1. Which of the following is an equation of the plane tangent to the surface x2 + y2 - 3z = 2 at the point (-2, -4, 6) ?

 A. x +y +z -2 = 0 B. -2x -4y +6z = 0 C. -2x -4y +6z -2 = 0 D. 4x +8y +3z = 0 E. 4x +8y +3z +22 = 0
solution

2. Let C be the curve in R3 defined by the parametric equations

x(t) = cos(et)

y(t) = sin(et)

z(t) = et

for t in [0,2]. What is the length of C?

 A. B.  C. D. E. solution

3. Which of the following is an equation of the normal line to the graph of
y2 + 6y - x = 4 at the point (3,1)?

 A. 8x - y = 23 B. x + 8y = 11 C. x - 8y = -5 D. 8x + y = -25 E. 8x + y = 25
solution

4. Which of the following is an equation of the tangent plane to the surface
z = x2 + y2x - 2 at the point (1,1,0)?

 A. 3x + 2y + z = 5 B. 3x + 2y - z = 5 C. 3x + 2y - z = 4 D. 2x + 2y -z = 4 E. 2x + 2y + z = 4
solution

5. Which of the following is an equation of the line normal to y = ex at (1,e)?

 A. y = e + e-1 - e-1x B. y = e - e-1 + e-1x C. y = ex D. y = 2e - ex E. y = 2e + ex
solution

6. Which of the following is an equation of the plane tangent to the surface xyz2 = 1 at the point (1,1,-1)?

 A. x + y - 2z = 4 B. x + y + z = 1 C. x + y - z = 3 D. x + y + 2z = 0 E. x + y - 2z = 3
solution

7. The equations below define a line L in R3

x + y = 1

x + z = 1

Which of the following equations defines a plane that is perpendicular to L?

 A. x + y = 1 B. x + z = 1 C. y - z = 0 D. 2x + y + z = 2 E. x - y - z = 0
solution

8. The coordinates of an object moving in R3 at time t are x = t, y = t2, z = (2/3)t3. What is the distance traveled by the object between t = 0 and t = 3 ?

 A. 21 B. 27 C. 57 D. 75 E. 3(46)1/2
solution

9. Let T be the closed region in the first quadrant of the xy-plane bounded by x = 0, x = 1, y = x, and y = x3 + 1. What is  A. 14/3 B. 52/9 C. 56/9 D. 20/3 E. 172/9
solution

10. What is the cosine of the angle between the vectors  A. -3/4 B. 1/150 C. 31/2/15 D. 1/2 E. 2
solution

11. The position vector to a curve in R3 is given by  What is the unit tangent vector at t = Pi/3 ?

 A. B. C. D. E. solution

12. Let,  Which of the following is a unit normal vector to S at t = Pi/6 ?

 A. B. C. D. E. solution

13. Let S be the closed region in the xy-plane whose boundary is the parallelogram with vertices (0,0), (2,0), (3,1), and (1,1).  A. 8 B. 28/3 C. 32/3 D. 73/6 E. 15
solution

14. The coordinates of an object moving through R3 are  for time t > 0, where a, b, and c are constants. What is the speed of the object at time t?

 A. B. C. D. E. solution

15. What is the magnitude of the projection of the vector <1,2,3> onto the vector <4,3,2> ?

 A. (14)1/2/16 B. 16/(406)1/2 C. 16/(29)1/2 D. 16/(14)1/2 E. 16
solution

16. What is the volume of the closed region in R3 bounded by z = 9 - x2 - y2 and z = 0 ?

 A.  B. C. D. E. solution

17. What is an equation of the plane tangent to the surface  at the point (-6,3,1) ?

 A. 2x + 6y - z = 5 B. 2x + 6y - z = -5 C. x + y + z = -2 D. x + y - z = -4 E. z = 1
solution

18. The line normal to the surface 3x + y2 - z2 = 0 at the point (3,0,3) also intersects the surface at what other point?

 A. (-3/4,0,21/2) B. (0,0,0) C. (3,0,-3) D. (21/4,0,-3/2) E. (27/4,0,-9/2)
solution

19. What is the cosine of the angle between the vectors <-2,1,2> and <6,3,-2> in R3 ?

 A. -13/10 B. -10/13 C. -13/21 D. 3/7 E. 19/21
solution