Question

Evaluate √3 to four decimal places by Newton's method

Answer 1

Step-by-step explanation:

The Newton-Raphson method approximates the roots of a function. So, we need a function whose root is the cube root we're trying to calculate.

Let's say we're trying to find the cube root of 3. And let's say that x is the cube root of 3. Therefore,

x3=3

For the Newton-Raphson method to be able to work its magic, we need to set this equation to zero.

x3−3=0

Now we will recall the iterative equation for Newton-Raphson.

xn+1=xn−f(xn)f'(xn)

Substituting for f(x)=x3−3 gives us:

xn+1=xn−(xn)3−33⋅(xn)2

Now, we pick an arbitrary number, (the closer it actually is to 3√3 the better) for x0. Let's use x0=0.5. Then we substitute each previous number for xn back into the equation to get a closer and closer approximation to a solution of x3−3=0.

x1=0.5