Question

1. use newtons method to determine ( x_1 ) for ( f(x)=x^{3}-7x^{2}+8x - 3 ) if ( x_0 = 5 ).

Explanation:

Step1: Find the derivative of \(f(x)\)

The derivative of \(f(x)=x^{3}-7x^{2}+8x - 3\) is \(f^{\prime}(x)=3x^{2}-14x + 8\) using the power rule \((x^{n})^\prime=nx^{n - 1}\).

Step2: Calculate \(f(x_{0})\) and \(f^{\prime}(x_{0})\)

When \(x_{0}=5\), \(f(5)=5^{3}-7\times5^{2}+8\times5-3=125-175 + 40-3=-13\)
\(f^{\prime}(5)=3\times5^{2}-14\times5 + 8=75-70 + 8=13\)

Step3: Apply Newton's method formula \(x_{n+1}=x_{n}-\frac{f(x_{n})}{f^{\prime}(x_{n})}\)

For \(n = 0\), \(x_{1}=x_{0}-\frac{f(x_{0})}{f^{\prime}(x_{0})}\)
Substitute \(x_{0}=5\), \(f(5)=-13\) and \(f^{\prime}(5)=13\) into the formula:
\(x_{1}=5-\frac{-13}{13}=5 + 1=6\)

Answer:

\(6\)