A particle is moving along a curve so that its position at time $t$ is $(x(t), y(t))$, where $x(t) = t^2 - 4t + 8$ and $y(t)$ is not explicitly given. Both $x$ and $y$ are measured in meters, and $t$ is measured in seconds. It is known that $\frac{dy}{dt} = te^{t-3} - 1$. (a) Find the speed of the particle at time $t = 3$ seconds. (b) Find the total distance traveled by the particle for $0 \leq t \leq 4$ seconds. (c) Find the time $t$, $0 \leq t \leq 4$, when the line tangent to the path of the particle is horizontal. Is the direction of motion of the particle toward the left or toward the right at that time? Give a reason for your answer. (d) There is a point with $x$-coordinate 5 through which the particle passes twice. Find each of the following. (i) The two values of $t$ when that occurs (ii) The slopes of the lines tangent to the particle's path at that point (iii) The $y$-coordinate of that point, given $y(2) = 3 + \frac{1}{e}$
A particle is moving along a curve so that its position at time $t$ is $(x(t), y(t))$, where $x(t) = t^2 - 4t + 8$ and $y(t)$ is not explicitly given. Both $x$ and $y$ are measured in meters, and $t$ is measured in seconds. It is known that $\frac{dy}{dt} = te^{t-3} - 1$.\\
(a) Find the speed of the particle at time $t = 3$ seconds.\\
(b) Find the total distance traveled by the particle for $0 \leq t \leq 4$ seconds.\\
(c) Find the time $t$, $0 \leq t \leq 4$, when the line tangent to the path of the particle is horizontal. Is the direction of motion of the particle toward the left or toward the right at that time? Give a reason for your answer.\\
(d) There is a point with $x$-coordinate 5 through which the particle passes twice. Find each of the following.\\
(i) The two values of $t$ when that occurs\\
(ii) The slopes of the lines tangent to the particle's path at that point\\
(iii) The $y$-coordinate of that point, given $y(2) = 3 + \frac{1}{e}$