Answer to Question #265033 in Calculus for Hema

Question #265033

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)



1
Expert's answer
2021-11-16T10:39:32-0500

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-cos\u03b8-(1-cos(\\pi\/3))=\\frac{sin(\\pi\/3)}{1-cos(\\pi\/3)}(\u03b8-sin\u03b8-(\\pi\/3-sin(\\pi\/3)))"

"1\/2-cos\u03b8=\\sqrt3(\u03b8-sin\u03b8-\\pi\/3+\\sqrt3\/2)"

"cos\u03b8+\\sqrt3(\u03b8-sin\u03b8)=\\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)






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