Math 2A  Vector Calculus  Fall ’07  Chapter 10 Test    Name____________________

Show your work for credit.  Do not use a calculator.  Write all responses on separate paper.

 

1.      Find a parameterization of the line segment from  to  where .

2.      Find a vector function that represents the curve of intersection of the elliptical cylinder  with the plane x + y + z = 1.

3.      Find the length of the curve  on the interval .

4.      Reparametrize the curve , with respect to arclength measured from the point (1,0,1) in the direction of increasing t.

5.      Find an equation for the osculating plane of the curve  
at the point (2, 3, 4).

6.      A particle starts at (0,0,100) and has initial velocity  and moves with acceleration .  Find its position function and determine where it intersects
the xy plane.

7.      Find an equation for the osculating circle at the vertex of the parabola .

8.      Find the tangential and normal components for the acceleration of a particle whose position function is .

9.      Describe the surface:

10.  Find a parametric representation for the part of the hyperboloid  that lies to the right of the plane x = 1.

Math 2A  Vector Calculus  Fall ’07  Chapter 10 Test Solutions.

 

1.      Find a parameterization of the line segment from  to  where .
ANS: 

2.      Find a vector function that represents the curve of intersection of the elliptical cylinder  with the plane x + y + z = 1.
ANS:   will do the trick. 

 

3.      Find the length of the curve  on the interval .
ANS: 

4.      Reparametrize the curve , with respect to arclength measured from the point (1,0,1) in the direction of increasing t.
ANS: 
Thus  and substituting for 2t we have

5.      Find an equation for the osculating plane of the curve  
at the point (2, 3, 4).
ANS:  and  are both in the osculating plane, so their cross product  is normal and choosing d  in the equation  to fit the given point we have

6.      A particle starts at (0,0,100) and has initial velocity  and moves with acceleration .  Find its position function and determine where it intersects
the xy plane.
ANS:  The velocity is  and  so the position is  Thus it’ll hit the ground when  

7.      Find an equation for the osculating circle at the vertex of the parabola .
ANS:  Taking  we have at the vertex,   and  1 so the radius is the reciprocal = 1 and the center is (0,1,0) and the equation is (x  1)² + (y  1)² = 1.

8.      Find the tangential and normal components for the acceleration of a particle whose position function is .
ANS:  .  Now  so  and .

9.      Describe the surface:
ANS:  This is the portion of the cylinder  of radius 1 with axis  between .

10.  Find a parametric representation for the part of the hyperboloid  that lies to the right of the plane x = 1.
ANS:   can be parameterized simply by