Respuesta :

Answer:

Using the Bisection Method [tex]\sqrt{2} \approx 1.4106[/tex]

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 [tex]c=\frac{a+b}{2}[/tex]
  3. if [tex]b-c\leq \epsilon[/tex] then accept c as the root and stop
  4. if [tex]f(a)f(c)\leq 0[/tex] 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 [tex]x^2-2=0[/tex]
  • ε = 0.01

Applying the steps of the Bisection Method you get:

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

[tex]f(1)=1^2-2=-1\\f(2)=2^2-2=2[/tex]

2. Define [tex]c=\frac{1+2}{2}=1.5[/tex]

3. [tex]2-1.5\geq 0.01[/tex]

4. Because [tex]f(1)\cdot f(1.5) = -0.25[/tex] we set 1.5 as the new b.

The bisection algorithm is detailed in the following table.

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

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