Year 11 Maths - Methods Applications of Differentiation and Antidifferentiation

Newtons Method to Solve Equations

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Questions

Question 1

29195

Starting with \(x=3\), and using Newton's Method with one application, find, correct to two decimal places and approximation to the solution to the equation \(x^3-20=0\)

Answer

2.74

Worked Solution

\begin{align}
f(x) &=x^{3}-20 \\
f^{\prime}(x) &=3 x^{2} \\
x_{n+1} &=x_{n}-\frac{f\left(x_{n}\right)}{f^{\prime}\left(x_{n}\right)} \\
x_{n}=&3 \\
f(3)&=7\;\; f^{\prime}(3)=27 \\
\therefore x_{n+1}&=3-\frac{7}{27} \\
&=2 \cdot 74
\end{align}

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