Question

Solve for a positive root of x-cos x =0 by regular falsi method.

Answer 1

Answer:

 if(x) = 0 has root between x = a and x = b, then write the first approximate root by the method of false position. 2. Find an approximate value of the root of x3 -3x+1 = 0 lying between 1 and 2 by regula falsi method. 3. Write Newton’s formula to find the cube root of N. 4. State fixed point theorem (or) If g(x) is continous in [a,b], then under what condition in [a,b]?. 5. Write down the condition for the convergence of Gauss – Seidel iteration scheme. 6. State any two differences between direct and iterative methods for solving system of equations.

Answer 2

Answer:

The positive root is 0.74.

Step-by-step explanation:

Given the equation $x-cos x =0$

The regula-falsi method is the method used to estimate the real root of an equation $f(x) = 0$.

• If there are two points a and b between which the root lies, then, $x = \frac{bf(a)-af(b)}{f(a)-f(b)}$

Let $f(x)=x-cos x$

For $x = 0,$  $f(0) = -0-cos0=-1$

For $x = 1,$  $f(1) =1-cos(1)= 0.45969$

Therefore, the root lies between 0 and 1

Let $a = 0, f(a) = -1$ and $b = 1; f(b) = 0.45969$

Substituting the values in the above formula,

$x_1 = \dfrac{1(-1)-0\times 0.45969}{(-1)-0.45969}= \dfrac{-1}{-1.45969}=0.68507$

and hence $f(x_1) =0.68507-cos(0.68507)= -0.08929$

Now, $x_1$ is $a$ and $f(x_1)$ is $f(a)$ to the next point.

$x_2 = \dfrac{1(-0.08929)-0.68507\times 0.45969}{(-0.08929)-0.45969}= 0.73629$

and hence $f(x_2) =0.73629-cos(0.73629)= -4.66039\times10^{-3}$

Now, $x_2$ is $a$ and $f(x_2)$ is $f(a)$ to the next point.

$x_3 = \dfrac{1(-4.66039\times10^{-3})-0.68507\times 0.45969}{(-4.66039\times10^{-3})-0.45969}=0.73894$

and hence $f(x_3) =0.73894-cos(0.73894)= -2.33926\times 10^{-4}$

Now,

$x_4 = \dfrac{1(-2.33926\times 10^{-4})-0.73894\times 0.45969}{(-2.33926\times 10^{-4})-0.45969}=0.73907$

and hence $f(x_4) =0.73907-cos(0.73907)= -1.17202\times 10^{-5}$

Now,

$x_5 = \dfrac{1(-1.17202\times 10^{-5})-0.73907\times 0.45969}{(-1.17202\times 10^{-5})-0.45969}=0.73908$

Therefore, correcting to two decimals, the positive root is 0.74.