use newtons method to approximate

Question

Use Newton's Method to approximate √10 to 6 significant figures starting with x0=3.5. Compare with the value obtained from a calculator.

 

√10≈________

 

Answer 1

Write f(x) = x² - 10. Newton's Method calls for the repeated computation of x - f(x)/f'(x) or Expression A: x - (x²-10)/(2x).

With x initially set to 3.5, compute the result of Expression A above and then repeat the computation of Expression A with the new result.

Continue to compute the value of Expression A for each new or succeeding result until the desired 6 significant figures are obtained.

Note that a Casio Programmable Calculator gives the pre-programmed value of √10 (= 3.16227766) with only 3 calculations of Expression A, starting at x = 3.5.