Question

Consider the curve represented by the parametric equations x(t)= 2+sin(t) and y(t)=1-cos(t) when answering the following questions.

A) Find Dy/Dx in terms of t

B) Find all values of t where the curve has a horizontal tangent.

C) Find all values of t where the curve has a vertical tangent.

D) Write an integral that represents the arc length of the curve on the interval 0 ≤ t ≤ 2π. Evaluate the integral.

Answer 1

a)

dy/dt = d(1-cos(t))/dt = sint
dx/dt = d(2+sin(t))/dt = cost

=> dy/dx = (dy/dt)/(dx/dt) = sint/cost = tan(t)

b)

Horizontal tangent implies that the slope of the line is 0

We already have the term that gives us the slope, which is dy/dx, which is tant.

For tan(t) = 0,
t = 0, pi, 2pi,.....

c)

Vertical tangent implies that the slope of the line is not defined, but for the given tan(t) slope this would imply that cos(t) is 0.

FOr cos(t) = 0,
t = (pi/2), (3pi/2),......

Answer 2

So for B and C, how can I show the answer if it continues on?

and could you help me with part D?