Answered: Consider the interval [0, 1]. (a) If…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter9: Systems Of Equations And Inequalities

Section: Chapter Questions

Problem 12T

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Question

Consider the interval [0, 1]. (a) If δ(x) = 1/9, find a δ(x)-fine tagged partition of [0, 1]. Does the choice of tags matter in this case? (b) Let δ(x) ={ 1/4 ifx = 0 /
x/3 if0< x ≤ 1. Construct a δ(x)-fine tagged partition of [0,1]. The tinkering required in Exercise 8.1.6 (b) may cast doubt on whether an arbitrary gauge always admits a δ(x)-fine partition. However, it is not too difficult to show that this is indeed the case.

Expert Solution

Step 1

The given interval [0, 1].

(a) $δ (x) = \frac{1}{9}$ , we have to find a $δ (x)$ -fine tagged partition of [0, 1]. Also tell whether the choice tags matter in this case or not?

(b) $δ (x) = \{\begin{cases} \frac{1}{4} & if x = 0 \\ \frac{x}{3} & if 0 < x \leq 1 \end{cases}$

We have to construct a $δ (x)$ -fine tagged partition of [0, 1].

Step 2

Some definitions:

1) Let $δ > 0$ . A partition P is $δ$ -fine if every subinterval $[x_{k - 1}, x_{k}]$ satisfies $x_{k} - x_{k - 1} < δ$ , that is the width of every subinterval is less than $δ$ .

2) A function $δ : [a, b] \to R$ is called a gauge on $[a, b]$ if $δ (x) > 0$ for all $x \in [a, b]$ .

3) Given a particular gauge $δ (x)$ , a tagged partition $(P, {\{c_{k}\}}_{k = 1}^{n})$ is $δ (x)$ -fine if every subinterval $[x_{k - 1}, x_{k}]$ satisfies $x_{k} - x_{k - 1} < δ (c_{k})$ , that is the width of every subinterval is less than $δ (c_{k})$ .

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