Applications of differentiation - the graph of a function and its derivatives
      Approximate solution to an equation, Newton's method (or the Newton-Raphson method)
         Use of Newton's method, example
Approximate solution to an equation, Newton's method (or the Newton-Raphson 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 intervalf ' (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 x-axis 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 x-axis 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 x1 is closer approximate value of the root than x = a or x = b.
The x-intercept 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, x1 is the approximate value of the root.
If the function value  f (x1) does not satisfy required accuracy then the same process can be repeated at (x1, f (x1)) and so on until a predetermined level of accuracy is reached,
Therefore, Newton's method generates a sequence of approximations x1, x2, x3, x4, . . . , until reached is desired level of accuracy. If we have found xn, 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) = x4 - 4x3 + 4x2 + 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) = 4x3 - 12x2 + 8x + 1    and      f '' (x) = 12x2 - 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 x0 = 3 as the starting point.
We can check the function value  f (xi) at each successive approximate value xi of the process, to estimate if given accuracy is reached, that is, whether the xi is close enough to represent the root of the function.
The approximate value  x5 = 2.492572713  we can take as the root of the given quartic with an error less than 10-8.
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