Question

The smallest positive zero of f(x)=xtan(x)+1.5 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. a) 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​. The root x1​ of the function after one improvement is (Note: Round your ariswer to 6 decimal places.)

Answer 1

To estimate f′(3.5) using the forward difference method, use a step size of 0.01. Then, use this value in the Newton-Raphson method to find an improved value of x0.

Step-by-step explanation:

To estimate f′(3.5) using the forward difference method, we need to use a step size of 0.01. We can calculate the forward difference using the formula:

f′(x) ≈ (f(x + h) - f(x)) / h

Substituting the values, we get:

f′(3.5) ≈ (f(3.51) - f(3.5)) / 0.01

Now, we can use this value in the Newton-Raphson method to find an improved value of x0. The formula for the Newton-Raphson method is:

x1 = x0 - f(x0) / f′(x0)

Substituting the values, we get:

x1 = 3.5 - f(3.5) / f′(3.5)

Calculate the values of f(3.51), f(3.5), and f′(3.5) using the given function to find the final value for x1.