Math 1262: Calculus II
Brown
Parametric Equations and Parametric Curves
1. Sketch the curve by plotting points. Indicate the direction in which the curve is traced as t increases.
y = t − cos t,
x = 2 cos t,
0 ≤ t ≤ 2π
2. Sketch the curves by plotting points. Indicate the direction in which the curve is traced as t increases.
Eliminate the parameter to find a Cartesian equation of the curve.
(a) x = 1 + t,
y = 5 − 2t,
−2 ≤ t ≤ 3
Solution. y = 7 − 2t
(b) x = t2 ,
y = t3
Solution. y 2 = x3
3. Sketch the curves by eliminating the parameter to find a Cartesian equation of the curve. Indicate the
direction in which the curve is traced as t increases.
(a) x = 4 cos θ,
y = 5 sin θ,
−π/2 ≤ θ ≤ π/2
x2
y2
+
=1
16 25
√
(b) x = ln t, y = t, t ≥ 1
Solution.
Solution. y = ex/2
4. Describe the motion of a particle with position (x, y) as t varies in the given interval.
x = sin t,
y = cos2 t,
−2π ≤ t ≤ 2π
Solution. The particle moves along the parabola y = 1 − x2 from (0, 1) to (1, 0) back to (0, 1) to (−1, 0)
back to (0, 1) to (1, 0) back to (0, 1) to (−1, 0) back to (0, 1).
5. Suppose a curve is given by the parametric equations x = f (t), y = g(t), where the range of f is [1, 4]
and the range of g is [2, 3]. What can you say about the curve?
Solution. The curve never leaves the rectangle [1, 4] × [2, 3] = {(x, y) | 1 ≤ x ≤ 4, 2 ≤ y ≤ 3}.
6. If a projectile is fired with an initial velocity of v0 meters per second at an angle α above the horizontal
and air resistance is assumed to be negligible, then its position after t seconds is given by the parametric
equations
1
x = (v0 cos α)t,
y = (v0 sin α)t − gt2
2
where g = 9.8m/s2 is the acceleration due to gravity.
(a) If a gun is fired with α = 30◦ and v0 = 500 m/s, when will the bullet hit the ground?
Solution. The bullet will hit the ground after t =
250
4.9
seconds.
(b) How far from the gun will the bullet travel?
√
Solution. It will travel x = 250 3( 250
4.9 ) meters.
(c) What is the maximum height reached by the bullet?
Solution. The maximum height is
1252
4.9
meters.
(d) Show that the path is parabolic by eliminating the parameter.
Solution. y = x tan α −
g
x2
2v0 cos2 α