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Ch. 2 - Limits and Continuity
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Hass 15th EditionCh. 2 - Limits and Continuity문제 2.3.5
2장, 문제 2.3.5

Centering Intervals About a Point


In Exercises 1–6, sketch the interval (a,b), on the x-axis with the point c inside. