Question

E(x) = 1x⁵ + 5x⁴ + 8x³ +9x + 3, − ∞ < < ∞.
You are asked to estimate one value of where the error is minimized, that is, E(x) = 0. Perform Newton’s Method to find that value.

Answer 1

Answer:

Now let’s look at how to calculate the approximations {x}_{0},{x}_{1},{x}_{2}\text{,…}. If {x}_{0} is our first approximation, the approximation {x}_{1} is defined by letting ({x}_{1},0) be the x-intercept of the tangent line to f at {x}_{0}. The equation of this tangent line is given by

y=f({x}_{0})+{f}^{\prime }({x}_{0})(x-{x}_{0}).

Therefore, {x}_{1} must satisfy

f({x}_{0})+{f}^{\prime }({x}_{0})({x}_{1}-{x}_{0})=0.

Solving this equation for {x}_{1}, we conclude that

{x}_{1}={x}_{0}-\frac{f({x}_{0})}{f\prime ({x}_{0})}.

Similarly, the point ({x}_{2},0) is the x-intercept of the tangent line to f at {x}_{1}. Therefore, {x}_{2} satisfies the equation

{x}_{2}={x}_{1}-\frac{f({x}_{1})}{f\prime ({x}_{1})}.

In general, for n>0,{x}_{n} satisfies

{x}_{n}={x}_{n-1}-\frac{f({x}_{n-1})}{f\prime ({x}_{n-1})}.

Next we see how to make use of this technique to approximate the root of the polynomial f(x)={x}^{3}-3x+1.

Hope it's helpful