Question

The graph of the parametric equations is called a cycloid.

x=θ-sinθ and y=1-cosθ for 0≤θ≤2π

(a) find dy/dx

(b) find an equation of the tangent to the cycloid at the point where θ=π/3

(c) at what point is the tangent horizontal?

(d) graph the cycloid and the tangent lines in parts (b) and (c)

Answer 1

a)

$dy/dx=\frac{dy}{d\theta}\frac{d\theta}{dx}$

$dy/d\theta=sin\theta$

$dx/d\theta=1-cos\theta$

$dy/dx=\frac{sin\theta}{1-cos\theta}$

b)

equation of the tangent:

$y-y_0=f'(x_0)(x-x_0)$

then, θ=π/3:

$1-cosθ-(1-cos(\pi/3))=\frac{sin(\pi/3)}{1-cos(\pi/3)}(θ-sinθ-(\pi/3-sin(\pi/3)))$

$1/2-cosθ=\sqrt3(θ-sinθ-\pi/3+\sqrt3/2)$

$cosθ+\sqrt3(θ-sinθ)=\pi/\sqrt3-1$

c)

the tangent horizontal if $f'(x)=0$

so,

$\frac{sin\theta}{1-cos\theta}=0$

$sin\theta=0$

$cos\theta\neq 1$

$\theta=\pi$

$y(\pi)=2,x(\pi)=\pi$

d)