191k views
5 votes
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.)

User Alvaropgl
by
8.8k points

1 Answer

6 votes

Final answer:

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.

User Skeptic
by
8.5k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories