8.3 Relative Rates of Growth

The function grows very fast. We could graph it on the chalkboard: If x is 3 inches, y is about 20 inches: We have gone less than half-way across the board horizontally, and already the y-value would reach the Andromeda Galaxy! At 64 inches, the y-value would be at the edge of the known universe! (13 billion light-years)

The function grows very slowly. If we graph it on the chalkboard it looks like this: We would have to move 2.6 miles to the right before the line moves a foot above the x-axis! 64 inches By the time we reach the edge of the universe again (13 billion light-years) the chalk line will only have reached 64 inches! The function increases everywhere, even though it increases extremely slowly.

Definitions: Faster, Slower, Same-rate Growth as Let f ( x ) and g ( x ) be positive for x sufficiently large. 1. f grows faster than g (and g grows slower than f ) as if or 2. f and g grow at the same rate as if

WARNING Please temporarily suspend your common sense.

According to this definition, does not grow faster than. Since this is a finite non-zero limit, the functions grow at the same rate! The book says that “ f grows faster than g ” means that for large x values, g is negligible compared to f. “Grows faster” is not the same as “has a steeper slope”!

Which grows faster, or ? This is indeterminate, so we apply L’Hôpital’s rule. Still indeterminate. grows faster than. We can confirm this graphically:

“Growing at the same rate” is transitive. In other words, if two functions grow at the same rate as a third function, then the first two functions grow at the same rate.

Example 4: Show that and grow at the same rate as. f and g grow at the same rate.

Order and Oh-Notation Definition f of Smaller Order than g Let f and g be positive for x sufficiently large. Then f is of smaller order than g as if We write and say “ f is little-oh of g.” Saying is another way to say that f grows slower than g.

Order and Oh-Notation Saying is another way to say that f grows no faster than g. Definition f of at Most the Order of g Let f and g be positive for x sufficiently large. Then f is of at most the order of g as if there is a positive integer M for which We write and say “ f is big-oh of g.” for x sufficiently large 

Binary Search: When searching a sorted list, the idea is to look at the element in the middle. If the key is equal to that, the search is finished. If the key is less than the middle element, do a binary search on the first half. If it's greater, do a binary search of the second half. Sequential Search: is a method for finding a particular value in a list, that consists of checking every one of its elements, one at a time and in sequence, until the desired one is found.

Webster’s dictionary contains ≈ 26,000 words that begin with the letter “a”. Using the binary search method what would be the maximum number of searches to find a match for a word beginning with the letter “a”? Using the sequential search method, what would be the maximum number of searches to find a match for a word beginning with the letter “a”? 26,000