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

Problema 4.18.1

Capitolo 4, Problema 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

Guida verificata passo dopo passo

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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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.

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Evaluating Composed Functions

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.

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Derivatives

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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Percorso guidato

4:26

Evaluating Composed Functions

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{Use of Tech} Special curves The following classical curves have been studied by generations of mathematicians. Use analytical methods (including implicit differentiation) and a graphing utility to graph the curves. Include as much detail as possible.

x⁴ - x² + y² = 0 (Figure-8 curve)

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

ƒ(x) = 3x³ + 3x² / 2 - 2x

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ƒ(x) = (4x³/3) + 5x² - 6x on [0,5]

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f(x) = eˣ/(e²ᵉ + 1)

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