Question

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

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.

(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.

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