A6 - MATH 222 HOMEWORK 6 DUE 1 P ROBLEMS Problem 1.1 There...

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Unformatted text preview: MATH 222 HOMEWORK 6 DUE OCTOBER 31, 2011 1. P ROBLEMS Problem 1.1. There is only one point where the plane tangent to the surface z = x2 + 2xy + 2y 2 − 6x + 8y is horizontal. Find it. Problem 1.2. Find an equation of the plane tangent to the surface z = f (x, y ) at the indicated point P . a.) z = x2 + y 2 , P = (3, 4, 25), b.) z = x3 − y 3 , P = (3, 2, 19), c.) z 2 = x2 + y 2 , P = (3, −4, 5). Problem 1.3. Find the local maxima, minima, and saddle points of the given surfaces. a.) z = 2x2 + 8xy + y 4 , 2 2 b.) z = e2x−4y−x −y , 2 2 c.) z = (1 + 2x2 )e−x −y . Problem 1.4. Calculate the directional derivative of f at P in the direction of v . a.) f (x, y ) = arctan(y/x), P = (−3, 3), v = (3, 4). b.) f (x, y, z ) = xy + yz + zx, P = (1, −1, 2), v = (1, 1, 1). Problem 1.5. Suppose a body has mass m and velocity v . Its kinetic energy is given by 1 K = 2 mv 2 . Assume that m = 10 and v = 25. Use differentials to estimate the increase in kinetic energy if both mass and velocity increase by 10%. Problem 1.6. Suppose that a, b, and c are positive constants. Find the maximum volume of 2 2 2 a box inscribed inside the ellipsoid x2 + y2 + z2 = 1. Assume that the edges of the box are a b c parallel to the x-y -z axis. 1 ...
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