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sample_mid_2

# sample_mid_2 - z = ln x 2 y at(2-3 π 7 Let f x y = p x 2 y...

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Math 237 - Sample Midterm 2 1. Short Answer Problems a) Let f : R 2 R . State the precise definition of lim ( x,y ) ( a,b ) f ( x, y ) = L . b) Let f : R 2 R . If f x and f y are both continuous at ( a, b ), then what are two things you can say about f at ( a, b )? c) Let f ( x, y ) = x 2 + xy + y 3 . What is the greatest rate of change of f at ( - 1 , 1)? d) If f : R 2 R has continuous second partial derivatives and Hf ( x, y ) = 0 0 0 0 for all points ( x, y ) in a neighborhood of ( a, b ) then what can you conclude about the accuracy of L ( a,b ) ( x, y )? Justify your answer. 2. Let f ( x, y ) = y 2 - x 2 . a) Sketch the domain and state the range of f ? b) Sketch the level curves and cross sections of z = f ( x, y ). 3. Prove that if f : R 2 R is differentiable at ( a, b ), then D u f ( a, b ) = f ( a, b ) · u . 4. Let g : R 2 R and let f ( x, y ) = g ( u, v ) where u = u ( x, y ) = xe xy and v = v ( x, y ) = x + y 2 . Find 2 f ∂x∂y . What assumptions do you need to make about g so that you can apply the chain rule? 5. Use the second degree Taylor polynomial to approximate ( - 0 . 99) 3 (1 . 02) 2 . 6. Find the equation of the tangent plane to the surface sin
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Unformatted text preview: z = ln( x 2 + y ) at (2 ,-3 , π ). 7. Let f ( x, y ) = p ( x 2 + y 2 ). Use Taylor’s Theorem to show that for x ≥ 1, y ≥ 1 | R 1 , (1 , 1) ( x, y ) | ≤ 2-3 / 2 ³ ( x-1) 2 + ( y-1) 2 ´ 8. Determine if each of the following limits exist. Evaluate the limits that exist. a) lim ( x,y ) → (0 , 0) x 2 y 4 x 4 + y 8 . b) lim ( x,y ) → (0 , 0)-2 x 2 + x 2 y 2-2 y 2 x 2 + y 2 . 9. Let f ( x, y ) = µ x 7 / 3 y 2 / 3 x 2 + y 2 , ( x, y ) 6 = (0 , 0) , ( x, y ) = (0 , 0) . a) Determine if f ( x, y ) is continuous at (0 , 0). b) Determine all points where f is diﬀerentiable. c) Based on your answer in part b), what can you conclude about the continuity of both f x and f y at (0 , 0)? d) Find the directional derivative of f at (0 , 0) in the direction of the vector (1 , 1)....
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