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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
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Tutti i libri di testoHass 15th EditionCh. 4 - Applications of DerivativesProblema 4.6.2
Capitolo 4, Problema 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.

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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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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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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.
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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

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Finding Extrema from Graphs


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


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

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Finding Critical Points


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


g(x) = √(2x − x²)

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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

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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]

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