To determine
To find:
Root of the given function correct to 6 decimal places, using Newton’s Method.
Answer
The approximate root is 5.2907148
Explanation
Given:
fx=x-42-lnx.
Graph of f is shown below:
Since the graph of f cuts the x-axis, at exactly two points, 2.9, 5.2 therefore f has two zeros.
The Newton’s formula is:xn+1=xn-fxnf'xn.
f'x=2x-4-1x. Let the initial approximation for first root be x0=3.0, then
x1=x0-fx0f'x0=2.9577376.
x2=x1-fx1f'x1=2.9585162.
x3=x2-fx2f'x2=2.9585165.
x4=x3-fx3f'x3=2.9585165.
Since x3=x4 upto 6 decimal places, first approximate root is 2.9585165.
As we have seen that there are exactly two zeros of f, so let’s consider approximation for second root, x0=5.3, then
x1=x0-fx0f'x0=5.2907548.
x2=x1-fx1f'x1=5.2907148.
x3=x2-fx2f'x2=5.2907148.
Since x2=x3 upto 6 decimal places, the approximate root is 5.2907148.
Conclusion:
The approximate root is 5.2907148