1. Relative Rates of Growth

2. 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)

3. The function grows very slowly.
64 inches
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!
The function increases everywhere, even though it increases extremely slowly.

4. 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

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

6. WARNING
WARNING
WARNING
Careful about relying on your intuition.

7. 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”!

8. “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.

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

10. Show that grows faster than
Show that grows at the
same rate as

11. 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.

12. Order and Oh-Notation p 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
Saying is another way to say that f grows no faster than g. p