Page 80 -

L=length(p);
v=x.^[(L-1):-1:0];
y=sum(p.*v);

In-Class Exercises
Pb. 3.13 Show that, for the polynomial p defined by Eq. (3.9), the product of
a a
n
the roots is (−1 ) 0 , and the sum of the roots is − n −1 .
a a
n n
Pb. 3.14 Find graphically the real roots of the polynomial p = [1 3 2
1 0 3].

3.5 Examples with the Trigonometric Functions
A time-dependent cosine function of the form:

x = cos(ω t + )φ (3.11)
a
appears often in many applications of electrical engineering: a is called the
amplitude, ω the angular frequency, and φ the phase. Note that we do not
have to have a separate discussion of the sine function because the sine func-
tion, as shown in the Supplement, differs from the cosine function by a con-
stant phase. Therefore, by suitably changing only the value of the phase
parameter, it is possible to transform the sine function into a cosine function.
In the following example, we examine the period of the different powers of
the cosine function; your preparatory task is to predict analytically the rela-
tionship between the periods of the two curves given in Example 3.2 and then
verify your answer numerically.

Example 3.2
3
Plot simultaneously, x (t) = cos (t) and x = cos(t) on t ∈ [0, 6π].
1
2
Solution: To implement this task, edit and execute the following script M-file:
t=0:.2:6*pi; % t-array
a=1;w=1; % desired parameters
x1=a*(cos(w*t))^3; % x1-array constructed

© 2001 by CRC Press LLC