Section 9.1Parametric Curves tkiryl（9.1节参数曲线tkiryl）.pdfVIP

Section 9.1 Parametric Curves 2010 Kiryl Tsishchanka Parametric Curves Suppose that x and y are both given as functions of a third variable t (called a parameter) by the equations x = f (t), y = g (t) (called parametric equations). Each value of t determines a point (x, y), which we can plot in a coordinate plane. As t varies, the point (x, y) = (f (t), g(t)) varies and traces out a curve C, which we call a parametric curve. EXAMPLE: Sketch and identify the curve defined by the parametric equations x = cos t, y = sin t 0 ≤ t ≤ 2π Solution: If we plot points, it appears that the curve is a circle (see the figure below and page 6). We can confirm this impression by eliminating t. If fact, we have x2 + y2 = cos2 t + sin2 t = 1 2 2 Thus the point (x, y) moves on a unit circle x + y = 1 . EXAMPLE: Sketch and identify the curve defined by the parametric equations x = sin 2t, y = cos 2t 0 ≤ t ≤ 2π 1 Section 9.1 Parametric Curves 2010 Kiryl Tsishchanka EXAMPLE: Sketch and identify the curve defined by the parametric equations x = sin 2t, y = cos 2t 0 ≤ t ≤ 2π Solution: If we plot points, it appears that the curve is a circle (see the Figure below). We can confirm this impression by eliminating t. If fact, we have x2 + y2 = sin2 2t + cos2 2t = 1 Thus the point (x, y) moves on a unit circ