2. Let k be the final digit in your student number and let y = 3k + 2. The purpose of this question is to use Newton's method to obtain an approximation to √√7, so let f(x) = x² - y. (a) Rewrite the equation f(x) = 0 to express x in terms of y. (b) Let the initial guess x0 = k + 1. Use this value of xo and equation (1) to calculate x1 and x2. (c) Compare your value of x2 to the value you get for √ by using a calculator. Without changing the initial guess, how could you use Newton's Method to obtain a better approximation? [3 marks] In this assignment we will introduce a standard method to find approximate solutions to nonlinear equations. Suppose that f is a nonlinear differentiable function and that we wish to solve the equation f(x) = 0. Recall that solving f(x) = 0 is equivalent to finding an x-intercept of this graph and that such an equation may have no solutions or many solutions. The method we outline here, called Newton's Method, is an approach for finding just one solution (if such a solution exists). Newton's Method starts with an initial guess for the solution, 20, and improves upon it as follows: 1. Calculate the tangent line to y = f(x) at (xo, f(xo)) 2. Find the x-intercept of the tangent line and set this equal to 1. This process is illustrated graphically in figure 1. If the conditions are right, a₁ will be a better approximation to the solution than the previous one, 20, and repeating this process with the new approximation ought to improve it again. Iterating this process will give a sequence of approximations which should approach the actual solution. Y Tangent at (xo, f(x0)) Actual solution (unknown) (xo, f(x0)) I 21 20 y = f(x) Figure 1: The first step of Newton's Method. The next step would involve calculating a tangent line at (x1, f(x1)) and finding the x-intercept x2.
2. Let k be the final digit in your student number and let y = 3k + 2. The purpose of this question is to use Newton's method to obtain an approximation to √√7, so let f(x) = x² - y. (a) Rewrite the equation f(x) = 0 to express x in terms of y. (b) Let the initial guess x0 = k + 1. Use this value of xo and equation (1) to calculate x1 and x2. (c) Compare your value of x2 to the value you get for √ by using a calculator. Without changing the initial guess, how could you use Newton's Method to obtain a better approximation? [3 marks] In this assignment we will introduce a standard method to find approximate solutions to nonlinear equations. Suppose that f is a nonlinear differentiable function and that we wish to solve the equation f(x) = 0. Recall that solving f(x) = 0 is equivalent to finding an x-intercept of this graph and that such an equation may have no solutions or many solutions. The method we outline here, called Newton's Method, is an approach for finding just one solution (if such a solution exists). Newton's Method starts with an initial guess for the solution, 20, and improves upon it as follows: 1. Calculate the tangent line to y = f(x) at (xo, f(xo)) 2. Find the x-intercept of the tangent line and set this equal to 1. This process is illustrated graphically in figure 1. If the conditions are right, a₁ will be a better approximation to the solution than the previous one, 20, and repeating this process with the new approximation ought to improve it again. Iterating this process will give a sequence of approximations which should approach the actual solution. Y Tangent at (xo, f(x0)) Actual solution (unknown) (xo, f(x0)) I 21 20 y = f(x) Figure 1: The first step of Newton's Method. The next step would involve calculating a tangent line at (x1, f(x1)) and finding the x-intercept x2.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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My last digit of student number is 4.
![2. Let k be the final digit in your student number and let y = 3k + 2. The purpose of this question is to
use Newton's method to obtain an approximation to √√7, so let f(x) = x² - y.
(a) Rewrite the equation f(x) = 0 to express x in terms of y.
(b) Let the initial guess x0 = k + 1. Use this value of xo and equation (1) to calculate x1 and x2.
(c) Compare your value of x2 to the value you get for √ by using a calculator. Without changing
the initial guess, how could you use Newton's Method to obtain a better approximation?
[3 marks]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F78d1386b-6fe1-438f-8c94-3f41a9b64a70%2F5aaf8900-8418-4a4d-aca9-e3318872d5ff%2Fr0iswzr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2. Let k be the final digit in your student number and let y = 3k + 2. The purpose of this question is to
use Newton's method to obtain an approximation to √√7, so let f(x) = x² - y.
(a) Rewrite the equation f(x) = 0 to express x in terms of y.
(b) Let the initial guess x0 = k + 1. Use this value of xo and equation (1) to calculate x1 and x2.
(c) Compare your value of x2 to the value you get for √ by using a calculator. Without changing
the initial guess, how could you use Newton's Method to obtain a better approximation?
[3 marks]
Transcribed Image Text:In this assignment we will introduce a standard method to find approximate solutions to nonlinear
equations.
Suppose that f is a nonlinear differentiable function and that we wish to solve the equation
f(x) = 0.
Recall that solving f(x) = 0 is equivalent to finding an x-intercept of this graph and that such an equation
may have no solutions or many solutions. The method we outline here, called Newton's Method, is an
approach for finding just one solution (if such a solution exists).
Newton's Method starts with an initial guess for the solution, 20, and improves upon it as follows:
1. Calculate the tangent line to y = f(x) at (xo, f(xo))
2. Find the x-intercept of the tangent line and set this equal to 1.
This process is illustrated graphically in figure 1.
If the conditions are right, a₁ will be a better approximation to the solution than the previous one, 20,
and repeating this process with the new approximation ought to improve it again. Iterating this process
will give a sequence of approximations which should approach the actual solution.
Y
Tangent at (xo, f(x0))
Actual solution
(unknown)
(xo, f(x0))
I
21 20
y = f(x)
Figure 1: The first step of Newton's Method.
The next step would involve calculating a tangent line at
(x1, f(x1)) and finding the x-intercept x2.
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