The displacement (in feet) of a particle moving in a straight line is given by $ s = \frac{1}{2}t^2 - 6t + 23 $, where $ t $ is measured in seconds.

(a) Find the average velocity over each time interval:

(i) $ [4, 8] $ (ii) $ [6, 8] $

(iii) $ [8, 10] $ (iv) $ [8, 12] $

(b) Find the instantaneous velocity when $ t = 8 $.

(c) Draw the graph of $ s $ as a function of $ t $ and draw the secant lines whose slopes are the average velocities in part (a). Then draw the tangent line whose slope is the instantaneous velocity in part (b).