Relative Rates of Growth The function grows very

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 gonethe less than half At 64 inches, y-value -way the edge boardof the wouldacross be at the horizontally, and already the known universe! y-value would reach the (13 billion light-years) Andromeda Galaxy!

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

Grows faster than any polynomial function. ln x Grows slower than any polynomial function.

WARNING Careful about relying on your intuition.

According to this definition, than. does not grow faster 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”!

“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.

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

Show that same rate as grows faster than grows at the

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 Saying slower than g. and say “f is little-oh of g. ” is another way to say that f grows

Order and Oh-Notation 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 for x sufficiently large We write and say “f is big-oh of g. ” Saying is another way to say that f grows no faster than g. p