Two particles move along the $x$-axis. For $0 \leq t \leq 8$, the position of particle $P$ at time $t$ is given by $x_P(t) = \ln\left(t^2 - 2t + 10\right)$, while the velocity of particle $Q$ at time $t$ is given by $v_Q(t) = t^2 - 8t + 15$. Particle $Q$ is at position $x = 5$ at time $t = 0$. (a) For $0 \leq t \leq 8$, when is particle $P$ moving to the left? (b) For $0 \leq t \leq 8$, find all times $t$ during which the two particles travel in the same direction. (c) Find the acceleration of particle $Q$ at time $t = 2$. Is the speed of particle $Q$ increasing, decreasing, or neither at time $t = 2$? Explain your reasoning. (d) Find the position of particle $Q$ the first time it changes direction.
Two particles move along the $x$-axis. For $0 \leq t \leq 8$, the position of particle $P$ at time $t$ is given by $x_P(t) = \ln\left(t^2 - 2t + 10\right)$, while the velocity of particle $Q$ at time $t$ is given by $v_Q(t) = t^2 - 8t + 15$. Particle $Q$ is at position $x = 5$ at time $t = 0$.\\
(a) For $0 \leq t \leq 8$, when is particle $P$ moving to the left?\\
(b) For $0 \leq t \leq 8$, find all times $t$ during which the two particles travel in the same direction.\\
(c) Find the acceleration of particle $Q$ at time $t = 2$. Is the speed of particle $Q$ increasing, decreasing, or neither at time $t = 2$? Explain your reasoning.\\
(d) Find the position of particle $Q$ the first time it changes direction.