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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 4.8.29
Chapter 4, Problem 4.8.29

{Use of Tech} Finding intersection points Use Newton’s method to approximate all the intersection points of the following pairs of curves. Some preliminary graphing or analysis may help in choosing good initial approximations.


y = 1/x and y = 4 - x²

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Step 1: Set the equations equal to each other to find the intersection points. This means solving 1/x = 4 - x².
Step 2: Rearrange the equation to form a single function f(x) = 4 - x² - 1/x. We need to find the roots of this function, which represent the intersection points.
Step 3: Use preliminary graphing or analysis to estimate initial guesses for the intersection points. Graph both y = 1/x and y = 4 - x² to visually identify where they might intersect.
Step 4: Apply Newton's method, which uses the formula x_{n+1} = x_n - f(x_n)/f'(x_n). Calculate the derivative f'(x) = -2x + 1/x².
Step 5: Iterate using Newton's method starting from your initial guesses until the values converge to a satisfactory approximation of the intersection points.

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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 solutions to equations. It starts with an initial guess and refines it using the function's derivative, converging to a root. The formula involves evaluating the function and its derivative at the current approximation, allowing for successive approximations that ideally get closer to the actual solution.
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Intersection Points

Intersection points occur where two curves meet, meaning their y-values are equal for the same x-value. To find these points, one typically sets the equations of the curves equal to each other and solves for x. The solutions can then be substituted back into either original equation to find the corresponding y-values, yielding the coordinates of the intersection points.
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Critical Points

Graphical Analysis

Graphical analysis involves plotting functions to visually identify features such as intersection points, behavior, and trends. By sketching the curves, one can gain insights into where they might intersect, which aids in selecting effective initial guesses for numerical methods like Newton's. This visual approach can simplify complex problems and enhance understanding of the relationships between functions.
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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→π/2⁻ (π/2 - x) sec x

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Textbook Question

{Use of Tech} Write the formula for Newton’s method and use the given initial approximation to compute the approximations x₁ and x₂.


f(x) = e⁻ˣ - x; x₀ = ln 2

Textbook Question

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


∫ (4√x - (4 /√x)) dx

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) = (x+4)/(4-x)

Textbook Question

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


g(s) = 1 / (s² + 1)

Textbook Question

Suppose f is differentiable on (-∞,∞), f(1) = 2, and f'(1) = 3. Find the linear approximation to f at x = 1 and use it to approximate f (1.1).