PARAMETRIC EQUATIONS, VECTOR FUNCTIONS, CURVILINEAR MOTION
Fig. 34-5
Fig. 34-6
34.6
Sketch the curve with the parametric equations x = sec t, y = tan t.
X
2 = y
2 + l. Hence, x
2 — y
2 = l. Thus, the curve is a rectangular hyperbola with the perpendicular
asymptotes y = ±x. See Fig. 34-6.
34.7
Sketch the curve with the parametric equations x = sin t, y = cos 2t.
y = cos 2t = 1 — 2 sin
2 1 = 1 — 2x
2 , defined for \x\ ^ 1. Thus, the curve is an arc of a parabola, with vertex at
(0,1), opening downward, and with the _y-axis as axis of symmetry (Fig. 34-7).
Fig. 34-7
Fig. 34-8
34.8
Sketch the curve with the parametric equations x = t + 1 It, y = t - 1 It.
x
2 = t
2 + 2 + 1/1
2 , y
2 = t
2 -2+l/t
2 . Subtracting the second equation from the first, we obtain the
hyperbola Jt
2 -y
2 = 4 (Fig. 34-8).
34.9
Sketch the curve with the parametric equations * = 1 + t, y = l-t.
x + y = 2. Thus, we have a straight line, going through the point (1,1) and parallel to the vector (1, —1); see
Fig. 34-9.
Fig. 34-9
Fig. 34-10
275