Ch. 4 - Applications of the Derivative

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 4 - Applications of the Derivative

문제 4.8.17

4장, 문제 4.8.17

{Use of Tech} Finding roots with Newton’s method For the given function f and initial approximation x₀, use Newton’s method to approximate a root of f. Stop calculating approximations when two successive approximations agree to five digits to the right of the decimal point after rounding. Show your work by making a table similar to that in Example 1.

f(x) = tan x - 2x; x₀ = 1.2

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Step 1: Understand Newton's Method. Newton's method is an iterative process to approximate the roots of a real-valued function. The formula for the next approximation is given by:

x_n + 1 = x_n - \frac{f (x_n)}{f' (x_n)}

Step 2: Calculate the derivative of the function. Given

f (x) = \tan (x) - 2 x

, the derivative is

f' (x) = \frac{1}{\cos^2 (x)} - 2

Step 3: Set up the iterative process. Start with the initial approximation

x_0 = 1.2

. Calculate

f (x_0)

and

f' (x_0)

Step 4: Compute the next approximation using the formula:

x_1 = x_0 - \frac{f (x_0)}{f' (x_0)}

. Repeat this process to find

x_2

x_3

, etc., until two successive approximations agree to five decimal places.

Step 5: Create a table to organize your results. For each iteration, record the values of

n

x_n

f (x_n)

f' (x_n)

, and the next approximation

x_n + 1

. Continue until the stopping criterion is met.

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Newton's Method

Newton's Method is an iterative numerical technique used to find approximate roots of a real-valued function. It starts with an initial guess and refines it using the formula x₁ = x₀ - f(x₀)/f'(x₀), where f' is the derivative of f. This process continues until the difference between successive approximations is sufficiently small, indicating convergence to a root.

Convergence Criteria

In numerical methods, convergence criteria determine when to stop the iterative process. For Newton's Method, this often involves checking if the absolute difference between two successive approximations is less than a specified tolerance level, such as 0.00001 for five decimal places. This ensures that the approximations are accurate enough for practical purposes.

Function and Derivative Evaluation

To apply Newton's Method, it is essential to evaluate both the function f(x) and its derivative f'(x) at each iteration. For the given function f(x) = tan x - 2x, the derivative f'(x) = sec² x - 2 must be computed. Accurate evaluation of these functions is crucial for obtaining correct approximations of the root.