Problems are listed in approximate order of difficulty. A single dot (•) indicates straigh...

Problems are listed in approximate order of difficulty. A single dot (•) indicates straightforward problems involving just one main concept and sometimes requiring no more than substitution of numbers in the appropriate formula. Two dots (••) identify problems that are slightly more challenging and usually involve more than one concept. Three dots (•••) indicate problems that are distinctly more challenging, either because they are intrinsically difficult or involve lengthy calculations. Needless to say, these distinctions are hard to draw and are only approximate.

•• (Section 15.4) For very large numbers, the factorial of N can be estimated by Stirling’s approximation , which can also be written as . Often, to simplify calculations, a further approximation is made and the formula is written as ln (N!) N ≈ N ln − N. (a) Plot ln(N!) versus N for , and on this same plot, graph the long version of Stirling’s an proximation. (b) Make a graph showing how the long and short versions of Stirling’s approximation compare over the range of 10 to 100. Consider carefully what plot will show this comparison to best effect.