An object moving along a curve in the $xy$-plane has position $(x(t), y(t))$ at time $t$ with $$\frac{dx}{dt} = \cos\left(t^3\right) \text{ and } \frac{dy}{dt} = 3\sin\left(t^2\right)$$ for $0 \leq t \leq 3$. At time $t = 2$, the object is at position $(4,5)$. (a) Write an equation for the line tangent to the curve at $(4,5)$. (b) Find the speed of the object at time $t = 2$. (c) Find the total distance traveled by the object over the time interval $0 \leq t \leq 1$. (d) Find the position of the object at time $t = 3$.
An object moving along a curve in the $xy$-plane has position $(x(t), y(t))$ at time $t$ with
$$\frac{dx}{dt} = \cos\left(t^3\right) \text{ and } \frac{dy}{dt} = 3\sin\left(t^2\right)$$
for $0 \leq t \leq 3$. At time $t = 2$, the object is at position $(4,5)$.
(a) Write an equation for the line tangent to the curve at $(4,5)$.
(b) Find the speed of the object at time $t = 2$.
(c) Find the total distance traveled by the object over the time interval $0 \leq t \leq 1$.
(d) Find the position of the object at time $t = 3$.