Chapter 9: Parametric Equations, Polar Coordinates and Conic Sections

Let c and d be constants, and for t ∈ [a, b] let f (t) and g(t) be differentiable functions with continuous first derivatives. Prove that the arc length of the parametric curve given by x = f (t), y(t) = g(t) for t ∈ [a, b] is equal to

the arc length of the parametric curve defined by x = c + f (t), y(t) = d + g(t) for t ∈ [a, b] for every c and d in R.

Find and prove a formula for the distance between the

points $(r_1,{\theta }_1)$ and $(r_2,{\theta }_1)$ when they are plotted in a polar

coordinate system.

Let k > 0 be a constant, and let f (t) and g(t) be differentiable functions of t with continuous first derivatives for every t ∈ [a, b]. Prove that the arc length of the curve defined by the parametric equations $x=kf\left(t\right),y=kg\left(t\right),t\in [a,b]$

is k times as long as the arc length of the curve defined by the parametric equations $x=f\left(t\right),y=g\left(t\right),t\in [a,b]$

What is the arc length of the curve defined by the equations

$x=f\left(kt\right),y=g\left(kt\right),t\in [a/k,b/k]?$

In Example 5 we saw that the cycloid

$x=r\theta -r\sin \theta ,y=r-r\cos \theta ,\theta \in \mathrm{ℝ}$

has a horizontal tangent line at each odd multiple of$\pi$. Show that the cycloid has a vertical tangent at each even multiple of $\pi$by showing that $\underset{\theta \to 2\mathrm{li}}{\lim }\frac{dy}{dx}$does not exist wherever k is an integer.

Sketch the curves defined by the given sets of parametric equations. Indicate the direction of motion on each curve.$x=\sin ht,y=\cos ht,t\in R$

Three noncollinear points determine a unique circle. Do three noncollinear points determine a unique ellipse? If so, explain why. If not, provide three noncollinear points that are on two distinct ellipses.

Sketch the curves defined by the given sets of parametric equations. Indicate the direction of motion on each curve.

$x=\sin ht$,$y=\cos h^{2}t$,$t\in \mathfrak{R}$

Use the results of Exercise 3 to analyze the direction of motion for the parametric curves given by the equations in Exercises 5–8

$x=t^3-t,y=t^3+t^{3}t\in \mathrm{ℝ}$

Fill in the blanks to complete each of the following theorem statements:

Let $r=f\left(\theta \right)$be a differentiable function of $\theta$such that $f\text{'}\left(\theta \right)$is continuous for all $\theta \in \left[\alpha ,\beta \right]$Furthermore, assume that $r=f\left(\theta \right)$is a one-to-one function from $\left[\alpha ,\beta \right]$to the graph of the function. Then the length of the polar graph of $r=f\left(\theta \right)$on the interval $\left[\alpha ,\beta \right]$is..............

Explain why the graphs of $\theta =\frac{\pi }{2}$and $\theta =-\frac{\pi }{2}$are identical in a polar coordinate system.