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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 4 - Applications of DerivativesProblem 4.6.8c
Chapter 4, Problem 4.6.8c

Dependence on Initial Point
8. Using the function shown in the figure, and, for each initial estimate x_0, determine graphically what happens to the sequence of Newton’s method approximations
Graph of a function y=f(x) with a peak at x=1.5 and a trough at x=5, showing behavior of Newton's method at x_0=2.
c. x_0=2

Verified step by step guidance
1
Newton's method is an iterative process used to find successively better approximations to the roots (or zeroes) of a real-valued function.
The formula for Newton's method is: x_{n+1} = x_n - \(\frac{f(x_n)}{f'(x_n)}\). This formula uses the function value and its derivative at the current approximation to find the next approximation.
For the initial estimate x_0 = 2, locate this point on the graph. At x = 2, observe the function value f(2) and the slope of the tangent line, which is given by the derivative f'(2).
Graphically, draw the tangent line at x = 2. The point where this tangent line intersects the x-axis is the next approximation, x_1.
Repeat the process using x_1 as the new initial estimate. Continue iterating until the sequence converges to a root or until the approximations stabilize.

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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. The formula x_{n+1} = x_n - f(x_n)/f'(x_n) updates the guess based on the function's slope, allowing convergence to a root, provided the initial guess is sufficiently close.
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Euler's Method

Convergence and Divergence

In the context of iterative methods like Newton's, convergence refers to the process where successive approximations get closer to the actual root of the function. Divergence occurs when the approximations move away from the root, often influenced by the choice of the initial estimate. Understanding these behaviors is crucial for predicting the success of the method.
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Graphical Interpretation

Graphical interpretation involves analyzing the function's graph to understand the behavior of Newton's Method visually. By plotting the function and its tangent lines at various initial estimates, one can observe how the approximations evolve. This visual approach helps in identifying regions of convergence and divergence, enhancing comprehension of the method's effectiveness.
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Related Practice
Textbook Question

Finding Antiderivatives

In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.

-(1/3)x⁻⁴ᐟ³

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Textbook Question

Finding Antiderivatives

In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.

√x + 1/√x

Textbook Question

Finding Antiderivatives

In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.

x² − 2x + 1

Textbook Question

Applications


Suppose that f(x) = d/dx (1 − √x) and g(x) = d/dx (x + 2).


Find:


∫[−f(x)] dx

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Textbook Question

[Technology Exercises] When solving Exercises 14–30, you may need to use appropriate technology (such as a calculator or a computer).

27. Converging to different zeros Use Newton's method to find the zeros of f(x)=4x^4-4x^2 using the given starting values.

c. x_0 = 0.8 and x_0 = 2, lying in (√2/2, ∞)

Textbook Question

Finding Antiderivatives

In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.

sin πx − 3sin 3x

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