Textbook Question
Finding Extrema from Graphs
In Exercises 1–6, determine from the graph whether the function has any absolute extreme values on [a, b]. Then explain how your answer is consistent with Theorem 1.
Ch. 4 - Applications of Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
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 guidance1
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.
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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 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).
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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.
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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.
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Related Practice
Textbook Question
Each of Exercises 67–88 gives the first derivative of a continuous function y=f(x). Find y'' and then use Steps 2–4 of the graphing procedure described in this section to sketch the general shape of the graph of f.
82. y' = sin t, for 0 ≤ t ≤ 2π
Textbook Question
109. Suppose the derivative of the function y = f(x) is
y'=(x-1)^2(x-2).
At what points, if any, does the graph of f have a local minimum, local maximum, or
point of inflection? (Hint: Draw the sign pattern for y'.)
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.
∫(1 − cot²x) dx
Textbook Question
Roots (Zeros)
Show that the functions in Exercises 19–26 have exactly one zero in the given interval.
f(x) = x⁴ + 3x + 1, [−2, −1]
Textbook Question
Initial Value Problems
Solve the initial value problems in Exercises 71–90.
d³y/dx³ = 6; y″(0) = −8, y′(0) = 0, y(0) = 5