Question

It is given that

i.       Show that

   ii.       Show by calculation that the value of a lies between 1.0 and 1.5.

  iii.       Use an iterative formula, based on the equation in part (i), to find the value of a correct to 3  decimal places. Give the result of each iteration to 5 decimal places.

Solution

     i.
 

We are given that;

Rule for integration of  is:

Rule for integration of , or ;

Rule for integration of  is:

Taking logarithm of both sides;

Since  for any ;

 

   ii.
 

We are required to show by calculation that the x-coordinate of a lies between 1.0 and 1.5. 

We need to use sign-change rule.

To use the sign-change method we need to write the given equation as .

From (ii) we have;

Therefore;

If the function  is continuous in an interval  of its domain, and if   and have  opposite signs, then  has at least one root between  and .

We can find the signs of at and as follows;

Since and have opposite signs for function , the function has  root between and .

 

  iii.
 

Iteration method can be used to find the root of the given equation using sequence defined by;

If the sequence given by the inductive definition , with some initial value , converges to a limit , then  is the root of the equation .

Therefore, if , then  is a root of .

We use as initial value.

It is evident that .

Hence, is a root of .

The root given correct to 2 decimal places is 1.343.