Chapter 1.6 Cyclic Models 1
Chapter 1.6
Chapter 1.6 Cyclic Models 2
From the time of the ancient Sumerians angles have been measured in units of
degrees, in which 360 comprise a full circle. This relationship is an artificial construct
based upon the Mesopotamian base 60 number system.
A different angle measurement is based upon the familiar fact that the circumference of a circle is 2p times its radius. That is, 2p radii can be wrapped around the
circumference to complete a circle. An angle measurement of 360 degrees, therefore, is equivalent to 2p radians. One radian is
Chapter 1.6 Cyclic Models 3
one radian
arc length = 1 radius
2p radians = 360 1 radian 57
Chapter 1.6 Cyclic Models 4
unit circle (radius = 1) x
Consider a circle with a radius of one (unit circle). A ray emanating from the origin will intersect the unit circle in a point, and form an angle, x, with the horizontal axis.
Chapter 1.6 Cyclic Models 5
The coordinates of the point of intersection of ray with circle are the lengths of the
dotted horizontal and vertical line segments shown in the figure.
unit circle (radius = 1) x
Chapter 1.6 Cyclic Models 6
The length of the horizontal leg is defined
as the cosine of angle x, represented by
cos x.
The length of the vertical leg is defined as the sine of angle x, represented by sin x.
cosine
sine unit circle (radius = 1)
Chapter 1.6 Cyclic Models 7 A graph of the sine function follows. The
units along the input axis are in multiples of p, ranging between -2 p and 2 p.
Chapter 1.6 Cyclic Models 8 The sine function deviates from the horizontal axis by 1. We say the sine function has an amplitude of 1, and an average value of 0.
Chapter 1.6 Cyclic Models 9 The sine function
has a maximum output value of 1.
The sine function has a minimum
Chapter 1.6 Cyclic Models 10 The output values of the sine function are the same after every 2p (~6.3) units. We say the sine function has a period of 2p.
Chapter 1.6 Cyclic Models 11 In many applications, output values tend to display periodic behavior. However, the period, amplitude, average value, and
maximum and minimum output values of this behavior will likely not be the same as that for the sine function.
How might we use a sine function to model these data?
We will do so by adding parameters to the sine function.
Chapter 1.6 Cyclic Models 12 The sine model has the following
form:
�
(
�
)
=
� ���
(
��
+
�
)
+
�
where a, b, c, and d are the parameter constants, a, b > 0
The parameters:
• modify the amplitude and period, and • horizontally & vertically translate the
Chapter 1.6 Cyclic Models 13
�
(
�
)
=
� ���
(
��
+
�
)
+
�
• a is the amplitude • the period is
• is the horizontal shift,
left, if c > 0; right, if c < 0 • d is the vertical shift
up, if d > 0; down, if d < 0 d is also the average value
Chapter 1.6 Cyclic Models 14
month rad month rad
1 42 7 165
2 61 8 150
3 95 9 113
4 129 10 73
5 150 11 44
6 172 12 30
Example: Monthly UV-B radiation in Southern CA. Radiation in units of kJ/m2. Input aligned to Jan = 1.
Chapter 1.6 Cyclic Models 16 Problem:
It would seem reasonable that UV data should exhibit periodic
behavior, as it does.
Use the TI-84 to fit a sinusoidal model to the UV data.
01p110
Chapter 1.6 Cyclic Models 17
� (�) ≅70.145 ���(.514 �− 1.640)+100.630
Chapter 1.6 Cyclic Models 18 What is the amplitude of this model?
What is the period of the model? What is the average output value? What is the maximum output value? What is the minimum output value?
Chapter 1.6 Cyclic Models 19 amplitude 70.145
�
(
�
)
≅
70.145
���
(
.514
�
−
1.640
)
+
100.630
�
(
�
)
=
� ���
(
��
+
�
)
+
�
The amplitude is the leading coefficient, a, 70.145 kJ/m2.
It is the deviation of the max and min from the average output value.
Chapter 1.6 Cyclic Models 20 period 12.224
�
(
�
)
≅
70.145
���
(
.514
�
−
1.640
)
+
100.630
�
(
�
)
=
� ���
(
��
+
�
)
+
�
The period is . The model value for b is
~.514. So the period is months.
Chapter 1.6 Cyclic Models 21 average output 100.630
�
(
�
)
≅
70.145
���
(
.514
�
−
1.640
)
+
100.630
�
(
�
)
=
� ���
(
��
+
�
)
+
�
The average output
value is the value of d, 100.630 kJ/m2.
Chapter 1.6 Cyclic Models 22 average output 100.6
�
(
�
)
≅
70.145
���
(
.514
�
−
1.640
)
+
100.630
�
(
�
)
=
� ���
(
��
+
�
)
+
�
The maximum output is the average plus the
amplitude, that is, d + a. d + a 100.630 + 70.145 Maximum output
170.775 kJ/m2.
Chapter 1.6 Cyclic Models 23 average output 100.6
�
(
�
)
≅
70.145
���
(
.514
�
−
1.640
)
+
100.630
�
(
�
)
=
� ���
(
��
+
�
)
+
�
The minimum output is the average minus the amplitude, that is, d - a. d - a 100.630 - 70.145 Minimum output
30.485 kJ/m2.
Chapter 1.6 Cyclic Models 24
� (�) ≅70.145 ���(.514 �− 1.640)+100.630
average value 100.6 amplitude 70.145
period 12.224 max output 170.775
Chapter 1.6 Cyclic Models 25
Summary of Models
� (�)=� �+�
Linear
• a = slope, b = vertical axis intercept • a > 0, increasing function,
• a < 0, decreasing function,
lim � − � (�) =− lim � � ( �)= lim � − � (�) = lim � � ( �)=−
Exponential , a, b > 0:
• f(x) > 0 for all input values
• f(0) = a (vertical axis intercept)
• for inputs which differ by a unit, the ratio of their outputs is b, that is, b is a constant multiplier
• concave up with no inflection points • b > 1: increasing
• 0 < b < 1: decreasing
Chapter 1.6 Cyclic Models 26
, x > 0
Logarithmic
• b > 0: increasing function, concave down • b < 0: decreasing function, concave up • no inflection point
Logistic
• L > 0 and is the limiting max value • B > 0, function always increasing • B < 0, function always decreasing • one inflection point
lim
�
� (�)=
lim
� 0
� (�)=−
lim
� 0 � ( �)=
lim
�
� (�)=−
� (�)= �
1+ � ��� (−�� )
0< � ( � )<�
lim � + � (�)=� lim � −
� (� ) =0
lim
� +
� ( �)=0
lim
� −
� (�) =�
Chapter 1.6 Cyclic Models 27
Quadratic
• a > 0: concave up parabola with a minimum value • a < 0: concave down parabola with a maximum value • no inflection point
Cubic • a > 0: ,
• a < 0: ,
• one inflection point
• not appropriate if data does not have inflection point
� (�)=� �2+��+�
lim � ± � (�) = lim � ± � (�) =−
Chapter 1.6 Cyclic Models 28
Cyclic
• a is the amplitude • the period is
• d is the average value
• d + a is the maximum value • d - a is the minimum value
