For what values of t is the slope of the curve undefined for the curve defined by x = 2t + 2 cos(t), y = 2t - 2sin(t) with 0 <= t <= 4pi?

x = 2t + 2cos(t)

=> dx/dt = d/dt(2t) + d/dt(2cos(t))

given the chain rule d/dx(cos(u)) = -sin(u) * d/dx(u)

=> dx/dt = d/dt(2t) − 2sin(t) * d/dt(t)

using the power rule

=> dx/dt = 2 − 2sin(t)

=> dt/dx = 1/(2 − 2sin(t))

y = 2t − 2sin(t)

=> dy/dt = d/dt(2t) − d/dt(2sin(t))

given the chain rule d/dx(sin(u)) = cos(u) * d/dx(u)

=> dy/dt = d/dt(2t) − 2cos(t) * d/dt(t)

using the power rule

=> dy/dt = 2 − 2cos(t)

now the slope is dy/dx

and dy/dx = dy/dt * dy/dx

=> dy/dx = (2 − 2cos(t)) * (1/(2 − 2sin(t)))

=> dy/dx = 2(1 − cos(t)) * (1/[2(1 − sin(t))])

=> dy/dx = (1 − cos(t))/(1 − sin(t))

so the slope will be undefined when 1 − sin(t) = 0

=> sin(t) = 1

=> t = 2πn + sin⁻¹(1) and t = 2πn + π −sin⁻¹(1) where n ∈ interger

=> t = 2πn + π/2 where n ∈ interger

now withing the interval of 0 ≤ t ≤ 4π

=> t = π/2, 5π/2

x = 2t + 2 cos(t), y = 2t - 2 sin(t) with 0 <= t <= 4pi

dx/dt = 2 - 2 sin t

dy/dt = 2 - 2 cos t

slope = dy/dx = (dy/dt)/(dx/dt) =

= (2 - 2 cos t)/(2 - 2 sin t) =

= (cos t - 1)/(sin t - 1)

slope is undefined where denominator is zero

sin t - 1 = 0

sin t = 1

t = pi/2, t = pi/2 + 2pi = 5pi/2

PS

at these points the tangent is vertical

the point is called "cusp"