To determine To prove that a continuously differentiable function g ( x ) satisfying | g ' ( x ) | < 1 on a closed interval cannot have two fixed points on that interval. Explanation Given information : The function g ( x ) is a continuously differentiable function. Concept used : Check convergence on two different fixed points. Proof : Assume that there are two fixed points r 1 , r 2 for the continuously differentiable function g ( x ) = y in the closed interval [ a , b ] . If r 1 is the fixed point, it is clear that the fixed point iteration will converge to r 1

3rd Edition

ISBN: 9780134696454

Author: Sauer, Tim

Publisher: Pearson,

See similar textbooks

Concept explainers

Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in ter…

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which …

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the …

Videos

Question

Chapter 1.2, Problem 26E

To determine

To prove that a continuously differentiable function g(x) satisfying |g'(x)|<1 on a closed interval cannot have two fixed points on that interval.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 1.2, Problem 25E

Chapter 1.2, Problem 27E

Students have asked these similar questions

Find the saddle points

Problem #5 Section A of my math class has 110 students. Section B of my math class has 80 students. a). If I randomly select 15 students from the combined classes, in a way that the order of my selection does not matter, what is the probability that all 15 students can from Section A? b). If I randomly select 15 students from the combined classes, in a way that the order of my selection does not matter, what is the probability that all 15 students can from Section B? c). If I randomly select 15 students from the combined classes, in a way that the order of my selection does not matter, what is the probability that all 7 students come from section A and 8 students come from section B?

Problem #6 A special passcode to unlock your phone consists of 4 digits where repeated digits are not allowed. If someone were to randomly guess a 4 digit passcode, what is the probability that they guess your passcode on the first try?

Chapter 1 Solutions

Numerical Analysis

Ch. 1.1 - Use the Intermediate Value Theorem to find an...Ch. 1.1 - Use the Intermediate Value Theorem to find an...Ch. 1.1 - Consider the equations in Exercise 1. Apply two...Ch. 1.1 - Consider the equations in Exercise 2. Apply two...Ch. 1.1 - Consider the equation x4=x3+10 . a. Find an...Ch. 1.1 - Suppose that the Bisection Method with starting...Ch. 1.1 - Prob. 1CP Ch. 1.1 - Use the Bisection Method to find the root to eight...Ch. 1.1 - Use the Bisection Method to locate all solutions...Ch. 1.1 - Prob. 4CP

