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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?
Solution
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.