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math100_L3_midterm_97

# math100_L3_midterm_97 - z = f x y = ln p 16-x 2-y 2 Find...

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Math 100 L3 Introduction to Multivariable Calculus Mid-Term Test 1997 Each problem is 20 points. (1) (a) Find a vector n perpendicular to the plane P determined by the points A (1 , - 1 , 0) , B (2 , 1 , - 1) and C ( - 1 , 1 , 2) . Hence, find the equation of the plane P . (b) Use part (a) or otherwise, find the area of triangle with vertices A, B and C . (c) Determine whether the line x = 1 + t, y = - 1 + 2 t, z = - 3 + t. is parallel to the plane P in (a). (2) (a) Find parametric equation of the curve of intersection of the surface z = f ( x, y ) = x 2 + y 2 and the plane x = - 2. Identify the curve. (b) Find a vector equation of the line tangent to the graph of r ( t ) = ti + t 2 j at the point (1 , 1) on the curve. (3) (a) Sketch the domain of the function
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Unformatted text preview: z = f ( x, y ) = ln p 16-x 2-y 2 . Find also the range of f . (b) Describe the level surfaces of the function f ( x, y, z ) = ( x-2) 2 + y 2 + z 2 (4) (a) If z = 8 xy-2 x + 3 y , x = uv , y = u-v . Find z u in terms of u and v . (b) Find the normal vector to the surface z = f ( x, y ) = x 2 + y 2 at (0 , , 0). (5) (a) Near the point (1 , 2), is f ( x, y ) = x 2-xy + y 2-3 more sensitive to changes in x , or changes in y ? Give reasons for your answer. (b) Locate all relative extrema and saddle points of the function z = f ( x, y ) = 2 x 2-4 xy + y 4 + 16 . 1...
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