Question

find a real root of equation cos x=3x-1 correct to four decimal place by iteration method ​

Answer 1

   Given Equation:

   $cosx=3x-1$

   To find:

    We have to find the real root of the equation cosx=3x-1 correct to four    

     decimal places by iteration method.

    Solution:

• Let $f(x)=cosx-3x+1$
• here $f(0)=cos(0)-3(0)+1$ = $1-0+1$ =2

        ∴ $f(0)=2$

• And $f(\frac{\pi }{2} )=cos(\frac{\pi }{2} )-3(\frac{\pi }{2} )+1$ = $-3\frac{\pi }{2} +1$ =  negative value
• Hence the real root of the given equation lies between 0 and $\frac{\pi }{2}$

        We can write the given equation as $x=\frac{1}{3} (cosx+1)$ = $\phi(x)$

        Now $\phi'(x)=\frac{sinx}{3}$

• $|\phi'(x)|=\frac{1}{3}|sinx|$ < 1 ∀  $x$ ∈$(0,\frac{\pi }{2} )$

       Hence the iteration method can be applied

    ITERATION METHOD:

• start with $x$₀ = 0
• Then $x$₁ = $\phi$(
• Similarly $x$₂ = $\phi$(
• $x$₃ = $\phi$(
• $x$₄ = $\phi$(
• $x$₅ = $\phi$(
• $x$₆ = $\phi$(

• $x$₇ = $\phi$(

    Hence $x$₆ and

    ∴ The real root of the equation cosx=3x-1  correct to four decimal places          

     is      0.6071

Answer 2

Answer:

find a real root of equation cos x=3x-1 correct to four