An object moving along a curve in the $xy$-plane is at position $(x(t), y(t))$ at time $t$, where $$\frac{dx}{dt} = \sin^{-1}\left(1 - 2e^{-t}\right) \text{ and } \frac{dy}{dt} = \frac{4t}{1 + t^{3}}$$ for $t \geq 0$. At time $t = 2$, the object is at the point $(6, -3)$. (Note: $\sin^{-1} x = \arcsin x$) (a) Find the acceleration vector and the speed of the object at time $t = 2$. (b) The curve has a vertical tangent line at one point. At what time $t$ is the object at this point? (c) Let $m(t)$ denote the slope of the line tangent to the curve at the point $(x(t), y(t))$. Write an expression for $m(t)$ in terms of $t$ and use it to evaluate $\lim_{t \rightarrow \infty} m(t)$. (d) The graph of the curve has a horizontal asymptote $y = c$. Write, but do not evaluate, an expression involving an improper integral that represents this value $c$.
An object moving along a curve in the $xy$-plane is at position $(x(t), y(t))$ at time $t$, where
$$\frac{dx}{dt} = \sin^{-1}\left(1 - 2e^{-t}\right) \text{ and } \frac{dy}{dt} = \frac{4t}{1 + t^{3}}$$
for $t \geq 0$. At time $t = 2$, the object is at the point $(6, -3)$. (Note: $\sin^{-1} x = \arcsin x$)
(a) Find the acceleration vector and the speed of the object at time $t = 2$.
(b) The curve has a vertical tangent line at one point. At what time $t$ is the object at this point?
(c) Let $m(t)$ denote the slope of the line tangent to the curve at the point $(x(t), y(t))$. Write an expression for $m(t)$ in terms of $t$ and use it to evaluate $\lim_{t \rightarrow \infty} m(t)$.
(d) The graph of the curve has a horizontal asymptote $y = c$. Write, but do not evaluate, an expression involving an improper integral that represents this value $c$.