LIGAND BINDING, DATA FITTING, AND LEAST-SQUARES ESTIMATES OF MODEL PARAMETERS - An Invitation to Biomathematics - page 244

the line l 1 tangent to the graph of f (x)at(x 0, f (x 0 )) crosses the x-axis,
denoting this point x 1 (see Figure 8-5). The equation of the tangent
line l 1 is

f 0 ð

y f ð x 0 Þ¼ x 0 Þð x x 0 Þ

or (8-18)

¼ ð Þþ

f 0 ð

Þð Þ:

y f x 0 x 0 x x 0

Note that Eq. (8-18) is the truncated form of the Taylor series (8-16).
Now, because the tangent line l 1 crosses the x-axis at x

¼ x 1, we have

¼

y 0 at this point, and Eq. (8-18) becomes

f 0 ð 0 ¼ f ð x 0 Þþ x 0 Þð x 1 x 0 Þ:

Solving for x 1 , we obtain the following equation for the ''better
guess'':

ð Þ

f x 0

x 1 ¼ x 0

x 0 Þ :

(8-19)

f 0 ð

If the value x 1 is not the root, we consider the line l 2 tangent to the
graph of f (x)at(x 1, f (x 1 )). The point x 2 at which the line l 2 crosses the
x-axis is calculated from

x 1 Þ
f 0 ð

f ð

x 2 ¼ x 1

Þ ;

(8-20)

x 1

y

f(x)

l 1

l 2

x 0

x 2 x 1

x *

0

x

FIGURE 8-5.
Successive iterations for obtaining improved guesses for the point x* where f(x*)

¼ 0 by using

Newton's method.