Evaluate the following:
(i) Limit floor(abs(n))/abs(floor(n))
abs(n)→ ∞
503
(ii) ∫abs(floor(x)) dx
2012
(i) 1
On the +inf side, this is obvious. On the negative n side, when n is integral the value itself is 1 but for nonintegral values the numerator and denominator differ by 1 and as they get larger, this difference matters less and less and the ratio approaches 1.
(ii) Since the absolute value is used, all areas between the "curve" (i.e., the step function) and the xaxis count as positive. The leftmost rectangle is 1x2012 and the one immediately to the left of the origin is 1x1. So far we have area = Sigma{i=1 to 2012}(i).
On the positive side, the first rectangle has height zero and the last has height 502, so we now add Sigma{i=0 to 502}(i).
Sigma{i=1 to 2012}(i) + Sigma{i=0 to 502}(i) = 2025078 + 126253 = 2151331.

Posted by Charlie
on 20120817 14:36:08 