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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 4.26.1
Chapter 4, Problem 4.26.1

{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) = x²(x - 100) + 1

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Start by analyzing the function f(x) = x²(x - 100) + 1. Notice that it is a cubic polynomial, which means it can have up to three real roots. The term x²(x - 100) suggests that the roots might be around x = 0 and x = 100.
Graph the function f(x) to visually inspect where the roots might be located. Look for points where the graph crosses the x-axis, as these indicate potential roots. This will help you choose good initial approximations for Newton's method.
Choose initial approximations based on the graph. For example, if the graph suggests roots near x = 0 and x = 100, you might start with x₀ = 0 and x₀ = 100. If the graph shows another crossing point, choose an initial approximation near that point as well.
Apply Newton's method, which uses the formula xₙ₊₁ = xₙ - f(xₙ) / f'(xₙ). First, find the derivative f'(x) of the function f(x). For f(x) = x²(x - 100) + 1, use the product rule and power rule to differentiate.
Iterate using Newton's method: Calculate x₁ using your initial approximation x₀, then use x₁ to find x₂, and so on, until the values converge to a stable root. Repeat this process for each initial approximation to find all roots.

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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 if the initial guess is close to the actual root, making it effective for functions that are differentiable.
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Preliminary Analysis

Preliminary analysis involves examining the function's behavior to identify potential roots before applying numerical methods. This includes evaluating the function at various points, checking for sign changes, and analyzing critical points and asymptotes. This step helps in selecting good 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, helping to identify roots, intercepts, and overall trends. By plotting the function, one can observe where it crosses the x-axis, indicating potential roots. This visual approach complements analytical methods and aids in selecting effective starting points for iterative methods like Newton's.
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Related Practice
Textbook Question

Suppose f is differentiable on (-∞,∞) and the equation of the line tangent to the graph of f at x = 2 is y = 5x -3. Use the linear approximation to f at x = 2 to approximate f(2.01).

Textbook Question

Tree notch (Putnam Exam 1938, rephrased) A notch is cut in a cylindrical vertical tree trunk (see figure). The notch penetrates to the axis of the cylinder and is bounded by two half-planes that intersect on a diameter D of the tree. The angle between the two half-planes is Θ. Prove that for a given tree and fixed angle Θ, the volume of the notch is minimized by taking the bounding planes at equal angles to the horizontal plane that also passes through D.

Textbook Question

{Use of Tech} Growth rate of spotted owlets The rate of growth (in g/week) of the body mass of Indian spotted owlets is modeled by the function r(t) = 10,147.9e⁻²·²ᵗ/(37.98e⁻²·² + 1), where t is the age (in weeks) of the owlets. What value of t > 0 maximizes r? What is the physical meaning of the maximum value?

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) = x⁵/5 - x³/4 - 1/20

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Textbook Question

Finding antiderivatives. Find all the antiderivatives of the following functions. Check your work by taking derivatives.


ƒ(y) = - 2/y³

Textbook Question

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

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