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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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8.3 Relative Rates of Growth


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8.3 Relative Rates of Growth

Greg Kelly, Hanford High School, Richland, Washington

slide2

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:

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!

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The function increases everywhere, even though it increases extremely slowly.

64 inches

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!

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Definitions: Faster, Slower, Same-rate Growth as

1. fgrows faster than g (and ggrows slower than f )

as if

2. f and ggrow at the same rate as if

Let f (x) and g(x) be positive for x sufficiently large.

or

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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”!

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grows faster than .

Which grows faster, or ?

We can confirm this graphically:

This is indeterminate, so we apply L’Hôpital’s rule.

Still indeterminate.

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“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.

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Example 4:

Show that and grow at the same rate as .

f and g grow at the same rate.

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Definitionf 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

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Definitionf 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 thang.

Order and Oh-Notation

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