Calculus With Parametric Equations

MAT 229, Spring 2021

Week 14

Supporting materials

If you wish to get a different perspective on the notes below, try either of the following textbook sections.

Stewart’s Calculus
	Section 10.2: Calculus with parametric curves

Boelkins/Austin/Schlicker’s Active Multivariable Calculus
	Section 9.7: Derivatives and integrals of vector-valued functions

Review

Parametric equations are functional values of t equal to the x and y coordinates of a point in the plane:
	x=f(t)
y=g(t)

(We’ll soon generalize this to 3-space, adding a third coordinate --  z=h(t).)

The parameter in this case is t; and while we think of this as a variable, we also think of x and y as the variables in the problem (the coordinates of a point in the plane -- or “in space”, more generally). Frequently it’s useful to think of this as a motion, and of t as “time”. 
As time t varies over its domain, a point undergoes a corresponding motion in the plane.

Questions

Verify that the following parameterizations are all for the same curve.

x=2cos(3t),y=2sin(3t), 0≤t≤π/3

x[t_]:=2Cos[3t]
y[t_]:=2Sin[3t]
ParametricPlot[{x[t],y[t]},{t,0,Pi/3}]
(* Motion: as time t varies from 0 to π/3, the point moves from (2,0) to (-2,0) along the arc of a circle, creating a semi-circle. *)

x=t,y=(Sqrt[4-(t^2)]),-2≤t≤2

x[t_]:=t
y[t_]:=Sqrt[4-t^2]
ParametricPlot[{x[t],y[t]},{t,-2,2}]

x=2sin(t),y=2cos(t), -π/2≤t≤π/2

x[t_]:=2Sin[t]
y[t_]:=2Cos[t]
p1=ParametricPlot[{x[t],y[t]},{t,-Pi/2,Pi/2}];
Manipulate[
Show[p1,ListPlot[{{x[t],y[t]}}]],{t,-Pi/2,Pi/2}
]

(Video)

Tangent lines

Questions

If y=(x^2)+1, compute ((d y)/(d x))(Subscript[|, x=1]). What does this number represent?	(Video)

Consider the Tschirnhausen curve given by (y^2)=(x^3)+3(x^2).

Verify (1,2) is a point on this curve.

Use implicit differentiation to find an equation for the tangent line to this curve at (1,2).

Show[
ContourPlot[y^2==x^3+3x^2,{x,-3,3},{y,-3,3},Axes->True,Frame->False],
ListPlot[{{1,2}},PlotMarkers->{}],
Plot[9/4(x-1)+2,{x,0,2},PlotStyle->{Red}]
]

(Video)

Slopes of curves defined by parametric equations

Slope is rise over run. Infinitesimally, this is ((d y)/(d x)). For parametric equations, the chain rule implies ((d y)/(d x))=((d y/d t)/(d x/d t)).

It’s the usual thing: the slope of the tangent line at (x((Subscript[t, 0])),y((Subscript[t, 0]))); but it’s frequently written in the usual way, with m=((d y/d t)/(d x/d t))(Subscript[|, t=(Subscript[t, 0])]): 
	l(x)=y=m(x-x((Subscript[t, 0])))+y((Subscript[t, 0]))

Questions

If x=2(t^2)+1,y=3(t^3)-4, find ((d y)/(d x)).	(Video)

x[t_]:=2t^2+1
y[t_]:=3t^3-4
dydx[t_]:=y'[t]/x'[t]
dydx[t]

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 where t=π/6.	(Video)

tnot=Pi/6;
x[t_]:=2(t-Sin[t])
y[t_]:=2(1-Cos[t])
ParametricPlot[{x[t],y[t]},{t,0,2Pi}]
p1=ParametricPlot[{x[t],y[t]},{t,0,Pi/2},AspectRatio->1];
dydx[t_]:=y'[t]/x'[t]
tangentLine[t_]:=y[tnot]+dydx[tnot](x[t]-x[tnot])
Print[Simplify[y[tnot]+dydx[tnot](x-x[tnot])]];
p2=ParametricPlot[{x[t],tangentLine[t]},{t,0,2tnot},PlotStyle->{Red}];
p3=ListPlot[{{x[tnot],y[tnot]}},PlotMarkers->{}];
Show[p1,p2,p3]

x=cos(t),y=sin(2t) are parametric equations for a curve.

x[t_]:=Cos[t]
y[t_]:=Sin[2t]
dydx[t_]:=y'[t]/x'[t]
dydx[t]
ParametricPlot[{x[t],y[t]},{t,0,2Pi}]

Find all points on this curve that have horizontal tangents.

Find all points on this curve that have vertical tangents.

Is (0,0) on this curve?

(Video)

Curve length

Given parametric equations for a curve
	x=x(t),y=y(t), a≤t≤b
the length of the parametric curve is
	(∫_a^b)(Sqrt[((x'(t))^2)+((y'(t))^2)])d t.

Note that "the curve" is actually the trajectory of a motion, and one may have retraced one's steps. For example, you might have a curve which is a circle about the origin of radius 1, and the values of t may indicate a trip around the circle multiple times. Each time you do, the length will increase by 2π.

