All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Just Math > Calculus
Floor and Absolute (Posted on 2012-08-17) Difficulty: 3 of 5
Evaluate the following:
(i) Limit floor(abs(n))/abs(floor(n))
   abs(n)→ ∞

     503
(ii) abs(floor(x)) dx
   -2012

See The Solution Submitted by K Sengupta    
Rating: 5.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution solution Comment 2 of 2 |

(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 non-integral 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 x-axis 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 2012-08-17 14:36:08
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (14)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2024 by Animus Pactum Consulting. All rights reserved. Privacy Information