Thread: Calculus AP Question

A particle moves along the x axis so that at time t its position is given by x(t)= t^3 - 6t^2 + 9t + 11
a) What is velocity of particle at t=0
b) During what time intervals is partical moving to the left?
c) What is the total distance traveled by partical from t=0 to t=2

Hello, meg24209!

A particle moves along the x axis so that at time $\displaystyle t$ its position
is given by: .$\displaystyle x(t) \:= \:t^3 - 6t^2 + 9t + 11$

a) What is velocity of particle at $\displaystyle t=0$ ?

You're expected to know that the velocity is the first derivaitve.

$\displaystyle v(t) \;=\;x'(t) \;=\;3t^2-12t + 9$

At $\displaystyle t=0\!:\;\;v(0) \;=\;3\!\cdot\!0^2 - 12\!\cdot\!0 + 9 \;=\;9$

b) During what time intervals is particle moving to the left?

When is $\displaystyle v(t)$ negative?

We have: .$\displaystyle 3t^2 - 12t + 9 \:<\:0\quad\Rightarrow\quad (t-1)(t-3) \:<\:0$

And we find that the velocity is negative for: .$\displaystyle 1 \:<\:t\:<\:3$

c) What is the total distance traveled by particle from $\displaystyle t=0$ to $\displaystyle t=2$ ?

This could be considered a trick question . . .


At $\displaystyle t=0\!:\;x(0)\;=\;0^3-6\!\cdot\!0^2 + 9\!\cdot\!0 + 11 \;=\;11$

At $\displaystyle t=1\!:\;x(1)\;=\;1^3-6\!\cdot\!1^2 + 9\!\cdot\!1 + 11 \;=\;15$ . . . The particle has moved 4 units to the right.

At $\displaystyle t=2\!:\;x(2)\;=\;2^3-6\!\cdot2^2 + 9\!\cdot\!2 + 11\;=\;3$ . . . The particle has moved 12 units to the left.


Therefore, the particle travelled $\displaystyle 4 + 12 \:=\:16$ units.