Question

Find the length of the curve x=3cos(t),y=3sin(t),0≤t≤π.	(Video)

x[t_]:=3Cos[t]
y[t_]:=3Sin[t]
Integrate[Sqrt[x'[t]^2+y'[t]^2],{t,0,Pi}] 
Integrate[3,{t,0,Pi}] (* Half a circumference! *)

x[t_]:=3Cos[t]
y[t_]:=3Sin[t]
Integrate[3,{t,0,2 Pi}] (* A full circumference *)

x[t_]:=3Cos[t]
y[t_]:=3Sin[t]
Integrate[3,{t,0,4Pi}] (* Two trips around... *)

Note: Only on rare occasions like the above example, can these integrals be evaluated exactly. Because of the square root, usually these length integrals must often be integrated using numerical techniques.

Questions

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.

x[t_]:=2(t-Sin[t])
y[t_]:=2(1-Cos[t])
{x[2Pi],y[2Pi]}
ParametricPlot[{x[t],y[t]},{t,-3Pi,3Pi}]

Find the length of one arch of the above cycloid.

x[t_]:=2(t - Sin[t])
y[t_]:=2(1 - Cos[t])
curveLength=Integrate[Sqrt[x'[t]^2+y'[t]^2],{t,0,2Pi}]

(Video)

Questions

Consider the parametrically defined curve
	x=cos(3t)
y=sin(2t)

This curve starts repeating. What is its period?

x[t_]:=Cos[3t] (* periodic of period 2Pi/3 *)
y[t_]:=Sin[2t] (* periodic of period 2Pi/2=Pi *)
period=2Pi;
ParametricPlot[{x[t],y[t]},{t,0,period}]
Plot[{x[t],y[t]},{t,0,8Pi}]
Plot[{x[t],y[t]},{t,0,period}]

Set up the integral to find the length of this curve.

x[t_]:=Cos[3t] (* periodic of period 2Pi/3 *)
y[t_]:=Sin[2t] (* periodic of period 2Pi/2=Pi *)
curveLength=NIntegrate[Sqrt[x'[t]^2+y'[t]^2],{t,0,period}]

(Video)

Why the formula works

For a given curve defined by parametric equations x=x(t),y=y(t),a≤t≤b, divide [a,b] into n subintervals of equal length Δ t=((b-a)/n), 
	(Subscript[t, 0])=a,(Subscript[t, 1])=a+Δ t,(Subscript[t, 2])=a+2Δ t,(Subscript[t, 3])=a+3Δ t,…,(Subscript[t, n])=a+n Δ t=b
Approximate the length of the curve using the line segments from (x((Subscript[t, i])),y((Subscript[t, i]))) to (x((Subscript[t, i+1])),y((Subscript[t, i+1]))), as t goes from (Subscript[t, 0]) to (Subscript[t, n]).

f[t_]=t+Cos[t];
g[t_]=t*Sin[t];
GraphicsGrid[{{ParametricPlot[{f[t],g[t]},{t,-5,5},Ticks->None,PlotRange->{{-5,5.3},{-5,2.2}}],Show[ParametricPlot[{f[t],g[t]},{t,-5,5}],Graphics[Join[Table[Text[(Subscript[t, n]),{f[-5+(n)],g[-5+(n)]},{1,-1}],{n,0,10}],{Opacity[0.4],Red,Line[Table[{f[-5+(n-1)],g[-5+(n-1)]},{n,11}]]}]],Ticks->None,PlotRange->{{-5,5.3},{-5,2.2}}]}}]

Question

What is the length of the line segment from point (x((Subscript[t, i])),y((Subscript[t, i]))) to (x((Subscript[t, i+1])),y((Subscript[t, i+1])))?

An approximation for the length of the curve is the sum of the lengths of these individual line segments.
	Length≈√((((x((Subscript[t, 0]))-x((Subscript[t, 1]))))^2)+(((y((Subscript[t, 0]))-y((Subscript[t, 1]))))^2))+√((((x((Subscript[t, 1]))-x((Subscript[t, 2]))))^2)+(((y((Subscript[t, 1]))-y((Subscript[t, 2]))))^2))+…+√((((x((Subscript[t, n-1]))-x((Subscript[t, n]))))^2)+(((y((Subscript[t, n-1]))-y((Subscript[t, n]))))^2))
=(Subscript[Σ, i=1]^n)√((((x((Subscript[t, i-1]))-x((Subscript[t, i]))))^2)+(((y((Subscript[t, i-1]))-y((Subscript[t, i]))))^2))
The squared quantities are differences. Make them into difference quotients.
	=(Subscript[Σ, i=1]^n)√(((((x((Subscript[t, i-1]))-x((Subscript[t, i]))))^2)/(Δ (t^2)))Δ (t^2)+((((y((Subscript[t, i-1]))-y((Subscript[t, i]))))^2)/(Δ (t^2)))Δ (t^2))
=(Subscript[Σ, i=1]^n)√((((((x((Subscript[t, i-1]))-x((Subscript[t, i]))))^2)/(Δ (t^2)))+((((y((Subscript[t, i-1]))-y((Subscript[t, i]))))^2)/(Δ (t^2))))Δ (t^2))
=(Subscript[Σ, i=1]^n)Δ t√((((((x((Subscript[t, i-1]))-x((Subscript[t, i])))/(Δ t))))^2)+(((((y((Subscript[t, i-1]))-y((Subscript[t, i])))/(Δ t))))^2))
For large n this becomes a better approximation. In the limit as n->∞ this expression “melts” into the integral
	(∫_a^b)(Sqrt[((x'(t))^2)+((y'(t))^2)])d t.

Homework

IMath problems on the Calculus of Parametric Equations