Advanced Placement CalculusSequences, L’Hôpital’s Rule, and Improper Integrals

Chapter 9 Section 3Relative Rates of Growth

Essential Question: How are little-oh and big-oh notation used to discuss the rates of growth of functions?

Objectives: The student will be able to use little-oh and big-oh notation in determining, investigating, and comparing the rates of growth of functions.

Terms:

Big-oh - Big O notation

Binary search

Little-oh notation

Rates of Growth

Sequential Search

Theorems:

- � -1

Definition: 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 x →∞ if

limx→∞

f x( )g x( ) = ∞   or, equivalently, if       lim

x→∞

g x( )f x( ) = ∞

2. f and g grow at the same rate as x→∞ if                               

limx→∞

f x( )g x( ) = L ≠ 0,   where L  is  finite and  not  zero 

Graphing Calculator Skills:

None

- � -2

Transitivity of Growing Rates                                If f grows at the same rate as g as x →∞ and g grows at the same rate as h as as x →∞,  then f grows at the same rate as h as x →∞.

Definition: f of Smaller Order than g       Let f  and g be positive for x sufficiently large. Then f is smaller order than g as x →∞ if                

                                       limx→∞

f x( )g x( ) = 0

We write f = o g( )and we say "f  is little-oh of g."                          

Definition: f of at Most Order of g       Let f  and g be positive for x sufficiently large. Then f is of at most the order of g as x →∞ if there is a positive integer M for which                

                                       limx→∞

f x( )g x( ) ≤ M

for sufficiently large. We write f =O g( )and we say "f  is big-oh of g."                          

To find an item in a list of length n,     1.  A sequential list takes O n( )  steps;

     2.  A binary search takes O log2 n( )  steps;          

Sample Questions:

1. Determine whether which function grows faster as � .

�

2. Determine the order of the function from slowest growing to fastest growing as � .

�

3. Write an expression using two functions listed above that correctly uses little-oh notation. Write an expression using two functions listed above that correctly uses big-oh notation.

Homework: Pages 461 - 462 Exercises: 1, 5, 11, 13, 17, 21, 25, 29, 33, 35, 43, 45, 47, 49, and 51

Exercises: 2, 6, 12, 14, 16, 24, 26, 30, 32, 36, 38, 44, 46, 48, and 50

x→∞

ln x     or       1x

x→∞

3x        x3        log3 x       ex3

- � -3

SOLUTIONS TO SAMPLE QUESTIONS

1. Determine whether which function grows faster as � .x→∞

- � -4

L = limx→∞

ln x1x

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

L = limx→∞

x ln x( )L = ∞ ⋅ ln ∞( )( )

L = ∞⋅∞L = ∞

∴ The function of ln x grows

faster than the function 1x

.

O ln x( ) = 1n

ln x     or       1x

2. Determine the order of the function from slowest growing to fastest growing as � .x→∞

- � -5

3x        x3        log3 x       ex3

L = limx→∞

3x

x3⎛⎝⎜

⎞⎠⎟

L = limx→∞

3x ⋅ ln 3⋅13x2

⎛⎝⎜

⎞⎠⎟

L = limx→∞

3x ⋅ ln 33x2

⎛⎝⎜

⎞⎠⎟

L = limx→∞

3x ⋅ ln 3⋅1⋅ ln 36x

⎛⎝⎜

⎞⎠⎟

L = limx→∞

3x ⋅ ln 3( )26x

⎛

⎝⎜⎞

⎠⎟

L = limx→∞

3x ⋅ ln 3⋅1⋅ ln 3( )26

⎛

⎝⎜⎞

⎠⎟

L = limx→∞

3x ln 3( )36

⎛

⎝⎜⎞

⎠⎟

L =3 ∞( ) ln 3( )3

6

L =3 ∞( ) 1.0986( )3

6

L = ∞⋅1.32606

L = ∞6

L = ∞

∴ The function 3x grows faster than the function x3.

L = limx→∞

x3

log3 x⎛⎝⎜

⎞⎠⎟

L = limx→∞

x3

ln xln 3

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

L = limx→∞

x3 ⋅ ln 3ln x

⎛⎝⎜

⎞⎠⎟

L = limx→∞

3x2 ⋅ ln 31x

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

L = limx→∞

3x3 ⋅ ln 31

⎛⎝⎜

⎞⎠⎟

L = limx→∞

3x3 ⋅ ln 3( )L = 3 ∞( )3 ⋅ ln 3L = 3 ∞( ) ⋅ 1.0986( )L = ∞

∴ The function x3 grows faster than the function log3x.

L = limx→∞

3x

log3 x⎛⎝⎜

⎞⎠⎟

L = limx→∞

3x

ln xln 3

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

L = limx→∞

3x ⋅ ln 3ln x

⎛⎝⎜

⎞⎠⎟

L = limx→∞

3x ⋅ ln 3⋅1⋅ ln 31x

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

L = limx→∞

3x ⋅ ln 3( )21x

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

L = limx→∞

3x ⋅ x ⋅ ln 3( )21

⎛

⎝⎜⎞

⎠⎟

L = limx→∞

3x ⋅ x ⋅ ln 3( )2( )L = 3 ∞( ) ⋅ ∞( ) ⋅ ln 3( )2

L = 3 ∞( ) ⋅ ∞( ) ⋅ 1.0986( )2

L = ∞( ) ⋅ ∞( ) ⋅ 1.2069( )L = ∞

∴ The function 3x grows faster than the function log3x.