Ch. 1.1 - Prob. 5CP Ch. 1.1 - Use the Bisection Method to calculate the solution...Ch. 1.1 - Use the Bisection Method to find the two real...Ch. 1.1 - The Hilbert matrix is the nn matrix whose ijth...Ch. 1.1 - Prob. 9CP Ch. 1.1 - A planet orbiting the sun traverses an ellipse....Ch. 1.2 - Find all fixed points of the following gx . a. 3x...Ch. 1.2 - Find all fixed points of the following gx . x+63x2...Ch. 1.2 - Prob. 3E Ch. 1.2 - Show that -1, 0, and 1 are fixed points of the...Ch. 1.2 - For which of the following gx is r=3 a fixed...Ch. 1.2 - For which of the following is a fixed...Ch. 1.2 - Use Theorem 1.6 to determine whether Fixed-Point...Ch. 1.2 - Use Theorem 1.6 to determine whether Fixed-Point...Ch. 1.2 - Find each fixed point and decide whether...Ch. 1.2 - Find each fixed point and decide whether...Ch. 1.2 - Express each equation as a fixed-point problem...Ch. 1.2 - Consider the Fixed-Point Iteration xgx=x20.24 ....Ch. 1.2 - (a) Find all fixed points of. (b) To which of the...Ch. 1.2 - Which of the following three Fixed-Point...Ch. 1.2 - Which of the following three Fixed-Point...Ch. 1.2 - Which of the following three Fixed-Point...Ch. 1.2 - Check that and -1 are roots of. Isolate the term...Ch. 1.2 - Prove that the method of Example 1.6 will...Ch. 1.2 - Explore the idea of Example 1.6 for cube roots. Lf...Ch. 1.2 - Improve the cube root algorithm of Exercise 19 by...Ch. 1.2 - Prob. 21E Ch. 1.2 - Prob. 22E Ch. 1.2 - Assume that gx is continuously differentiable and...Ch. 1.2 - Assume that g is a continuously differentiable...Ch. 1.2 - Prob. 25E Ch. 1.2 - Prove that a continuously differentiable function ...Ch. 1.2 - Prob. 27E Ch. 1.2 - Prob. 28E Ch. 1.2 - Prob. 29E Ch. 1.2 - Prob. 30E Ch. 1.2 - Prob. 31E Ch. 1.2 - Find the set of all initial guesses for which the...Ch. 1.2 - Prob. 33E Ch. 1.2 - Prob. 1CP Ch. 1.2 - Prob. 2CP Ch. 1.2 - Calculate the square roots of the following...Ch. 1.2 - Calculate the cube roots of the following numbers...Ch. 1.2 - Prob. 5CP Ch. 1.2 - Prob. 6CP Ch. 1.2 - Prob. 7CP Ch. 1.3 - Find the forward and backward error for the...Ch. 1.3 - Find the forward and backward error for the...Ch. 1.3 - (a) Find the multiplicity of the root r=0 of...Ch. 1.3 - (a) Find the multiplicity of the root of. (b)...Ch. 1.3 - Find the relation between forward and backward...Ch. 1.3 - Let be a positive integer. The equation defining...Ch. 1.3 - Let be the Wilkinson polynomial. (a) Prove that ...Ch. 1.3 - Let fx=xnaxn1 , and set gx=xn . (a) Use the...Ch. 1.3 - Prob. 1CP Ch. 1.3 - Carry' out Computer Problem 1 for fx=sinx3x3 .Ch. 1.3 - Prob. 3CP Ch. 1.3 - Prob. 4CP Ch. 1.3 - Prob. 5CP Ch. 1.3 - Prob. 6CP Ch. 1.4 - Apply two steps of Newton’s Method with initial...Ch. 1.4 - Apply two steps of Newton’s Method with initial...Ch. 1.4 - Use Theorem 1.11 or 1.12 to estimate the error...Ch. 1.4 - Estimate as in Exercise 3. (a) ; , (b) ; , Ch. 1.4 - Consider the equation 8x412x3+6x2x=0 . For each of...Ch. 1.4 - Sketch a function f and initial guess for which...Ch. 1.4 - Let fx=x47x3+18x220x+8 . Does Newton’s Method...Ch. 1.4 - Prove that Newton’s Method applied to fx=ax+b...Ch. 1.4 - Show that applying Newton’s Method to fx=x2A...Ch. 1.4 - Find the Fixed-Point Iteration produced by...Ch. 1.4 - Use Newton’s Method to produce a quadratically...Ch. 1.4 - Suppose Newton’s Method is applied to the...Ch. 1.4 - (a) The function has a root at . If the error ...Ch. 1.4 - Let denote the Newton’s Method iteration for the...Ch. 1.4 - Each equation has one root. Use Newton’s Method to...Ch. 1.4 - Prob. 2CP Ch. 1.4 - Apply Newton’s Method to find the only root to as...Ch. 1.4 - Carry out the steps of Computer Problem 3 for (a)...Ch. 1.4 - Prob. 5CP Ch. 1.4 - Prob. 6CP Ch. 1.4 - Consider the function fx=esin3x+x62x4x31 on the...Ch. 1.4 - Prob. 8CP Ch. 1.4 - Prob. 9CP Ch. 1.4 - Set fx=54x6+45x5102x469x3+35x2+16x4 . Plot the...Ch. 1.4 - The ideal gas law for a gas at low temperature and...Ch. 1.4 - Prob. 12CP Ch. 1.4 - Prob. 13CP Ch. 1.4 - Prob. 14CP Ch. 1.4 - Prob. 15CP Ch. 1.4 - Prob. 16CP Ch. 1.4 - Consider the national population growth model...Ch. 1.5 - Prob. 1E Ch. 1.5 - Apply two steps of the Method of False Position...Ch. 1.5 - Apply two steps of Inverse Quadratic Interpolation...Ch. 1.5 - A commercial fisher wants to set the net at a...Ch. 1.5 - Prob. 5E Ch. 1.5 - If the Secant Method converges to, , and , then...Ch. 1.5 - Consider the following four methods for...Ch. 1.5 - Prob. 1CP Ch. 1.5 - Use the Method of False Position to find the...Ch. 1.5 - Prob. 3CP Ch. 1.5 - Prob. 4CP Ch. 1.5 - Prob. 5CP Ch. 1.5 - Prob. 6CP Ch. 1.5 - Write a MATLAB function file for f . The...Ch. 1.5 - Plot f on , . You may use the @ symbol as...Ch. 1.5 - Reproduce Figure 1.15. The MATLAB commands and...Ch. 1.5 - Solve the forward kinematics problem for the...Ch. 1.5 - Prob. 5SA Ch. 1.5 - Find a strut length p2 , with the rest of the...Ch. 1.5 - Calculate the intervals in p2 , with the rest of...Ch. 1.5 - Prob. 8SA

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

To prove that a continuously differentiable function g ( x ) satisfying | g ' ( x ) | &lt; 1 on a closed interval cannot have two fixed points on that interval.

Concept explainers

Videos

To prove that a continuously differentiable function g(x) satisfying |g'(x)|<1 on a closed interval cannot have two fixed points on that interval.

Want to see the full answer?

Chapter 1 Solutions

To prove that a continuously differentiable function g ( x ) satisfying | g ' ( x ) | < 1 on a closed interval cannot have two fixed points on that interval.