Quiz 11

Name:

MAT 229, Spring 2025

1. x=2(t-sin(t)),y=2(1-cos(t)) are parametric equations for a cycloid. Find an equation for the 	tangent line to this curve when t=π/6.

2. x=cos(t),y=sin(2t) are parametric equations for the “infinity curve”:

x[t_]:=Cos[t]
y[t_]:=Sin[2t]
dydx[t_]:=y'[t]/x'[t]
dydx[t]
Manipulate[
Show[
ParametricPlot[{x[t],y[t]},{t,0,2Pi}],
ListPlot[{{x[a],y[a]}},PlotMarkers->{}]
],
{a,0,2Pi}
]

a. Find all times for which this curve will have horizontal tangents.

b. Find all times for which this curve will have vertical tangents.

c. At what time does the curve pass through (0,0) on this curve?

3. The cycloid x=2(t-sin(t)),y=2(1-cos(t)) lies above the x-axis, but touches the axis occasionally. Determine the values of parameter t where it touches the x-axis, then find the length of one arch of the cycloid.