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