Question

17. I am using Newton’s method to find the negative root of f(x) = 3−x².

(a) What would be a good guess for x₁? Draw the line tangent to f(x) at your x₁ and explain why using Newton’s method would lead to the negative root of the function.

(b) What would be a bad guess for x₁? Draw the line tangent to f(x) at your x₁ and explain why using Newton’s method would not lead to the negative root of the function.

Answer 1

(a) We know, to find the roots of a solution using Newton Raphsons method

xn+1= xn- f(xn)/f'(xn)

Where xn= root after nth iteration

f'(xn)= value of the differentiation at xn.

Given f(x)=3-x2

f'(x)=-2x

So x1= x0+(3-x02)/2x0

A good guess value would be -1, since at x0=-1, x1=-2 which is negative. The tangent will lead to a negative root.

(b) 1 will be a bad root because at x0=1, x1=2 and hence it is not converging to a point, but diverging, since the actual root is plus and minus 1.732. But taking this tangent will lead to the positive solution.