265Lesson10Problems

265Lesson10Problems - T ( x,y,z ) = x + yz . A y is at the...

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MATH 265: MULTIVARIABLE CALCULUS: LESSON 10 PROBLEMS Problems are worth 20 points each. (1) Let f ( x,y ) = x cos y . (a) Calculate f ( x,y ). (b) If -→ u = 3 5 -→ ı - 4 5 -→ , find D -→ u f (2 , π 3 ). (2) Let g ( x,y,z ) = e x y + z ln y . Calculate the gradient of g . (3) Use the methods of this section to find an equation for the plane tangent to the surface defined by the equation 3 xy + z 2 - 4 = 0 at the point (1 , 1 , 1). (4) Let f ( x,y ) = x 2 + y and let -→ c ( t ) = h e t ,e - t i . (a) Compute d d t f ( -→ c ( t )) using the Chain Rule. (b) Compute d d t f ( -→ c ( t )) without using the Chain Rule, but instead first writing the composition f ( -→ c ( t )) as a function of t , then differentiating with respect to t in the usual Calc I way. (Of course, the two answers must be same!) (5) The temperature at the point ( x,y,z ) in space is given by
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Unformatted text preview: T ( x,y,z ) = x + yz . A y is at the point (1 , 2 , 1). In what direction should he begin to y to cool o as quickly as possible? Your answer should be a unit vector in the requested direction. (6) (Bonus question: worth 10 points. Total points for assignment not to exceed 100.) Suppose a tub has the shape of an elliptical paraboloid given by z = ax 2 + by 2 (where a,b are some positive constants). If a marble were released at the point (1 , 1 ,a + b ) on the inside surface of the tub, in what direction would it begin to roll? Your answer should be a unit vector in the requested direction. 1...
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