Given an equation f(x) = 0, a number
a such that f(a)
= 0 is called a root of the equation.� In other words, a root is a solution of the equation.

To find the number of roots of an equation, usually graphical method is used.

If y = f(x) is easy to sketch,
the number of x-intercepts is the number of roots.

Otherwise, rewrite the equation as g(x)
= h(x) where the graphs of y = g(x) and
y = h(x) are easy to sketch.�� Then the number
of intersection points between the two graphs is the number of roots of
the equation.

xistence Of Roots In An Interval

If�

the graph of y = f(x) is continuous
in [a, b] (ie, there is no break in the graph from a
to b), and

f(a) and f(b) are of opposite sign,

then the equation f(x) = 0 has a
root in the interval (a, b).

inear Interpolation

Theory:��Suppose we know that there is a root of the equation f(x) = 0 in the interval (a,
b), where b - a is small.� Then in the ideal situation, the chord joining the points
P(a, f(a)) and Q(b, f(b))
will be close to the curve y = f(x).� In this
case, the x-intercept, c, of the chord PQ will be
close to the root of the equation.

Formula:

af(b) -
bf(a)

c =�

�����

f(b) -
f(a)

ewton-Raphson Method

Theory:� Suppose we know that a root of f(x) = 0 is close to x = x_{1}.� Then in the ideal situation, the tangent to the curve at x = x_{1} will be close to y = f(x) in the surrounding of x = x_{1}.
Therefore, we can use the x-intercept, x_{2}, of
the tangent to be an approximation to the root of f(x) = 0.

Formula:

f(x_{1})

x_{2} = x_{1} -

���

f �(x_{1})

nder-/Over- Estimation

Whether linear interpolation or Newton-Raphson method will give an
under- or over- estimation depends greatly on the shape of the curve near
the root.

By computing the signs (ie + or -) of�
f �(x) and� f �(x) on an interval, we
can deduce the general shape of the curve y = f(x)
on the interval.� The following table shows all the four cases.

�

f �(x) > 0

f �(x) < 0

f �(x) > 0

f �(x) < 0

Just remember the following results and the table will become easy.

�

f �(x) > 0� �

the curve is increasing

f �(x) < 0� �

the curve is decreasing

�

f �(x) > 0� �

the curve concaves upwards

f �(x) < 0� �

the curve concaves downwards

�

Let's look at the case when f �(x) > 0 and f �(x)
> 0.

It is clear from the above diagram that linear interpolation (red
chord) produces an under-estimation whereas the Newton-Raphson method
(blue tangent) produces an overestimation.

Similarly, you may deduce the results for the other 3 cases.