Mathematics

Problem

Consider the differentiable functions

x(t)

and

y(t)

given by

x(t)=t^3-4t^2+6t, \qquad y(t)=\sin t+2t,

for

0\le t\le4

. The parametric curve

C

is traced once as

t

increases over this interval. a. Determine

\frac{dx}{dt}

and

\frac{dy}{dt}

for the curve

C

. Provide exact expressions in terms of

t

. b. Identify every value of

t

[0,4]

at which the tangent line to

C

is vertical, and explain your reasoning in complete sentences.