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# Exam1Solution - Math 2343 Practice Exam 1 Answer Key 1 The...

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Math 2343 Practice Exam 1 Answer Key 1. The c -level curve will be a circle of radius p c/ 2. 2. - 600 3. 3 x + 3 y - z = 4 4. 24 5. Saddle point at ( π 2 ). 6. Find all ﬁrst and second order partial derivatives of the function f ( x,y ) = e xy + cos x . Answer: f x ( x,y ) = ye xy - sin x , f y ( x,y ) = xe xy , f xx ( x,y ) = y 2 e xy - cos x , f yy ( x,y ) = x 2 e xy , f xy ( x,y ) = (1 + xy ) e xy . 7. At the point (1 , 2 , - 4), ﬁnd the vector equation of the normal line to the surface S : z = 3 x 2 y - 2 xy 3 + 6 x. Answer: x ( t ) = 1 + 2 t, y ( t ) = 2 - 21 t, z ( t ) = - 4 - t 8. Let f ( x,y ) = x 2 - y 2 x 2 + yx . Find the limit of f as ( x,y ) goes toward (0 , 0) along the curves: (a) y = x (b) y = x 2 . What can you conclude about lim ( x,y ) (0 , 0) f ( x,y )? Answer: (a) 0 (b) 1 Limit does not exist because there are diﬀerent values along diﬀerent curves. 9. Find all critical points and determine the local extrema of f ( x,y ) = x 3 - 6 xy + y 2 +8 y . Answer:

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Exam1Solution - Math 2343 Practice Exam 1 Answer Key 1 The...

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