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

Problem 4.8.13

Chapter 4, Problem 4.8.13

{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) = x² - 10; x₀ = 3

Verified step by step guidance

Step 1: Understand Newton's Method. Newton's method is an iterative process used 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. For the function f(x) = x^2 - 10, the derivative f'(x) is 2x.

Step 3: Set up the iterative formula using the initial approximation x₀ = 3. Substitute into the Newton's method formula: x_{n+1} = x_n - \(\frac{x_n^2 - 10}{2x_n}\).

Step 4: Perform the first iteration. Substitute x₀ = 3 into the formula to find x₁: x₁ = 3 - \(\frac{3^2 - 10}{2*3}\). Calculate this value to get the first approximation.

Step 5: Continue iterating. Use the result from the previous step as the new x_n and repeat the process until two successive approximations agree to five decimal places. Record each iteration in a table format to track the progress.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 this guess 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 sufficiently accurate for practical purposes.

Function and Derivative Evaluation

To apply Newton's Method, one must evaluate both the function f and its derivative f' at each iteration. For the given function f(x) = x² - 10, the derivative is f'(x) = 2x. Accurate evaluation of these expressions is crucial, as errors in calculation can lead to incorrect approximations of the root.

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Related Practice

Textbook Question

Second Derivative Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima.

f(x) = eˣ(x - 2)²

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Textbook Question

Maximum-area rectangles Of all rectangles with a perimeter of 10, which one has the maximum area? (Give the dimensions.)

Textbook Question

Graphing functions Use the guidelines of this section to make a complete graph of f.

f(x) = e⁻ˣ²/₂

Textbook Question

Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.

ƒ(x) = x/(x²+9)⁵ on [-2,2]

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Textbook Question

Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.

ƒ(x) = 3x³ + 3x² / 2 - 2x

Textbook Question

{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) = x ln (x + 1) -1 ; x₀ = 1.7