practice_2

practice_2 - AC . Find the area of ABCD . 7. Assume f (1 ,...

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Practice Midterm II for Math 10C Disclaimer: There may be problems on the midterm that do not appear on this practice midterm. Conversely, there may be problems on this practice midterm that are not on the midterm. It is highly recommended that you complete the homework problems before taking this practice midterm. Click here for answers . 1. Find an equation of the plane that contains the points ( 1 , 1 , 0), (0 , 1 , 1), and (2 , 0 , 1). 2. Let f ( x, y )= ax + by + c be a linear function such that f (1 , 2) = 3 and the trace of its graph in the yz -plane is the line z = y +1. Find a , b ,and c . 3. Find the linear function f ( x, y ) whose graph is a plane perpendicular to the vector ~v =2 ~ i +3 ~ j ~ k and its trace in the xz -plane is the line z =2 x +3. 4. Evaluate the expression ~a · ( ~ b × ~c ) for ~a =2 ~ i ~ k , ~ b =2 ~ i , ~c = ~ j + ~ k . 5. Find the area of the triangle ABC ,where A =(1 , 0 , 0), B =(0 , 2 , 1), C =(1 , 0 , 2). 6. Let ABCD be a parallelogram in the 3-space such that AB = ~ i 2 ~ j , AD = ~ j + ~ k ,and AB + AD =
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Unformatted text preview: AC . Find the area of ABCD . 7. Assume f (1 , 2) = 10 , f x (1 , 2) = 5 , f y (1 , 2) = 2 . Let g ( x, y ) = xf ( x, y ) 3 xy . (a) Find g (1 , 2), g x (1 , 2) and g y (1 , 2). (b) Estimate g (0 . 9 , 2) and g (1 , 2 . 1). 8. Compute the partial derivatives f x and f y for f ( x, y ) = 2 xy sin( x 2 + y ). 9. Evaluate f x (1 , 2) and f y (1 , 2) for f ( x, y ) = 2 x 2 3 xy + y 2 . Answers 1. x + 2 y z = 1. 2. a = 4, b = 1, c = 1. 3. z = 2 x + 3 y + 3. 4. 2. 5. 5. 6. 6. 7. (a) g (1 , 2) = 4, g x (1 , 2) = 9, g y (1 , 2) = 5; (b) g (0 . 9 , 2) 3 . 1, g (1 , 2 . 1) 3 . 5. 8. f x = 2 y sin( x 2 + y ) + 4 x 2 y cos( x 2 + y ), f y = 2 x sin( x 2 + y ) + 2 xy cos( x 2 + y ). 9. f x (1 , 2) = 2, f y (1 , 2) = 1....
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practice_2 - AC . Find the area of ABCD . 7. Assume f (1 ,...

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