Advanced Placement Calculus Sequences, L’Hôpital’s Rule, and Improper Integrals Chapter 9 Section 3 Relative 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 gx () be positive for x sufficiently large. 1. f grows faster than g (and g grows slower than f) as x →∞ if lim x→∞ f x () gx () = ∞ or, equivalently, if lim x→∞ gx () f x () = ∞ 2. f and g grow at the same rate as x →∞ if lim x→∞ f x () gx () = L ≠ 0, where L is finite and not zero
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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
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