1. Consider the two vectors !v = 5!i + 8!j − !k and w
! = 3!i − 2!j + 6!k.
(4 points)
(a) Find a unit vector in the direction of the vector w.
!
(a)
(9 points)
(b) Find the projection of !v on to the vector w.
!
(Hint: This projection is also called !vparallel .)
(b)
2. Consider the vectors !a = "c, −3, c# and !b = "c, 2, 1#.
(5 points)
(a) Compute the dot product !a · !b.
(a)
(7 points)
(b) What value(s) of c make !a perpendicular to !b?
(b)
(12 points) 3. On the axes below, sketch a contour diagram for the function f (x, y) = x3 − y + 2 with four
labeled contours.
4. For the function f (x, y) = (3xy + 2x)3 :
(6 points)
(a) Compute fx (1, 0).
(a)
(6 points)
(b) Compute fy (−1, 1).
(b)
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5. Consider the three points P = (1, 2, 3), Q = (−1, 3, 4) and R = (0, 1, −1).
(5 points)
(a) Compute !a, the vector from P to Q, and !b, the vector from P to R.
(6 points)
(b) Calculate the cross-product !a × !b.
(b)
(4 points)
(c) Give an equation for the plane containing the points P , Q, and R.
(c)
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(12 points) 6. Is the function
f (x, y) =
2x + 3y
x2 + y 2
continuous on the rectangle 1 ≤ x ≤ 2, 2 ≤ y ≤ 3? Why or why not?
(12 points) 7. For the function z = 12 (x2 + 4y 2 ), find the equation of the tangent plane at the point (2, 1, 4).
7.
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8. Let f (x, y) = xy + y 3 .
(4 points)
(a) Find the directional derivative f!u (1, 2), where !u = 35!i − 45!j. (Note: ||!u|| = 1.)
(a)
(4 points)
(b) In what direction does the maximum rate of change of f at (1, 2) occur?
(b)
(4 points)
(c) What is the maximum rate of change of f at (1, 2)?
(c)
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