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