|
Calculus III |
Review Test II |
Summer 2006 |
1. Eliminate the parameter, then solve for y, and sketch the graph for each of the parametric functions below.
a. 
b. 
2. Find 
and 
(formula given on test) for 
.
3
a Find the arc length for the parametric equation
. Show the set-up and the result.
b Find the arc length for the parametric equation
.Show the set-up and the result.
4 Sketch the following two polar curves, by finding a chart of their polar coordinates on 
at intervals of
.
a 
b 
5. Find a chart of coordinates on , which contains the key points necessary for finding the area of the region common to the interior of the polar curves 
and 
. Set-up, but do not solve the integrals.
6. Start with the Law of Cosines
, find an alternate form of 
by squaring 
. Then, derive the formula for the dot product of vectors 
and
, 
, that involves the cosine of the angle between vectors and .
7.
a. Find the angle between the vectors a = (1,2) & b = (3,4), expressed in
form.
b. Normalize the vectors a and b.
8. If a Force of 10 Newtons is applied at an angle of 30 degrees and moves an object 0.5 meters, what is the amount of work done?
9.
a. Find the vector sum for the vectors u = (1,2,3) and v = (5,7,9)
b. Find the vector from u to v.
.
c. What is the angle between the vectors u and v?
10.
a. Determine whether the points (1,-1,5), (0,-1,6), and (3,-1,3) lie on a straight line.
b. Find the direction cosines of the point (3,-1,5).