MATH 214 Lecture NotesSection 10.1-10.21Parametric curvesA curveCin plane can be represented byparametric equationsx=f(t),y=g(t), t∈[a, b]wherefandgare functions on the interval [a, b].Each value oftdetermines a point(x, y) = (f(t), g(t)) in the plane.Astvaries over [a, b], the point (f(t), g(t)) varies andtraces out the curveC.The curve has initial point (f(a), g(a)) and terminal point (f(b), g(b)).Example: What values is represented by the following parametric equations?(a)x= cost, y= sint,0≤t≤2π.(b)x= tan2θ, y= secθ,-π2< θ <π2.1
Example: Find parametric equations for the circle with center (a, b) and radiusr.2Calculus with parametric curves2.1TangentsSupposefandgare differentiable functions and we want to find the tangent line at a pointon the curvex=f(t), y=g(t) whereyis also a differentiable function ofx. Then the ChainRule givesdydt=dydx·dxdtIfdxdt6= 0, we can solve fordydx:dydx=dydtdxdt,ifdxdt6= 0.Example: A curveCis defined by the parametric equationsx=t2, y=t3-3t.(a) Show thatChas two tangents at the point (3,0) and find their equations.