Physical interpretation of the derivative
A radar detects the position of a ship at every moment therefore one can know the trajectory of the ship, which turns out to be: $x(t)=\sin (2t) +t$
Answer the following:
a) Find the average speed and the distance where the ship is after the first hour of trajectory ($1h=3600s$)
b) Find the instantaneous speed when $t=10s$ and $t=100s$
a) To compute the covered distance we are going to see the initial and final positions.
$$x(0)=0m; \ x(3600s)=3600m$$
$$\Delta x=3600m$$
We find the average speed in this interval: $$v_m=\dfrac{\Delta x}{\Delta t}=\dfrac{3600m}{3600s}=1m/s$$
b) We can compute the generic instantaneous speed and then we will substitute this for the required moments.
$$v(t)=x'(t)=2 \cos(2t)+1$$
Therefore,
$$v(10s)=2,88 \ m/s$$
$$v(100s)=-0,88 \ m/s$$
a) $v_m=1m/s$
$$\Delta x=3600m$$
b) $v(10s)=2,88 \ m/s$
$$v(100s)=-0,88 \ m/s$$