Page 179 - DMTH201_Basic Mathematics-1

Basic Mathematics – I

Notes (b) The Maclaurin series of an odd function includes only odd powers.
(c) The Fourier series of a periodic even function includes only cosine terms.
(d) The Fourier series of a periodic odd function includes only sine terms.

6.2 Rational Function

Rational functions and the properties of their graphs such as domain, vertical and horizontal
asymptotes, x and y intercepts are explored using an applet. The investigation of these functions
is carried out by changing parameters included in the formula of the function. Each parameter
can be changed continuously which allows a better understanding of the properties of the
graphs of these functions.

6.2.1 Definition and Domain of Rational Functions

A rational function is defined as the quotient of two polynomial functions.
f(x) = P(x) / Q(x)
Here are some examples of rational functions:
g(x) = (x + 1) / (x 1)
2
h(x) = (2x + 1) / (x + 3)
The rational functions to explored are of the form
f(x) = (ax + b)/(cx + d)
where a, b, c and d are parameters that may be changed, using sliders, to understand their effects
on the properties of the graphs of rational functions defined above.

6.2.2 Exponential Function
The function e .
t
The exponential function is denoted mathematically by e and in matlab by Exponent t. This
t
function is the solution to the world’s simplest, and perhaps most important, diferential equation,
y_ = ky
This equation is the basis for any mathematical model describing the time evolution of a quantity
with a rate of production that is proportional to the quantity itself.
Such models include populations, investments, feedback, and radioactivity. We are using t for
the independent variable, y for the dependent variable, k for the proportionality constant, and
dy
y
dt
for the rate of growth, or derivative, with respect to t. We are looking for a function that is
proportional to its own derivative.
Let’s start by examining the function
y = 2 t
t
t
We know what 2 means if t is an integer, 2 is the t-th power of 2.
1
2
0
2 –1 = 1/2; 2 = 1; 2 = 1; 2 = 4
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