Chapter 7: Q 80 (page 593)

Use the result of Exercise $79$ to approximate the square roots in Exercises $80 - 83$ . In each case, start with $x_{0} = 1$ and stop when $|x_{k + 1} - x_{k}| < 0.001 .$
$\sqrt{2}$

Short Answer

Expert verified

The required approximate value is :-

$\sqrt{2} = 1.414214$ .

Step by step solution

Step 1. Given Information

We have given the following square root :-

$\sqrt{2}$ .

We have to find the value of $\sqrt{2}$ by using the recursion formula with initial approximation $x_{0} = 1$ . Also we have to stop our process of find iteration while $|x_{k + 1} - x_{k}| < 0.001$ .

Step 2. To find the value of 2

In the previous example we find the following recursion formula :-

$x_{k + 1} = \frac{1}{2} (x_{k} + \frac{a}{x_{k}})$ , to find the approximate value of $\sqrt{a}$ .

Here $a = 2$ and $x_{0} = 1$ , then the first iteration is :-

$x_{1} = \frac{1}{2} (1 + \frac{2}{1}) \Rightarrow x_{1} = \frac{1}{2} \times 3 \Rightarrow x_{1} = 1.5$

then second iteration is :-

$x_{2} = \frac{1}{2} (1.5 + \frac{2}{1.5}) \Rightarrow x_{2} = \frac{1}{2} (2.8334) \Rightarrow x_{2} = 1.41667$

Then third iteration is :-

$x_{3} = \frac{1}{2} (1.41667 + \frac{2}{1.41667}) \Rightarrow x_{3} = 1.414216$

Then fourth iteration is :-

$x_{4} = \frac{1}{2} (1.414216 + \frac{2}{1.414216}) \Rightarrow x_{4} = 1.414214$

Here $|x_{4} - x_{3}| = |1.414214 - 1.414216| = 0.000002 < 0.001$

So $x = 1.414214$ is the required approximate value of $\sqrt{2}$ .

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Use the result of Exercise $79$ to approximate the square roots in Exercises $80 - 83$ . In each case, start with $x_{0} = 1$ and stop when $|x_{k + 1} - x_{k}| < 0.001 .$
$\sqrt{2}$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. To find the value of 2

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Use the result of Exercise 79to approximate the square roots in Exercises 80-83. In each case, start with x0=1and stop when xk+1-xk<0.001.2

Short Answer

Step by step solution

Step 1. Given Information

Step 2. To find the value of 2

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Statistics

Decision Maths

Applied Mathematics

Theoretical and Mathematical Physics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Use the result of Exercise $79$ to approximate the square roots in Exercises $80 - 83$ . In each case, start with $x_{0} = 1$ and stop when $|x_{k + 1} - x_{k}| < 0.001 .$
$\sqrt{2}$