0021&0022 MA261 Quiz 4
Spring 2011
Problem 1. The position function of a particle is given by
r(t) = 3 sin t, 3 cos t, 4t .
(1) Find the velocity v(t).
Solution. v(t) = r(t) = 3 cos t, 3 sin t, 4 .
(2) Find the speed |(t)|.
v
Solution. |v(t)| = 32 + 42 = 5.
(3) Find the acceleration a(t).
Solution. a(t) = v(t) = 3 sin t, 3 cos t, 0 .
Problem 2. Consider the function f of two variables
x2
f (x, y ) = 2 + y 2 .
2
(1) Draw the level curves corresponding to f (x, y ) = 0, f (x, y ) = 1 and f (x, y ) = 4,
respectively?
(2) the Draw graph of f (x, y ) and name the surface.
Solution.
2
When f (x, y ) = 0, we have x2 + y 2 = 0. Only the point (0, 0) makes it hold. Thus,
2
the level curve is just a point.
2
When f (x, y ) = 1, we get x2 + y 2 = 1. This is an ellipse with its major axis on the
2
x axis.
2
2
When f (x, y ) = 4, we het x2 + y2 = 1. This is still an ellipse but larger than the
4
2
previous one.
The level curves and graph of the function are similar to those for Example 11 in
Section 14.1 in the textbook. So they are omitted here.
1