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22 MIDTERM 1A

# 22 MIDTERM 1A - lim x,y →(0 0 y 4-x 4 y 2 x 2 3 Problem...

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1 MIDTERM I NAME MATH 20E (please print) TA’S NAME (please print) FORM A Problem 1. (16 pts.) Find the equation of: (a) the line through the point P = (1 , 2 , 0) and perpendicular to vector v = (1 , - 1 , 2) . (b) the plane through the points P 1 = (1 , 2 , 0) , P 2 = (1 , 1 , 0) and P 3 = (0 , 1 , - 1) . Problem 2. (14 pts.) Let v 1 = (1 , 2 , 1) ,v 2 = (1 , 0 , - 1) ,v 3 = (0 , 2 , 1) be three given vectors in R 3 . Find: (a) the area of the parallelogram spaned by v 1 and v 2 , (b) the volume of the parallelepiped spaned by v 1 ,v 2 ,v 3 .

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2 Problem 3. (14 pts.) For the matrices A and B compute detA and AB : A = 1 0 2 2 1 - 2 - 1 1 2 , B = - 1 0 2 2 1 0 1 2 - 1 Problem 4. (14 pts.) Compute the limit:

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Unformatted text preview: lim ( x,y ) → (0 , 0) y 4-x 4 y 2 + x 2 3 Problem 5. (14 pts. ) Let f ( x,y,z ) = cosxyz e x 2 + y 2 + z 2 . (a) Compute the gradient of f (b) Is f diﬀerentiable in its domain? Explain your answer! Problem 6. (14 pts) Find the equation of the tangent plane to the surface x 2 + z 2 + yx + 2 = 0 at the point (-1 , 4 ,-1) . 4 Problem 7. (14 pts) Find the second order Taylor formula for the function f ( x 1 ,x 2 ,x 3 ) = x 2 e x 3 + x 1 x 2 at the point X = (0 , , 0) ....
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22 MIDTERM 1A - lim x,y →(0 0 y 4-x 4 y 2 x 2 3 Problem...

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