A secant line on a curve is just a straight line that connects two points on the curve. So for (a), where you are given h = 0.5, you are looking for the slope of the line between ( 8, f(8) ) and ( 8.5, f(8.5) ).
Since you have the function f(x) = x\^2 / 2 - 1, finding both f(8) and f(8.5) is just plug and chug. Then, the slope of the line between those two points is just the normal Delta-y / Delta-x business.
The repetitions then involve changing h. You'll notice that "h" is getting smaller and smaller, and what you're doing is taking "aim" at what the slope of the line would be if the second point actually collapsed right back on top of the first. In other words, you're learning how to discover the slope of a line at a SINGLE point by following the trend as the second point gets closer and closer to the first. It's cool stuff.
Also, WebWork is awesome :-)