Question

8 The Newton-Raphson method. This is a technique which was developed independently approximately 300 years ago by two Isaac Newton and Joseph Raphson. This is an iterative (repetitive) technique which produces successively better approximations for the roots (or zeros) of a real function. Using this technique, if we cannot solve an equation, we can find a very accurate approximation to its roots. Say we cannot solve some equation f(x)- 0. We can investigate its roots by drawing the graph of y-f(x). Then the solutions of this equation are given by the x-intercepts of this graph. y-f(x) Tangent at Tangent at x 而 Consider the graph of y-f(x) (drawn in red above). Its x-intercept is the point marked in red on the ar-axis (whose value we cannot find) We approximate this value as x. (i.e. we make an intelligent guess at it) We then draw a tangent to the curve at x = xo-the blue line on the diagram-and we find where this line cuts the x-axis (the point x on the diagram. This point x is a better approximation to the required solution. We then repeat this process. ie. We now draw at tangent to the curve at x = the green line on the diagram- and we find where this line cuts the x-axis (the point x2 on the diagram. This point x is an even better approximation to the required solution. By repeating this process many times we can get very accurate approximations to the value of the point where the graph of y=f(x) cuts the x-axis. (a) (i) Write down the equation of the tangent to y= f(x) at the point x (ii) Show carefully that this tangent cuts the x-axis at the point x, where

Answer 1

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