7.4 Relative Rates of Growth

Transcription

1 7.4 Relative Rates of Growth

2 Domination This section is about comparing functions to see which dominate as x. Definition Let f (x) and g(x) be positive for some sufficiently large x. 1 f grows faster than g as x if f (x) x g(x) =

3 Domination This section is about comparing functions to see which dominate as x. Definition Let f (x) and g(x) be positive for some sufficiently large x. 1 f grows faster than g as x if f (x) x g(x) = 2 f grows slower than g as x if f (x) x g(x) = 0

4 Domination This section is about comparing functions to see which dominate as x. Definition Let f (x) and g(x) be positive for some sufficiently large x. 1 f grows faster than g as x if f (x) x g(x) = 2 f grows slower than g as x if f (x) x g(x) = 0 3 f grows at the same rate as g as x if f (x) x g(x) = L

5 Which grows faster: e x or x?

6 Which grows faster: e x or x? Hint: Use L Hopital s Rule

7 Which grows faster: e x or x? Hint: Use L Hopital s Rule e x grows faster than x

8 Which grows faster: e x or x 20?

9 Which grows faster: e x or x 20? e x grows faster than x 20

10 Which grows faster: e x or x 20? e x grows faster than x 20 Can we generalize this?

11 Which grows faster: 2 x or 4 x?

12 Which grows faster: 2 x or 4 x? 2 x grows slower than 4 x

13 Which grows faster: 2 x or 4 x? 2 x grows slower than 4 x Can we generalize this?

14 Which grows faster: ln(x) or x?

15 Which grows faster: ln(x) or x? ln(x) grows slower than x

16 Which grows faster: log 3 x or log 2 x?

17 Which grows faster: log 3 x or log 2 x? log 3 (x) grows at the same rate as log 2 (x)

18 Which grows faster: log 3 x or log 2 x? log 3 (x) grows at the same rate as log 2 (x) log 3 (x) = log(x) log(3)

19 Which grows faster: log 3 x or log 2 x? log 3 (x) grows at the same rate as log 2 (x) log 3 (x) = log(x) log(3) log 2 (x) = log(x) log(2)

20 Which grows faster: log 3 x or log 2 x? log 3 (x) grows at the same rate as log 2 (x) x log 3 (x) = log(x) log(3) log 2 (x) = log(x) log(2) log(x) log(3) log(x) log(2) =

21 Which grows faster: log 3 x or log 2 x? log 3 (x) grows at the same rate as log 2 (x) x log 3 (x) = log(x) log(3) log 2 (x) = log(x) log(2) log(x) log(3) log(x) log(2) = log(2) log(3)

22 Comparisons Sometimes it is easier to show functions grow at the same rate by comparing them to a common function.

23 Comparisons Sometimes it is easier to show functions grow at the same rate by comparing them to a common function. Consider x vs. 2x Differentiating would be a pain, so we wouldn t want to use L Hopital s Rule, but we could use something simple, like x. x x2 + 1 = x

24 Comparisons Sometimes it is easier to show functions grow at the same rate by comparing them to a common function. Consider x vs. 2x Differentiating would be a pain, so we wouldn t want to use L Hopital s Rule, but we could use something simple, like x. x x2 + 1 x = x x 2 x x 2 = 1

25 Comparisons Sometimes it is easier to show functions grow at the same rate by comparing them to a common function. Consider x vs. 2x Differentiating would be a pain, so we wouldn t want to use L Hopital s Rule, but we could use something simple, like x. x x2 + 1 x = x x 2 x x 2 = 1 x 3 2x3 + 1 = x

26 Comparisons Sometimes it is easier to show functions grow at the same rate by comparing them to a common function. Consider x vs. 2x Differentiating would be a pain, so we wouldn t want to use L Hopital s Rule, but we could use something simple, like x. x x2 + 1 x = x x 2 x x 2 = 1 x 3 2x3 + 1 x = x 3 2x 3 x x 3 = 3 2

27 Comparisons Sometimes it is easier to show functions grow at the same rate by comparing them to a common function. Consider x vs. 2x Differentiating would be a pain, so we wouldn t want to use L Hopital s Rule, but we could use something simple, like x. x x2 + 1 x = x x 2 x x 2 = 1 x 3 2x3 + 1 x 3 2x 3 = x x x 3 = 3 2 Since they both grow at the same rate as the same function, we conclude that x and 3 2x grow at the same rate.

28 Notation Definition A function f is of smaller order than g as x if We indicate this by writing f (x) x g(x) = 0 f = o(g) which is read f is little oh of g.

29 Notation Definition A function f is of smaller order than g as x if We indicate this by writing f (x) x g(x) = 0 f = o(g) which is read f is little oh of g. x = o(e x )

30 Notation Definition Let f (x) and g(x) be positive for sufficiently large x. Then f is at most the order of g as x if there is a positive integer M for which f (x) g(x) M for x sufficiently large, We indicate this by writing which is read f is big oh of g. f = O(g)

31 Notation Definition Let f (x) and g(x) be positive for sufficiently large x. Then f is at most the order of g as x if there is a positive integer M for which f (x) g(x) M for x sufficiently large, We indicate this by writing which is read f is big oh of g. f = O(g) x = O(e x ) because x e x 0 as x, i.e. we can select an M.