# mm1 - [15 points Compute the gradient ∇ f for the...

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Calculus IV [2443–004] Midterm I For full credit, give reasons for all your answers. Q1]. ..[15 points] Draw the level curves f = 0, f = 1, f = 4, and f = - 1 for the function f ( x,y ) below. Also, sketch the graph of f in a neighborhood of the origin. f ( x,y ) = x 2 - y 2 Does the following limit exist? Give reasons for your answer. lim ( x,y ) (0 , 0) x 2 - y 2 x 2 + y 2 Q2]. ..[10 points] If x = t sin s and y = t cos s ﬁnd 2 ∂s∂t f ( x,y ) Q3]. ..[10 points] Write down the diFerential for the function f ( x,y ) = ye x + y 2 Use the diFerential above to estimate the value of f ( - 3 . 92 , 2 . 05). Q4].
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Unformatted text preview: ..[15 points] Compute the gradient ∇ f for the function f ( x,y,z ) = ye-x 2 sin z ±ind the direction in which f is increasing most rapidly at the point (0 , 1 ,π/ 3). What is the maximum rate of change of f at the point (0 , 1 ,π/ 3)? ±ind an equation for the tangent plane to the level surface of f at the point (0 , 1 ,π/ 3). Q5]. ..[10 points] ±ind and classify the critical points of the function f ( x,y ) = 2 x 3-6 xy-3 y 2...
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