Math252Sols3F07

# Math252Sols3F07 - QUIZ 3(SECTIONS 16.3-16.9 SOLUTIONS MATH...

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QUIZ 3 (SECTIONS 16.3-16.9) SOLUTIONS MATH 252 – FALL 2007 – KUNIYUKI SCORED OUT OF 125 POINTS MULTIPLIED BY 0.84 105% POSSIBLE 1) Let f x , y , z ( ) = 3 x 2 y + z 3 . Find f x x , y , z ( ) . (4 points) f x x , y , z ( ) = D x 3 x 2 y + z 3 ( ) = D x 3 x 2 y + z 3 ( ) 1/2 = 1 2 3 x 2 y + z 3 ( ) 1/2 D x 3 x 2 y "#" + z 3 "#" = 1 2 3 x 2 y + z 3 ( ) 1/2 3 y 2 x ( ) = 3 xy 3 x 2 y + z 3 2) Let f r , s ( ) = cos rs ( ) . Find f r r , s ( ) and use that to find f rs r , s ( ) . (6 points) f r r , s ( ) = D r cos rs ( ) = sin rs ( ) D r r s "#" = sin rs ( ) s = s sin rs ( ) f rs r , s ( ) = D s s sin rs ( ) We will use a Product Rule for Differentiation. = D s s ( ) sin rs ( ) + s D s sin rs ( ) ( ) = 1 sin rs ( ) + s cos rs ( ) D s r "#" s = 1 sin rs ( ) + s cos rs ( ) r = sin rs ( ) rs cos rs ( )

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3) Assume that f is a function of x and y . Write the limit definition of f y x , y ( ) using the notation from class. (4 points) f y x , y ( ) = lim h 0 f x , y + h ( ) f x , y ( ) h 4) Let f x , y ( ) = 3 xy 2 4 y 3 + 5 . Use differentials to linearly approximate the change in f if x , y ( ) changes from 4, 2 ( ) to 3.98, 1.96 ( ) . (12 points) f x x , y ( ) = D x 3 x y 2 "#" 4 y 3 "#" + 5 = 3 y 2 f x 4, 2 ( ) = 3 2 ( ) 2 = 12 dx = new x old x = 3.98 4 = 0.02 f y x , y ( ) = D y 3 x "#" y 2 4 y 3 + 5 = 3 x 2 y ( ) 12 y 2 = 6 xy 12 y 2 f y 4, 2 ( ) = 6 4 ( ) 2 ( ) 12 2 ( ) 2 = 96 dy = new y old y = 1.96 − − 2 ( ) = 0.04 The approximate change in f is given by: df = f x 4, 2 ( ) dx + f y 4, 2 ( ) dy = 12 0.02 ( ) + 96 0.04 ( ) = 4.08 Note: Actual change ≈ − 4.01315
5) Let f , f 1 , f 2 and f 3 be differentiable functions such that w = f x , y , z ( ) , x = f 1 u , v ( ) , y = f 2 u , v ( ) , and z = f 3 u , v ( ) . Use the Chain Rule to write an expression for w v . (5 points) w v = w x x v + w y y v + w z z v 6) Find z x if z = f x , y ( ) is a differentiable function described implicitly by the equation e xyz = xz 4 . Use the Calculus III formula given in class. Simplify. (9 points) First, isolate 0 on one side: e xyz xz 4 Let this be F x , y , z ( )   = 0 Find z x . When using the formula, treat x , y , and z as independent variables.

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