Transcript Efficiency of Algorithms - H-SC

Real-Valued Functions
of a Real Variable and
Their Graphs
Lecture 43
Section 9.1
Wed, Apr 18, 2007
Functions

We will consider real-valued functions that
are of interest in studying the efficiency of
algorithms.
Power functions
 Logarithmic functions
 Exponential functions

Power Functions
A power function is a function of the form
f(x) = xa
for some real number a.
 We are interested in power functions where
a  0.

The Constant Function f(x) =
1
2
1.5
1
0.5
2
4
6
8
10
The Linear Function f(x) = x
10
8
6
4
2
2
4
6
8
10
The Quadratic Function f(x)
= x2
100
80
60
40
20
2
4
6
8
10
The Cubic Function f(x) = x3
600
500
400
300
200
100
2
4
6
8
10
Power Functions xa, a  1

The higher the power of x, the faster the
function grows.
 xa
grows faster than xb if a > b.
The Square-Root Function
3
2.5
2
1.5
1
0.5
2
4
6
8
10
The Cube-Root Function
2
1.5
1
0.5
2
4
6
8
10
The Fourth-Root Function
1.75
1.5
1.25
1
0.75
0.5
0.25
2
4
6
8
10
Power Functions xa, 0 < a <
1

The lower the power of x (i.e., the higher
the root), the more slowly the function
grows.
 xa

grows more slowly than xb if a < b.
This is the same rule as before, stated in
the inverse.
Power Functions
x3
4
x2
3
x
2
1
x
0.5
1
1.5
2
Multiples of Functions
15
x2
12.5
10
3x
2x
7.5
x
5
2.5
1
2
3
4
Multiples of Functions
Notice that x2 eventually exceeds any
constant multiple of x.
 Generally, if f(x) grows faster than cg(x), for
any real number c, then f(x) grows
“significantly” faster than g(x).
 In other words, we think of g(x) and cg(x)
as growing at “about the same rate.”

Logarithmic Functions
A logarithmic function is a function of the
form
f(x) = logb x
where b > 1.
 The function logb x may be computed as
(ln x)/(ln b).

The Logarithmic Function
f(x) = log2 x
6
4
2
10
-2
20
30
40
50
60
Growth of the Logarithmic
Function

The logarithmic functions grow more and
more slowly as x gets larger and larger.
f(x) = log2 x vs. g(x) = x1/n
x1/2
log2 x
4
x1/3
2
5
-2
10
15
20
25
30
Logarithmic Functions vs.
Power Functions

The logarithmic functions grow more slowly
than any power function xa, 0 < a < 1.
f(x) = x vs. g(x) = x log2 x
x log2 x
4
3
x
2
1
0.5
1
1.5
2
2.5
3
f(x) vs. f(x) log2 x
The growth rate of log x is between the
growth rates of 1 and x.
 Therefore, the growth rate of f(x) log x is
between the growth rates of f(x) and x f(x).

f(x) vs. f(x) log2 x
x2 log2 x
50
x2
40
x log2 x
30
20
x
10
2
4
6
8
Multiplication of Functions
If f(x) grows faster than g(x), then f(x)h(x)
grows faster than g(x)h(x), for all positivevalued functions h(x).
 If f(x) grows faster than g(x), and g(x)
grows faster than h(x), then f(x) grows
faster than h(x).

Exponential Functions
An exponential function is a function of the
form
f(x) = ax,
where a > 0.
 We are interested in power functions where
a  1.

The Exponential Function
f(x) = 2x
15
12.5
10
7.5
5
2.5
1
2
3
4
The Exponential Function
f(x) = 2x
4x
80
3x
60
40
2x
20
1
2
3
4
Growth of the Exponential
Function

The larger the base, the faster the function
grows

ax grows faster then bx, if a > b > 1.
f(x) = 2x vs. Power Functions
(Small Values of x)
5
4
3
2x
2
1
0.5
1
1.5
2
f(x) = 2x vs. Power Functions
(Large Values of x)
3500
2x
3000
2500
x3
2000
1500
1000
500
5
10
15
20
Growth of the Exponential
Function

Every exponential function grows faster
than every power function.

ax grows faster than xb, for all a > 1, b > 0.
Rates of Growth of
Functions

The first derivative of a function gives its
rate of change, or rate of growth.
Rates of Growth of Power
Functions
 is increasing, if a  1,
d a
a 1 
x  ax  is constant,if a  1,
dx

is decreasing, if 0  a  1.
 
Rates of Growth of
Logarithmic Functions
d
1
log b x  
is decreasing , if b  1.
dx
x ln b
Rates of Growth of
Exponential Functions
 
d x
a  a x ln a
dx