Skip to main content
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 4.6.2
Chapter 4, Problem 4.6.2

Root Finding
2. Use Newton's method to estimate the one real solution of x^3 +3x + 1 = 0. Start with x_0 = 0 and then find x_2.

Verified step by step guidance
1
Step 1: Understand Newton's Method. It is an iterative method to approximate the roots of a real-valued function. The formula is: x_{n+1} = x_n - f(x_n) / f'(x_n).
Step 2: Identify the function and its derivative. Here, f(x) = x^3 + 3x + 1. Calculate the derivative: f'(x) = 3x^2 + 3.
Step 3: Start with the initial guess x_0 = 0. Calculate f(x_0) and f'(x_0). Substitute these into the Newton's method formula to find x_1.
Step 4: Use the result from Step 3 to find x_1. Substitute x_1 back into the formula to calculate x_2 using the same process: x_2 = x_1 - f(x_1) / f'(x_1).
Step 5: Continue the iteration process if needed, but for this problem, you only need to find up to x_2. Ensure each step is calculated accurately to improve the approximation.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3m
Was this helpful?

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 approximate the roots of a real-valued function. Starting with an initial guess, x_0, the method uses the function and its derivative to generate a sequence of approximations that converge to a root. The formula is x_{n+1} = x_n - f(x_n)/f'(x_n).
Recommended video:
06:30
Disk Method Using y-Axis

Derivative

The derivative of a function measures how the function's output value changes as its input changes. It is essential in Newton's Method as it helps determine the slope of the tangent line at a given point, which is used to find the next approximation. For the function f(x) = x^3 + 3x + 1, the derivative is f'(x) = 3x^2 + 3.
Recommended video:
05:44
Derivatives

Convergence of Iterative Methods

Convergence refers to the process of approaching a final value as iterations proceed. In the context of Newton's Method, convergence means that the sequence of approximations gets closer to the actual root. The choice of the initial guess, x_0, and the nature of the function affect the speed and success of convergence.
Recommended video:
06:30
Disk Method Using y-Axis
Related Practice
Textbook Question

Checking the Mean Value Theorem


Which of the functions in Exercises 7–12 satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers.


f(x) = {2x − 3, 0 ≤ x ≤ 2

6x − x² − 7, 2 < x ≤ 3

205
views
Textbook Question

Finding Extrema from Graphs


In Exercises 11–14, match the table with a graph.


171
views
Textbook Question

99. In Exercises 99 and 100, the graph of f' is given. Determine x-values corresponding to inflection points for the graph of f.

144
views
Textbook Question

Finding Critical Points


In Exercises 41–50, determine all critical points and all domain endpoints for each function.


g(x) = √(2x − x²)

197
views
Textbook Question

Finding Indefinite Integrals


In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.


∫(2x³ − 5x + 7) dx

16
views
Textbook Question

Checking the Mean Value Theorem


Which of the functions in Exercises 7–12 satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers.


f(x) = x²ᐟ³, [−1, 8]

163
views