Question

Use the Bisection Method to find √ 2 correct to roughly within 10^−2 . (Hint: Consider f(x) = x^2 − 2 on [1, 2])

Answer 1

Answer:

Using the Bisection Method $\sqrt{2} \approx 1.4106$

Step-by-step explanation:

These are the steps for the Bisection Method:

Suppose we need a root for f(x) = 0 and we have an error tolerance of ε

1. Find two numbers a and b at which the function has different signs
2. Define $c=\frac{a+b}{2}$
3. if $b-c\leq \epsilon$ then accept c as the root and stop
4. if $f(a)f(c)\leq 0$ then set c as the new b. Otherwise, set c as the new a. return to step 1.

We know from the information given that

• The function is $x^2-2=0$
• ε = 0.01

Applying the steps of the Bisection Method you get:

1. There is a root between [1,2] because:

$f(1)=1^2-2=-1\\f(2)=2^2-2=2$

2. Define $c=\frac{1+2}{2}=1.5$

3. $2-1.5\geq 0.01$

4. Because $f(1)\cdot f(1.5) = -0.25$ we set 1.5 as the new b.

The bisection algorithm is detailed in the following table.

Note that after 7 steps we have $b-c=0.0078 \leq 0.01$ hence the required root approximation is c = 1.4106