Need help with this calculus problem

Question

a) Which way is the particle moving vertically at time t = 2 seconds, up or down?

b) At what rate is the y-coordinate changing at time t = 2 seconds?

Answer 1

The movement of this particle is described by a parametric equation.

You are given that the x-coordinate is decreasing at a rate of 1 unit/sec. That is

x´(t) = -1

We find the antiderivative/integral of x´(t) = -1 to get

x(t) = -t + c

At t = 2, we are told that x = π/6. Therefore

x(2) = -2 + c = π/6

which implies that c = π/6 + 2, so

x(t) = -t + π/6 + 2

We are told that y(x) = cos(x)

Into that x variable we substitute our function for x in terms of t. Thus,

y(t) = cos(-t + π/6 + 2)

Differentiate using the chain rule to get y´(t) and plug in t=2 to answer both parts of the question.