Math 1010 Tutorial
Quiz 7
Week 7
The solutions to Quiz 7 are below. I might be showing more steps than necessary, this is to aid your
understanding.
1. x = 1 − t2 , y = t − 2, −2 ≤ t ≤ 2
2
(a) Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the
direction in which the curve is traced as t increases.
Solution: First, we will determine a few points on the curve. Since t = −2 and t = 2 are the
endpoints, we should include these. Thus a few points on the curve are
t -2 -1
0
1
2
x -3
0
1
0 -3
y -4 -3 -2 -1
0
Thus we get the sketch for the curve as shown above.
1
(b) Eliminate the parameter to find a Cartesian equation of the curve.
Solution: From y = t − 2 we get t = y + 2. Also, since −2 ≤ t ≤ 2, we have −4 ≤ y ≤ 0.
Substituting into x = 1 − t2 , we then have
x = 1 − (y + 2)2
−4≤y ≤0
This matches our sketch since this corresponds to a parabola with x as the dependent variable.
1
2. Given x = t sin t and y = t2 + t, find
Solution: We have
and
Thus
dy
dx .
dy
= 2t + 1
dt
dx
= t cos t + sin t
dt
dy
dy/dt
2t + 1
=
=
dx
dx/dt
t cos t + sin t
1 (bonus)
3. Sketch the region in the plane consisting of points whose polar coordinates satisfy 0 ≤ r < 2 and
π ≤ θ ≤ 3π/2.
Solution: 0 ≤ r < 2 indicates a region that is a disk of radius 2, but not including the edge.
π ≤ θ ≤ 3π/2 indicates the third quadrant.
Together, we get the region in the sketch.