Domanda del libro di testo
Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
-sec²(3x/2)
31
views
Ch. 4 - Applications of Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
TOC.tableOfContents
- 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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 4, Problema 4.6.27c
[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, ∞)
Guida verificata passo dopo passo1
Step 1: Recall Newton's method formula: x_{n+1} = x_n - f(x_n) / f'(x_n). This iterative method is used to approximate the zeros of a function by starting with an initial guess x_0.
Step 2: Compute the derivative of the given function f(x) = 4x^4 - 4x^2. Using the power rule, f'(x) = 16x^3 - 8x.
Step 3: Substitute the given initial values x_0 = 0.8 and x_0 = 2 into the Newton's method formula. For each starting value, calculate f(x_0) and f'(x_0), then compute the next iteration x_1 using the formula x_{n+1} = x_n - f(x_n) / f'(x_n).
Step 4: Repeat the iterative process for each starting value (x_0 = 0.8 and x_0 = 2) until the difference between successive approximations (|x_{n+1} - x_n|) is smaller than a chosen tolerance (e.g., 10^-6). This ensures convergence to a zero of the function.
Step 5: Analyze the results. For x_0 = 0.8, the method should converge to a zero in the interval (√2/2, ∞). Similarly, for x_0 = 2, the method should converge to another zero in the same interval. Use technology (e.g., a calculator or computer) to perform the iterations efficiently.
Risposta video verificata per un problema simile:
Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
6mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Newton's Method
Newton's Method is an iterative numerical technique used to find approximate solutions to equations, particularly for finding roots (or zeros) of functions. The method starts with an initial guess and refines it using the formula x_{n+1} = x_n - f(x_n)/f'(x_n), where f' is the derivative of f. This process continues until the approximation converges to a desired level of accuracy.
Video consigliato:
07:33
Euler's Method
Zeros of a Function
The zeros of a function, also known as roots, are the values of x for which the function f(x) equals zero. Finding these points is crucial in various applications, including solving equations and analyzing the behavior of functions. In the context of calculus, identifying zeros helps in understanding the function's graph and its intersections with the x-axis.
Video consigliato:
6:37Zero and Negative Rules
Convergence
Convergence in numerical methods refers to the process by which a sequence of approximations approaches a specific value, often the true solution of an equation. In the context of Newton's Method, convergence is influenced by the choice of the initial guess and the nature of the function. A good initial guess can lead to rapid convergence, while a poor choice may result in divergence or slow convergence.
Video consigliato:
Percorso guidato
04:06Improper Integrals: Infinite Intervals Example 2
Pratica correlata
Domanda del libro di testo
Theory and Examples
Sketch the graph of a differentiable function y = f(x) that has a local maximum at (1, 1) and a local minimum at (3, 3).
217
views
Domanda del libro di testo
Finding displacement from an antiderivative of velocity
a. Suppose that the velocity of a body moving along the s-axis is
ds/dt = v = 9.8t − 3.
iii. Now find the body’s displacement from t = 1 to t = 3 given that s = s₀ when t = 0.
29
views
Domanda del libro di testo
Analyzing Functions from Derivatives
Answer the following questions about the functions whose derivatives are given in Exercises 1–14:
c. At what points, if any, does f assume local maximum or minimum values?
f′(x) = (x − 1)(x + 2)(x − 3)
256
views
Domanda del libro di testo
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
c. x_0=2
194
views
Domanda del libro di testo
Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
2 - 5 / x²
33
views