Chapter 1 Topics in Analytic Geometry

MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 17Ifafunctionf isone-to-one,thenithasaninverse functionf −1 . Inthiscasetheequationy = f −1 (x) is equivalent to x = f(y). We can express the graph of f −1 in parametric formby introducing the parameter t = y; this yields the parametric equationsx = f(t), y = tFor example, the graph of f(x) = x 5 +x+1 can be represented parametrically asx = t, y = t 5 +t+1and the graph of f −1 can be represented parametrically asx = t 5 +t+1, y = tTangent Line to Parametric CurvesWe will be concerned with curves that are given by parametric equationsx = f(t), y = g(t)in which f(t) and g(t) have continuous first derivatives with respect to t. It can be provedthat if dx/dt ≠ 0, then y is a differentiable function of x, in which case the chain rule impliesthatdydx = dy/dt(2.2)dx/dtThis formula makes it possible to find dy/dxdirectly fromthe parametric equations withouteliminating the parameter.Example 2.4 Find the slope of the tangent line to the curveat the point where t = π/3.Solution .........x = t−3sint, y = 4−3cost (t ≥ 0)It follows from Formula (2.2) that the tangent line to a parametric curve will be horizontalat those points where dy/dt = 0 and dx/dt ≠ 0, since dy/dx = 0 at such points.Atpointswheredx/dt = 0anddy/dt ≠ 0, therightsideof (2.2)hasanonzeronumeratorand a zero denominator; we will agree that the curve has infinite slope and a verticaltangent line at such point.At points where dx/dt and dy/dt are both zero, the right side of (2.2) becomes anindeterminate form; we call such points singular points.Example 2.5 A curve C is defined by the parametric equationsx = t 2and y = t 3 −3t.Find the points on C where the tangent is horizontal or vertical.