A particle moves in the $xy$-plane so that the position of the particle at any time $t$ is given by $$x(t) = 2e^{3t} + e^{-7t} \text{ and } y(t) = 3e^{3t} - e^{-2t}.$$ (a) Find the velocity vector for the particle in terms of $t$, and find the speed of the particle at time $t = 0$. (b) Find $\frac{dy}{dx}$ in terms of $t$, and find $\lim_{t \rightarrow \infty} \frac{dy}{dx}$. (c) Find each value $t$ at which the line tangent to the path of the particle is horizontal, or explain why none exists. (d) Find each value $t$ at which the line tangent to the path of the particle is vertical, or explain why none exists.
A particle moves in the $xy$-plane so that the position of the particle at any time $t$ is given by
$$x(t) = 2e^{3t} + e^{-7t} \text{ and } y(t) = 3e^{3t} - e^{-2t}.$$
(a) Find the velocity vector for the particle in terms of $t$, and find the speed of the particle at time $t = 0$.
(b) Find $\frac{dy}{dx}$ in terms of $t$, and find $\lim_{t \rightarrow \infty} \frac{dy}{dx}$.
(c) Find each value $t$ at which the line tangent to the path of the particle is horizontal, or explain why none exists.
(d) Find each value $t$ at which the line tangent to the path of the particle is vertical, or explain why none exists.