Practice_Final - polar transformation x = r cos , y = r sin...

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PRACTICE FINAL FOR CALCULUS III (1) (a) Find a vector function represents the curve of intersection of the cylinder x 2 + y 2 = 16 and the plane x + z = 5; (b) Find the curvature of the above curve at the point (4 , 0 , 1). (2) Find the linear approximation of the function f ( x,y,z ) = x 3 p y 2 + z 2 at the point (2 , 3 , 4) and use it to estimate the number (1 . 98) 3 p (3 . 01) 2 + (3 . 97) 2 . (3) Let f ( x,y ) = ( x 2 - 3 y 2 x 2 + y 2 if ( x,y ) 6 = (0 , 0) 1 if ( x,y ) = (0 , 0) . (a) Is f continuous at (0 , 0)? Justify your answer. (b) Do ∂f ∂x (0 , 0) and ∂f ∂y (0 , 0) exist? What are the values of the partial deriva- tives at (0 , 0) if they exist? (4) Let P = (1 , - 2 , 1) be a point on the surface S defined by the equation z = 3 x 2 - y 2 + 2 x . (a) Find the equation of the tangent plane to S at the point P ; (b) Find the parametric equations of the normal line to S through P . (5) If v = x 2 sin y + ye xy , where x = s + 2 t and y = st , use the Chain Rule to find ∂v/∂s and ∂v/∂t when s = 0 and t = 1. (6) Suppose f ( x,y ) has continuous second order partial derivatives. Under the
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Unformatted text preview: polar transformation x = r cos , y = r sin , express 2 f x 2 + 2 f y 2 in terms of r , and the partial derivatives of f with respect to r and . (7) If z = y + f ( x 2-y 2 ), where f is dierentiable, show that y z x + x z y = x . (8) Find the direction in which f ( x,y,z ) = ze xy increases most rapidly at the point (0 , 1 , 2). What is the maximum rate increase. (9) Suppose a > 0 and a i > 0 for i = 1 ,...,n . Find the extremes of the function f ( x 1 ,...,x n ) = x a 1 1 x a 2 2 x a n n under the constraint x 1 + x 2 + + x n = a where x i > 0 for i = 1 ,...,n . (10) Find the points on the surface xy 2 z 3 = 2 that are closed to the origin. 1...
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This note was uploaded on 02/27/2011 for the course MATH 101 taught by Professor Y during the Spring '11 term at Columbia College.

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