∴SO FAR........log 3x < x3 < 3x

- � -6

L = limx→∞

e x3

3x⎛

⎝⎜⎞

⎠⎟

ln L( ) = ln limx→∞

ex13

3x⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

ln L( ) = limx→∞

ln ex13

3x⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

ln L( ) = limx→∞

ln ex13⎛

⎝⎜⎞⎠⎟− ln 3x( )⎛

⎝⎜⎞⎠⎟

ln L( ) = limx→∞

x13 ln e( )− x ln 3( )⎛

⎝⎜⎞⎠⎟

ln L( ) = limx→∞

x13 1( )− x ln 3( )⎛

⎝⎜⎞⎠⎟

ln L( ) = limx→∞

x13 1− x

23 ln 3( )⎛

⎝⎜⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

ln L( ) = ∞( )13 1− ∞( )

23 ln 3( )⎛

⎝⎜⎞⎠⎟

ln L( ) = ∞( ) 1− ∞( ) 1.0986( )( )ln L( ) = ∞( ) 1− ∞( )ln L( ) = ∞( ) −∞( )ln L( ) = −∞

eln L( ) = e−∞

L = 0∴ The function 3x grows faster

than the function e x3.

L = limx→∞

e x3

log 3x⎛

⎝⎜⎞

⎠⎟

L = limx→∞

e x3

ln xln 3

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

L = limx→∞

ex13 ⋅ ln 3ln x

⎛

⎝⎜⎜

⎞

⎠⎟⎟

L = limx→∞

13x−23 ⋅ex

13 ⋅ ln 3

1x

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

L = limx→∞

13x13 ⋅ex

13 ⋅ ln 3

1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

L = limx→∞

13x13 ⋅ex

13 ⋅ ln 3

⎛⎝⎜

⎞⎠⎟

L = 13∞( )

13 ⋅ex

∞( )⋅ ln 3

L = 13∞( ) ⋅e ∞( ) ⋅ 1.0986( )

L = 13∞( ) ⋅ ∞( ) ⋅ 1.0986( )

L = ∞

∴ The function e x3 grows faster

than the function log3 x.

- � -7

∴The order from slowest growing to fastest

growing is: log3 x, x3, e x3 , and 3x .

L = limx→∞

e x3

x3⎛

⎝⎜⎞

⎠⎟

L = limx→∞

ex13

x3⎛

⎝⎜⎜

⎞

⎠⎟⎟

L = limx→∞

13x−23ex

13

3x2

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

L = limx→∞

ex13

9x83

⎛

⎝⎜⎜

⎞

⎠⎟⎟

L = limx→∞

13x−23ex

13ex

13

24x53

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

L = limx→∞

ex13

72x73

⎛

⎝⎜⎜

⎞

⎠⎟⎟

L = limx→∞

13x−23ex

13ex

13

5043x43

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

L = limx→∞

ex13

504x2⎛

⎝⎜⎜

⎞

⎠⎟⎟

L = limx→∞

13x−23ex

13ex

13

1008x

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

L = limx→∞

ex13

3024x43

⎛

⎝⎜⎜

⎞

⎠⎟⎟

L = limx→∞

13x−23ex

13ex

13

4032x13

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

L = limx→∞

ex13

12096x

⎛

⎝⎜⎜

⎞

⎠⎟⎟

L = limx→∞

e x3

x3⎛

⎝⎜⎞

⎠⎟

!continued!

L = limx→∞

ex13

12096x

⎛

⎝⎜⎜

⎞

⎠⎟⎟

L = limx→∞

13x−23ex

13

12096

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

L = limx→∞

ex13

36288x23

⎛

⎝⎜⎜

⎞

⎠⎟⎟

L = limx→∞

13x−23ex

13

326883

x−13

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

L = limx→∞

ex13

36288x13

⎛

⎝⎜⎜

⎞

⎠⎟⎟

L = limx→∞

13x−23ex

13

326883

x−23

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

L = limx→∞

ex13

36288

⎛

⎝⎜⎜

⎞

⎠⎟⎟

L = limx→∞

e x3

x3⎛

⎝⎜⎞

⎠⎟

!continued again!

L = limx→∞

ex13

36288

⎛

⎝⎜⎜

⎞

⎠⎟⎟

L = e ∞( )13

36288

L = e∞

36288

L = ∞36288

L = ∞

∴ The function e x3 grows faster

than the function x3.

3. Write an expression using two functions listed above that correctly uses little-oh notation. Write an expression using two functions listed above that correctly uses big-oh notation.

- � -8

∴The order from slowest growing to fastest

growing is: log3 x, x3, e x3 and 3x .

Any of the following could be used for big-oh notation:

O x3( ) = log 3x O e x3( ) = x3 O 3x( ) = e x3

O e x3( ) = log 3x O 3x( ) = x3

O 3x( ) = log 3x

Any of the following could be used for big-oh notation:

o log 3x( ) = 3x o x3( ) = e x3o e x3( ) = 3x

o log 3x( ) = e x3o x3( ) = 3x

o log 3x( ) = x3