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Chapter 2: Problem 6

In Exercises \(1-6,\) sketch the interval \((a, b)\) on the \(x\) -axis with the point \(x_{0}\) inside. Then find a value of \(\delta > 0\) such that for all \( x, 0 < \left|x-x_{0}\right| < \delta \Rightarrow a < x < b\) $$ a=2.7591, \quad b=3.2391, \quad x_{0}=3 $$

Short Answer

Expert verified
The value of \( \delta \) is 0.2391.

Step by step solution

01

Understanding the Problem

We need to find a value for \( \delta > 0 \) such that any \( x \) within the open interval centered around \( x_0 = 3 \) is also within the interval \((a, b) = (2.7591, 3.2391)\). This means we want \( a < x < b \) and simultaneously, \( 0 < |x - x_0| < \delta \).
02

Analyzing the Bounds of the Interval

We start by analyzing the interval \( (a, b) \). We see that \( a = 2.7591 \) and \( b = 3.2391 \). The point \( x_0 \) must lie within this interval, which it does, since \( 2.7591 < 3 < 3.2391 \).
03

Determining the Distance from \( x_0 \) to the Interval Bounds

Next, we calculate the distance from \( x_0 \) to each of the interval bounds: - The distance from \( x_0 \) to \( a \) is \( 3 - 2.7591 = 0.2409 \).- The distance from \( x_0 \) to \( b \) is \( 3.2391 - 3 = 0.2391 \).
04

Choosing \( \delta \)

To ensure \( x \) stays within the interval \((a, b)\) whenever \( 0 < |x - x_0| < \delta \), we choose \( \delta \) to be the smallest of the distances from \( x_0 \) to each boundary, which is \( \delta = 0.2391 \).
05

Confirming the Solution

Check that for the chosen \( \delta = 0.2391 \), we have:- If \( |x - 3| < 0.2391 \), then the smallest \( x \) can be is \( 3 - 0.2391 = 2.7609 \) and the largest is \( 3 + 0.2391 = 3.2391 \). - Therefore, \( 2.7591 < x < 3.2391 \) which satisfies the interval condition. Hence, our choice of \( \delta \) is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Interval Notation
Let鈥檚 begin with understanding interval notation, which is a way to describe a set of numbers lying between two endpoints. In this context, we have the interval \((a, b)\), where the numbers are greater than \(a\) and less than \(b\). This notation helps us clearly communicate which numbers are included in the set. Specifically, when you see parentheses like \((a, b)\), it indicates an open interval. This means the endpoints \(a\) and \(b\) themselves aren鈥檛 included, just everything in between.
Interval notation is very useful in calculus because it simplifies how we express constraints and limits in a problem.
Open Interval
An open interval, such as \((a, b)\), includes all numbers between \(a\) and \(b\) but not the numbers \(a\) and \(b\) themselves. This is different from closed intervals, which do include their endpoints.
In the given exercise, \((a, b) = (2.7591, 3.2391)\), the interval is open, meaning \(2.7591\) and \(3.2391\) are not part of the interval. Instead, only numbers strictly between these values are included. Understanding this helps when determining what the constraints are around any given \(x_0\).
For an open interval, even if \(x_0\) is calculated to be very close to \(a\) or \(b\), it cannot actually be them.
Distance Calculation
To determine how far \(x_0\) is from the interval boundaries, we calculate the distance separately for each side. This involves simple subtraction:
  • The distance from \(x_0\) to the left boundary \(a\) is \(3 - 2.7591 = 0.2409\).
  • The distance from \(x_0\) to the right boundary \(b\) is \(3.2391 - 3 = 0.2391\).
These distances briefly illustrate how close \(x_0\) is to each edge of the interval.
These measurements help determine how much freedom we have when choosing a value for \(x\) that remains in the interval.
Choosing Delta
Choosing \(\delta\) is about ensuring that all \(x\) values near \(x_0\) remain within the interval \((a, b)\). The ideal \(\delta\) is the smallest distance between \(x_0\) and one of the interval's endpoints.
In this scenario, our calculated distances are \(0.2409\) to \(a\) and \(0.2391\) to \(b\). Therefore, the smallest distance, \(0.2391\), is chosen as \(\delta\).
This choice ensures that a point \(x\) will satisfy both \(0 < |x - x_0| < \delta\) and remain within the interval limits of \(2.7591 < x < 3.2391\). By selecting \(\delta = 0.2391\), we guarantee no overshooting occurs toward the interval boundaries.

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