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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 4.8.23
Chapter 4, Problem 4.8.23

{Use of Tech} Finding all roots Use Newton’s method to find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations.


f(x) = e⁻ˣ - ((x + 4)/5)

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Step 1: Understand the problem and the function. We need to find the roots of the function f(x) = e^(-x) - ((x + 4)/5). A root of the function is a value of x for which f(x) = 0.
Step 2: Perform a preliminary analysis. Analyze the behavior of the function by considering its limits as x approaches positive and negative infinity, and by finding the derivative to understand its increasing or decreasing nature.
Step 3: Graph the function f(x) = e^(-x) - ((x + 4)/5) to visually identify approximate locations of the roots. This will help in choosing good initial guesses for Newton's method.
Step 4: Apply Newton's method. Newton's method uses the formula x_(n+1) = x_n - f(x_n)/f'(x_n) to iteratively find a root. Calculate the derivative f'(x) = -e^(-x) - 1/5, and use it in the formula.
Step 5: Choose initial approximations based on the graph and apply Newton's method iteratively. For each initial guess, compute successive approximations until the change is sufficiently small, indicating convergence to a root.

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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 x_{n+1} = x_n - f(x_n)/f'(x_n), where f' is the derivative of f. This method converges quickly under suitable conditions, making it effective for finding roots when the function is well-behaved near the root.
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Preliminary Analysis

Preliminary analysis involves examining the function's behavior to identify potential roots before applying numerical methods. This can include evaluating the function at various points, checking for sign changes, and analyzing critical points and asymptotes. Such analysis helps in selecting appropriate initial approximations for Newton's Method, increasing the likelihood of convergence.
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Graphing Functions

Graphing functions provides a visual representation of their behavior, which is crucial for understanding where roots may lie. By plotting the function, one can observe intersections with the x-axis, indicating potential roots. Additionally, graphing can reveal the function's overall shape, helping to identify regions where the function is increasing or decreasing, which aids in selecting effective initial guesses for numerical methods.
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Related Practice
Textbook Question

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.


lim_x→ 0 (eˣ - sin x - 1) / (x⁴ + 8x³ + 12x²)

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Textbook Question

Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f'(x)dx.


f(x) = tan x

Textbook Question

{Use of Tech} Finding all roots Use Newton’s method to find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations.


f(x) = cos 2x - x² + 2x

Textbook Question

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


f(x) = cos² x on [-π,π]

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Textbook Question

23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.


∫ (3x ¹⸍³ + 4x ⁻¹⸍³ + 6) dx

Textbook Question

Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.


lim_x→ 0 (eˣ - 1) / (2x + 5)