Use Newtons method to find an approximation to the first pos

Use Newton’s method to find an approximation to the first positive zero of f(x)=xtan(x)-1.

(a) Solve the equation numerically using MATLAB:

        format long

        f=@(x) x*tan(x)-1

        sol = fzero(f,[0,1])

       Why are we guaranteed at least one solution in the interval [0,1]?

(b)   Find the derivative of f(x).

(c)    Using ½+10/n as your starting value, provide your iterative formula

(d)   Use MATLAB to write and execute a routine. How many iterations does it take for your answer to match the answer given in a? Do you have fast (quadratic) or slow (linear) convergence? Why? Display the intermediate approximate solutions as a table.

Solution

on script page

clear all

f=@(x)x*tan(x)-1

df=@(x)x*sec(x)^2+tan(x)

x0=input(\'enter initial guess\')

while abs(f(x0))>0.0001

x1=x0-(f(x0)/df(x0))

x0=x1

end

On work space

f =

    @(x)x*tan(x)-1

df =

    @(x)x*sec(x)^2+tan(x)

enter initial guess0.5

x0 =

    0.5000

x1 =

    1.1080

x0 =

    1.1080

x1 =

    0.9466

x0 =

    0.9466

x1 =

    0.8710

x0 =

    0.8710

x1 =

    0.8605

x0 =

    0.8605

x1 =

    0.8603

x0 =

    0.8603

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