
Applications
of differentiation  the graph of a function and its derivatives 
Approximate
solution to an equation, Newton's method (or the NewtonRaphson method) 
Use of
Newton's method, example 






Approximate
solution to an equation, Newton's method (or the NewtonRaphson method) 
The
mean value theorem can be applied to find approximate value of a root of
a function. 
Newton's
method is iterative method to approximate solution
to
the equation f (x) = 0
to
a desired accuracy. 
If
a
function f
continuous on the closed
interval [a,
b]
satisfies
following conditions, 

its function values at endpoints of the interval are of the opposite
signs 

the
first and second derivatives of which do not change the sign on the
interval,
f
' (x)
and f ''
(x)
is not
0 
then,
we can find approximate value of the root of the function within the
interval [a,
b]
using Newton's method. 
Since
f
' (x)
is not
0,
the graph of f
intersects
the xaxis
only once at the root of the function,
and because of f ''
(x)
is not
0
corresponding
arc
AB
is concave up or concave down within
the interval. 
Newton's
method begins
by drawing the line tangent to the graph at the endpoint of the interval
at which f (x)
· f ''
(x) > 0
taking the intercept of the tangent with the xaxis
as the approximate value of
the root of the
function. 
Thus,
each of two signs of the second derivative, f ''
(x) > 0
or f ''
(x) < 0
determines two cases, depending
whether
f
' (x) > 0
or f
' (x) < 0,
there are four possible cases as shows the figure below. 

Note
that, by satisfying f (x)
· f ''
(x) > 0
in each of four cases the intercept x_{1}
is closer approximate value
of
the root than x = a
or x = b. 
The
xintercept
of the tangent 
y
= f (b) + f
' (b)
(x 
b) 
at
the point B
of the curve, we get by solving y
= 0
for x, 

where,
x_{1}
is the
approximate value of
the root. 
If
the function value f (x_{1})
does not satisfy required accuracy then the same process can be
repeated at (x_{1}, f (x_{1})) and
so on until a predetermined level of accuracy is reached, 

Therefore,
Newton's method generates a sequence of approximations x_{1},
x_{2},
x_{3},
x_{4},
. . . , until reached is desired level of accuracy.
If
we have found x_{n},
the next approximation is
the solution of the equation 


Use
of Newton's method example 
Example:
Use
Newton's method to approximate the root of the function
f (x) = x^{4}

4x^{3} + 4x^{2} + x 
4, located
within the interval [2,
3], with an error
less than 0.0001. 
Solution:
Let's first chose the starting point x
that satisfies the necessary condition f (x)
· f ''
(x) > 0
to ensure
the use of
Newton's method the right way. 
Since
f
' (x) = 4x^{3}

12x^{2} + 8x + 1
and f
'' (x) = 12x^{2}

24x + 8, 
by
plugging the endpoints x =
2
and x =
3
of the given interval into f (x)
and f
'' (x) we
get, 
f (2) =

2
and f
'' (2) = 8
so that f (2)
· f ''
(2)
< 0
therefore, x =
2
fail 
while
f (3) =
8 and f
'' (3) = 44
so that f (3)
· f ''
(3)
> 0 so we use
x_{0} =
3
as the
starting point. 
We
can check the function value
f (x_{i})
at
each successive approximate value
x_{i}
of the process, to estimate if
given accuracy is reached, that is, whether the x_{i}
is close
enough to represent the root of the function. 

The
approximate value x_{5} =
2.492572713
we can take as the root of the given quartic with an error
less than
10^{}^{8}. 









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