I attached the missing part of the question.
Part A
Angular velocity is simply the rate at which angle changes, in other words, it is the first derivative of the angle function with respect to time:
[tex]\theta (t)=\gamma t+\beta t^3\\ w(t)=\frac{d\theta(t)}{dt}\\ w(t)=\gamma +3\beta t^2[/tex]
Part B
Initial value is the value at t=0. To find initial value we simply plug t=0 into the equation.
[tex]w(0)=\gamma +3\beta\cdot 0^2\\
w(0)=\gamma\\
w_0=\gamma= 0.422\frac{rad}{s}[/tex]
Part C
To find these values we simply need to plug in t=5 in the equation.
[tex]w(5)=\gamma +3\beta\cdot 5^2\\
w(5)=\gamma +75\cdot \beta=0.422+75 \cdot 1.35\cdot10^{-2}=1.43\frac{rad}{s}[/tex]
Part D
Average angular velocity is total angular displacement divided by time:
[tex]w_{av}=\frac{\Delta \theta}{\Delta t}[/tex]
[tex]\Delta \theta=\theta(5)-\theta(0)=\gamma\cdot5+\beta 5^3\\
\Delta \theta=0.422\cdot5+125\cdot1.35\cdot10^{-2}=3.7975 $rad[/tex]
The average angular velocity is:
[tex]w_{av}=\frac{\Delta \theta}{\Delta t}=\frac{3.7975 }{5}=0.7595\frac{rad}{s}[/tex]
