Section 10.2: Calculus With Parametric Equations

Review

Parametric equations are functional values for x and y coordinates
	x=f(t)
y=g(t)

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}]

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]
ParametricPlot[{x[t],y[t]},{t,-Pi/2,Pi/2}]

Tangent lines

Questions

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

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).

p1=Plot[{Sqrt[x^3+3x^2],-Sqrt[x^3+3x^2]},{x,-3,3}];
slope=9/4
p2=Plot[{slope(x-1)+2},{x,-5,5},PlotStyle->Dashed];
Show[p1,p2]

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)).

Questions

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

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.

x[t_]:=2(t-Sin[t])
y[t_]:=2(1-Cos[t])
p1=ParametricPlot[{x[t],y[t]},{t,0,Pi/4},AspectRatio->1];
dydx[t_]:=y'[t]/x'[t]
tangentLine[t_]:=y[Pi/6]+dydx[Pi/6](x[t]-x[Pi/6])
p2=ParametricPlot[{x[t],tangentLine[t]},{t,0,Pi/4}];
Show[p1,p2]

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?

Concavity

The curve y=f(x) is concave up wherever (((d^2)y)/(d (x^2)))>0, and it is concave down where (((d^2)y)/(d (x^2)))<0. For parametric equations using the chain rule
	(((d^2)y)/(d (x^2)))=(d/(d x))(((d y)/(d x)))=(((d/(d t))(((d y)/(d x ))))/((d x)/(d t)))=(((d/(d t))(((d y/d t)/(d x /d t))))/((d x)/(d t)))

Questions

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

x[t_]:=2t^2+1
y[t_]:=3t^3-4
dydx[t_]=y'[t]/x'[t]
dy2dx2[t_]=D[y'[t]/x'[t],t]/x'[t]

x=2(t-sin(t)),y=2(1-cos(t)) are parametric equations for a cycloid. Is the cycloid concave up or down at t=π/6?

x[t_]:=2(t - Sin[t])
y[t_]:=2(1 - Cos[t])
dydx[t_]=y'[t]/x'[t]
dydx[t]
dy2dx2[t_]=D[y'[t]/x'[t],t]/x'[t]
dy2dx2[Pi/6]
N[%]

x=cos(t),y=sin(2t) are parametric equations for a curve. For which values of t in [0,2π] is the curve concave up?

x[t_]:=Cos[t]
y[t_]:=Sin[2t]
dydx[t_]=y'[t]/x'[t]
dydx[t]
ParametricPlot[{x[t],y[t]},{t,0,2Pi}]
dy2dx2[t_]=D[y'[t]/x'[t],t]/x'[t]
Plot[dy2dx2[t],{t,0,2Pi}]

Curve length

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

Question

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

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 be integrated using numerical techniques.

Questions

The cycloid x=2(t-sin(t)),y=2(1-cos(t)) lies above the x-axis. Determine the values of parameter t where it touches the x-axis.

ParametricPlot[{2(t-Sin[t]),2(1-Cos[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}]

Questions

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

This curve starts repeating. What is its period?

x[t_]:=Cos[3t]
y[t_]:=Sin[2t]
ParametricPlot[{x[t],y[t]},{t,0,2Pi}]

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

curveLength=NIntegrate[Sqrt[x'[t]^2+y'[t]^2],{t,0,2Pi}]

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 curve with the line segments from (x((Subscript[t, i])),y((Subscript[t, i]))) to (x((Subscript[t, i+1])),y((Subscript[t, i+1]))).

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≈(Sqrt[(((x((Subscript[t, 0]))-x((Subscript[t, 1]))))^2)+(((y((Subscript[t, 0]))-y((Subscript[t, 1]))))^2)])+(Sqrt[(((x((Subscript[t, 1]))-x((Subscript[t, 2]))))^2)+(((y((Subscript[t, 1]))-y((Subscript[t, 2]))))^2)])+…+(Sqrt[(((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)(Sqrt[(((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)(Sqrt[((((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)(Sqrt[(((((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)(Sqrt[(((((x((Subscript[t, i-1]))-x((Subscript[t, i])))/(Δ t))))^2)+(((((y((Subscript[t, i-1]))-y((Subscript[t, i])))/(Δ t))))^2)])Δ t
For large n this becomes a better approximation. In the limit as n->∞ this expression goes to
	(∫_a^b)(Sqrt[((x'(t))^2)+((y'(t))^2)])d t.