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practice2 - MATH 20C PRACTICE PROBLEMS FOR MIDTERM II 1...

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MATH 20C - PRACTICE PROBLEMS FOR MIDTERM II 1. Find the critical points of the function f ( x, y ) = 2 x 3 - 3 x 2 y - 12 x 2 - 3 y 2 and determine their type i.e. local min/local max/saddle point. 2. Determine the global max and min of the function f ( x, y ) = x 2 - 2 x + 2 y 2 - 2 y + 2 xy over the compact region - 1 x 1 , 0 y 2 . 3. Using Lagrange multipliers, optimize the function f ( x, y ) = x 2 + ( y + 1) 2 subject to the constraint 2 x 2 + ( y - 1) 2 18 . 4. Consider the function w = e x 2 y where x = u v, y = 1 uv 2 . Using the chain rule, compute the derivatives ∂w ∂u , ∂w ∂v . 5. Consider the function f ( x, y ) = x 2 y 4 . (i) Carefully draw the level curve passing through (1 , - 1). On this graph, draw the gradient of the function at (1 , - 1). (ii) Compute the directional derivative of f at (1 , - 1) in the direction u = ( 4 5 , 3 5 ) . Use this calculation to estimate f ((1 , - 1) + . 01 u ) . (iii) Find the unit direction v of steepest descent for the function f at (1 , - 1). (iv) Find the two unit directions w for which the directional derivative D w f = 0 . 6. Consider the function f ( x, y ) = x 4 y 3 . (i) Write down the equation of the tangent plane at the graph of the function at the point (1 , 1 , 1) .
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(ii) Write down an expression for the change, Δ z , in z = f ( x, y ) depending on Δ x and Δ y , the change in x and y, respectively, near the point x = y = 1 . Is the function f ( x, y ) more sensitive to a change in x or to a change in y ? (iii) Using your answer to (ii), find the approximate value of f (1 . 01 , 1 . 01) . 7. Show that the surfaces z = 7 x 2 - 12 x - 5 y 2 and xyz 2 = 2 intersect orthogonally at the point (2 , 1 , - 1). That is, show that the tangent planes to the two surfaces are perpendicular. 8. Evaluate RR D 3 ydA, where D is the region bounded by xy 2 = 1 , y = x, x = 0 , y = 3 . 9. Evaluate Z π 0 Z π y sin( x ) x dxdy. 10. Find the volume of the region bounded on top by the plane z = x + 3 y + 1, on the bottom by the xy -plane, and on the sides by the planes x = 0 , x = 3 , y = 1 , y = 2. 11. ( Harder, solve only after looking at problems 1-10 ) Two paraboloids z = ( x - 2) 2 + ( y - 2) 2 and z = 20 - x 2 - y 2 intersect along a curve C . Find the point of C which is closest to the point (1 , 1 , 0).
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SOLUTIONS Problem 1. Find the critical points of the function f ( x, y ) = 2 x 3 - 3 x 2 y - 12 x 2 - 3 y 2 and determine their type i.e. local min/local max/saddle point. Are there any global min/max? Solution: Partial derivatives f x = 6 x 2 - 6 xy - 24 x, f y = - 3 x 2 - 6 y. To find the critical points, we solve f x = 0 = x 2 - xy - 4 x = 0 = x ( x - y - 4) = 0 = x = 0 or x - y - 4 = 0 f y = 0 = x 2 + 2 y = 0 .
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