Cracking The Ap Calculus ab Exam 2018

See the negative velocity? The sign of the velocity is important because it indicates the direction of the
particle. Make sure that you know the following:

When    the velocity    is  negative,   the particle    is  moving  to  the left.

When    the velocity    is  positive,   the particle    is  moving  to  the right.

When     the     velocity    and     acceleration    of  the     particle    have    the     same    signs,  the     particle’s  speed   is

increasing.

When     the     velocity    and     acceleration    of  the     particle    have    opposite    signs,  the     particle’s  speed   is

decreasing  (or slowing down).

When    the velocity    is  zero    and the acceleration    is  not zero,   the particle    is  momentarily stopped and

changing    direction.

Example 2: If the position of a particle is given by x(t) = t^3 − 12t^2 + 36t + 18, where t > 0, find the point
at which the particle changes direction.

The derivative is

x′(t)   =   v(t)    =   3t^2    −   24t +   36

Set it equal to zero and solve for t.

x′(t)   =   3t^2    −   24t +   36  =   0

t^2     −   8t  +   12  =   0

(t  −   2)(t    −   6)  =   0

So we know that t = 2 or t = 6.

You need to check that the acceleration is not 0: x′′(t) = 6t − 24. This equals 0 at t = 4. Therefore, the
particle is changing direction at t = 2 and t = 6.

Example 3: Given the same position function as in Example 2, find the interval of time during which the
particle is slowing down.

When 0 < t < 2 and t > 6, the particle’s velocity is positive; when 2 < t < 6, the particle’s velocity is
negative. You can verify this by graphing the function and seeing when it’s above or below the x-axis. Or,
try some points in the regions between the roots and outside the roots. Now, we need to determine the
same information about the acceleration.