Question

The smallest positive zero of f(x)=xtan(x)+1 is a measure of how quickly certain evanescent water waves decay, and its value, x0​, is near 3.5 . Note: Work to 6 decimal places of accuracy. Use the forward difference method to estimate f′(3.5) with a step size of 0.01 and use this value in an approximate version of the Newton-Raphson method to derive 1 improvement on x0​.

Answer 1

Using the forward difference method with a step size of 0.01 to estimate the derivative of the function at x=3.5, the Newton-Raphson method can be applied to obtain an improved approximation for the smallest positive zero of the function.

Step-by-step explanation:

The student is looking to improve an approximation for the smallest positive zero of the function f(x) = xtan(x) + 1 using the Newton-Raphson method. To do this, we first need to estimate f'(3.5) using the forward difference method with a step size of 0.01. Once we have this estimate, we can apply one iteration of the Newton-Raphson method to improve the current approximation of x0, which is around 3.5.

To begin, calculate f(3.5) and f(3.51) and then use the forward difference approximation:

• f'(3.5) ≈ (f(3.51) - f(3.5)) / 0.01

With this slope approximation, apply Newton-Raphson:

• x1 = x0 - f(x0) / f'(x0)

Here, x1 is the improved approximation.