Ch. 4 - Applications of the Derivative

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 4 - Applications of the Derivative

Problem 4.18.1

Chapter 4, Problem 4.18.1

{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

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 Newton's method is: x_{n+1} = x_n - \(\frac{f(x_n)}{f'(x_n)}\).

Step 2: Calculate the derivative of the function. Given f(x) = x \(\ln\)(x + 1) - 1, we need to find f'(x). Using the product rule and chain rule, f'(x) = \(\ln\)(x + 1) + \(\frac{x}{x + 1}\).

Step 3: Set up the iterative formula. Using the initial approximation x₀ = 1.7, apply Newton's method: x_{n+1} = x_n - \(\frac{x_n \ln(x_n + 1) - 1}{\ln(x_n + 1) + \frac{x_n}{x_n + 1}\)}.

Step 4: Perform the iterations. Start with x₀ = 1.7 and calculate x₁ using the formula from Step 3. Continue calculating x₂, x₃, etc., until two successive approximations agree to five decimal places.

Step 5: Create a table of values. For each iteration, record the value of x_n, f(x_n), f'(x_n), and the difference |x_{n+1} - x_n|. Stop when the difference is less than 0.00001, indicating that the approximations agree to five decimal places.

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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 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 sufficiently accurate 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) = x ln(x + 1) - 1, the derivative f'(x) must be computed to facilitate the iterative formula. Accurate evaluation of these expressions is crucial for the method's success and convergence to the correct 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

{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

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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

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) = (4x³/3) + 5x² - 6x on [0,5]

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Textbook Question

Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.

f(x) = eˣ/(e²ᵉ + 1)

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Textbook Question

Rectangles beneath a parabola A rectangle is constructed with its base on the x-axis and two of its vertices on the parabola y = 48 - x². What are the dimensions of the rectangle with the maximum area? What is the area?