ch 9 notes - Math 0230 Chapter 9 Calculus 2 Lectures...

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Math 0230Calculus 2LecturesChapter 9Parametric Equations and Polar CoordinatesNumeration of sections corresponds to the textJames Stewart, Essential Calculus, Early Transcendentals, Second edition.Section 9.1Parametric CurvesParametric equations:x=f(t),y=g(t), wheref(t) andg(t) are continuous functions. Theydefine a curve in thexy-plane whentruns along an interval [a, b]: (x, y) = (f(t), g(t)), whenatb. Initial point of the curve is (f(a), g(a)) and terminal point is (f(b), g(b)). In theparametric formyis not necessarily must be a function ofx.Example 1.Sketch the curvex=t-1,y=t2-3, when-2t4.
Example 2.(a) Sketch the curvex= cost,y= sint, when 0t2π.
(b) Sketch the curvex=h+rcost,y=k+rsint, when 0t2π.
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Section 9.2Calculus with Parametric CurvesTangentsx=f(t),y=g(t). Here we assume thatf(t) andg(t) are differentiable functions. y can beexpressed asy=x(t). Using the Chain Rule we getdydt=dydx·dxdtordydx=dy/dtdx/dt=y0x0.The derivative evaluated at a point (x0, y0), such thatx0=x(t0),y0=y(t0) for somet0, is aslope of the tangent line to the parametric curve at that point.Ifdydt= 0, then the tangent is horizontal. Ifdxdt= 0, then the tangent is vertical.For the second derivative,d2ydx2=ddxdydx=ddtdydxdxdtExample 1.The curvex= 2 cost,y= 3 sint, 0t2πis an ellipse.(a) Find an equation of the tangent line to the ellipse at a point wheret=π/3.

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