A particle starts at point A on the positive x-axis at time t = 0 and travels along the curve from A to B to C to D, as shown above. The coordinates of the particle's position ( x(t) , y(t) ) are differentiable functions of t, where
x'(t) =
and
y'(t) =
in not explicitly given. At time t=9, the particle reaches its final position at point D on the positive x-axis.

(a) At point C is positive? At point C is positive? Give a reason for each answer?
At point C is not positive, but negative because the particle is moving down from B to D.
At point C in not positive, but negative because the particle is moving backwards from B to D.

(b) The slope of the curve is undefined at point B. At what time t is the particle at point B?

(dy/dt) / (dx/dt) = (dy/dx), so if the slope is undefined than (dy/dx) is undefined which means the (dx/dt) = 0

, which means t = 3, dumb computer program.

(c) The line tangent to the curve at the point ( x(8), y(8)) has the equation y=(5/9)x-2. Find the velocity vector and the speed of the particle at the point B?

- Solve x'(t) for t = 8



and when this is further solved x'(8) = -4.5

- Solve y'(t) for t = 8

y'(t) = -2.5 , to make the ratio when divided out of (5/9)

(d) How far apart are points A and D, the initial and final positions respectively, of the particles

distance =