) = e x cos y + y tan x (d) f ( x,y ) = ( x + 1) 2 y 2 x 2 ( y 1) 2 (e) f ( x,y,z ) = x 2 y 3 z (f) f ( x,y,z ) = xy + xz + yz xyz 2. For each function below, find its derivative d −→ a f ( −→ x ), the linearization T −→ a f ( −→ x ), and the gradient grad f ( −→ a ) = −→ f ( −→ a ) at the given point −→ a . (a) f ( x,y ) = x 2 + 4 xy + 4 y 2 , −→ a = (1 , 2) (b) f ( x,y ) = cos( x 2 + y ) , −→ a = ( π, π 3 ) (c) f ( x,y ) = radicalbig x 2 + y 2 , −→ a = (1 , 1) (d) f ( x,y ) = x cos y y cos x, −→ a = ( π 2 , π 2 ) (e) f ( x,y,z ) = xy + xz + yz, −→ a = (1 , 2 , 3) (f) f ( x,y,z ) = ( x + y ) 2 ( x y ) 2 + 2 xyz, −→ a = (1 , 2 , 1) 3. (a) Use the linearization of f ( x,y ) = xy at −→ a = (9 , 4) to find an approximation to radicalbig (8 . 9)(4 . 2). (Give your approximation to four decimals.) (b) A cylindrical tin can is h = 3 inches tall and its base has radius r = 2 inches. If the can is made of tin that is 0
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Unformatted text preview: . 01 inches thick, use the di²erential of V ( r,h ) = πr 2 h to estimate the total volume of tin in the can. 4. If two resistors with respective resistance R 1 and R 2 are hooked up in parallel, the net resistance R is related to R 1 and R 2 by 1 R = 1 R 1 + 1 R 2 . (a) Show that the di²erential of R = R ( R 1 ,R 2 ), as a function of the two resistances, is given by dR = p R R 1 P 2 △ R 1 + p R R 2 P 2 △ R 2 . (b) If we know R 1 = 150 ohms and R 2 = 400 ohms, both with a possible error of 10%, what is the net resistance, and what is the possible error?...
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