8.3 Relative Rates of Growth. Greg Kelly, Hanford High School, Richland, Washington. At 64 inches, the y-value would be at the edge of the known universe! (13 billion light-years). The function grows very fast. We could graph it on the chalkboard: .
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Greg Kelly, Hanford High School, Richland, Washington
(13 billion light-years)
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!
The function increases everywhere, even though it increases extremely slowly.
The function grows very slowly.
If we graph it on the chalkboard it looks like this:
By the time we reach the edge of the universe again (13 billion light-years) the chalk line will only have reached 64 inches!
We would have to move 2.6 miles to the right before the line moves a foot above the x-axis!
1. fgrows faster than g (and ggrows slower than f )
2. f and ggrow at the same rate as if
Let f (x) and g(x) be positive for x sufficiently large.
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 ?
We can confirm this graphically:
This is indeterminate, so we apply L’Hôpital’s rule.
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 and grow at the same rate as .
f and g grow at the same rate.
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
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 thang.
Order and Oh-Notation