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Approximate a cubic root using Newton's method

Question 1 of 8

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Separate the iteration calculation from proof of the final rounding

f(x)=-x^{3}-6x^{2}-14x-13

I=[-5,-2],\qquad x_0=-2

The target is the unique root in I. Carry unrounded values between steps: displayed decimals are terminal roundings, never the next input. Use half-away-from-zero rounding

Calculate one Newton step exactly

(a)3 marks

Enter every coefficient in descending powers, including zero for an absent term

f'(x)=A_2x^{2}+A_1x+A_0

\(\text{coefficient of }x^{2}\) \(\text{coefficient of }x\) \(\text{constant coefficient}\)

(b)1 mark

Which recurrence uses the current tangent to update the estimate?

x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}

x_{n+1}=x_n+\frac{f(x_n)}{f'(x_n)}

x_{n+1}=\frac{f(x_n)}{f'(x_n)}

x_{n+1}=x_n-\frac{f(x_n)}{f'(x_0)}\quad\text{for every }n

Enter the exact function value, tangent gradient and next estimate. Fractions are accepted

\(f(x_0)\) \(f'(x_0)\) \(x_1\)