8.3 relative rates of growth. the function grows very fast. we could graph it on the chalkboard: if...
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8.3 Relative Rates of Growth
The function grows very fast.xy e
We could graph it on the chalkboard:
If x is 3 inches, y is about 20 inches:
3,20
At 10 inches,
1 mile
3
x
y
At 44 inches,
2 million light-years
x
y
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.lny xIf 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.
lny x
Definitions: Faster, Slower, Same-rate Growth as x
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 ifx
limx
f x
g x or
lim 0x
g x
f x
2. f and g grow at the same rate as ifx
lim 0x
f xL
g x
WARNINGWARNINGWARNING
Please temporarily suspend your common sense.
According to this definition, does not grow faster than .
2y xy x
2y x
y x
2lim lim 2 2x x
x
x
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 ?xe 2x
2lim
x
x
e
x
This is indeterminate, so we apply L’Hôpital’s rule.
lim2
x
x
e
x still
lim2
x
x
e
Still indeterminate.
grows faster than .xe 2x
approaches
We can confirm this graphically:
2
xey
x
“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 .
2 5f x x 2
2 1g x x
x
Let h x x
2 5limx
x
x
2
2
5limx
x
x
2
2 2
5limx
x
x x 2
5lim 1x x
10
2
2 1limx
x
x
2
2
2 1limx
x
x
2
2 1limx
x
x x
limx
f
g
40
limx
f h
h g
11
4 1
4 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 ifx
lim 0x
f x
g x
We write and say “f is little-oh of g.” of g
Saying is another way to say that f grows slower than g.
of g
Order and Oh-Notation
Saying is another way to say that f grows no faster than g.
Of 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 whichx
f xM
g x
We write and say “f is big-oh of g.” Of g
for x sufficiently large
p