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Gavin's Calc II Class Clarification: Sep 17

Question: How do we decide what functions to compare things with?
Excellent question. The first thing is to get a 'library' of integrals that we know converge or don't. The examples that we did in class are a good place to start with for this: we did integrals of 1/x2, 1/x, and e-x (all for integrals with limits going to infinity). The first and last of these converge, while the middle one doesn't. So the first thing we might remember is
  1. integrals of 1/xp from some number (greater than zero) to infinity converge if p>1.
  2. integrals of decaying exponentials (e-x, 2-x, etc.) from some number to infinity converge.
Thus, these functions then become good candidates for comparisons. Actually figuring out which functions are bigger or smaller than which of these isn't necessarily easy. A good place to start is by graphing them or plugging in some points. Also remember that logarithms (ln(x), for example) grow amazingly slowly.

Gavin's Calc II Clarification 990917
Last Modified: Sat Sep 18 13:18:42 CDT 1999
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