Calculus help?

Put  f (x) = x^4 − 2x^3 + 7x^2 − 8

First, check the value of  f (x)  for a few values of  x  in  [1, 2],  including the ends:

       x       f (x)

    1.0      -2.0

    1.2       0.7

    1.4       4.1

    1.6       8.3

    1.8     13.5

    2.0     20.0

The required root lies between  1.0  and  1.2;

and  f(x) is increasing at an increasing rate throughout. So we should choose  1.2  as our initial approximation.

f' (x) = 4x^3 - 6x^2 + 14x

The change from  old x  to  new x  is  -f (x) / f' (x).

Set this up in Excel, e.g.

       x                      f (x)                    f' (x)

   change

     1.2                    0.6976             15.072   -0.0462845

     1.153715499  0.017807593 14.308315952   -0.001244562     1.152470936  0.000012486 14.288255633   -0.000000874     1.152470063   0                     14.288241556

   0     1.152470063        0                14.288241556

   0

At this point, we see that the value of  x  has stabilised to  9  decimal places.  So we may quote the solution as

  x = 1.152470,  to six decimal places.

See Attached Table of Calculations