Mathematics

Problem

For

t \ge 0

, a particle is moving along a curve so that its position at time

t

(x(t), y(t))

. At time

t = 2

, the particle is at position

(1, 5)

. It is known that

\frac{dx}{dt} = \frac{\sqrt{t+2}}{e^t}

and

\frac{dy}{dt} = \sin^2 t

. a. Is the horizontal movement of the particle to the left or to the right at time

t = 2

? Explain your answer. Find the slope of the path of the particle at time

t = 2

. b. Find the

x

-coordinate of the particle's position at time

t = 4

. c. Find the speed of the particle at time

t = 4

. Find the acceleration vector of the particle at time

t = 4

. d. Find the distance traveled by the particle from time

t = 2

t = 4