Aim of the applet

Interpret geometrically the approximations of a root constructed by means of the algorithm. Analyze the convergence according to the information of entry.

Input data

Via keyboard:


• Function of x.
• Value of tolerance.
• Value of x₀.

Via graph:


• The function can drag for the graph.
• The point x₀ can move along the function domain.
• Value of x₀.

To perform the algorithm calculations press the button "Calculate result table".

Output data

Graphical representation of the Newton-Rhapson algorithm.

Table with values of each step of algorithm, the final approximation and final relative error.

Observations

By defining a specific value x₀, point on the graph is fixed and you must press the button "Reset x₀" to move this. This move to zero for default.

A window of error:


• When the function has no roots.
• When the maximum error value is negative.
• When the derivative of f is zero at some point.

First steps

• Approximates a solution x²-4=0 with an error less than 10^-9. Change with the mouse the value of x₀ to approach the other solution. Is there a trouble point in this search?
• Approximates a solution x³ - x - 1=0. Modify x₀ with the mouse to get situations in which the algorithm does not converge.
• Find an interval [a, b], for which f (a) f (b) is less than 0, where f(x)=x² + 2x - 5 + log(x). If x₀ is in this interval, approximates a root of f with a relative error below 10^-6. Decrease the error to 10^-9, How many steps do you need?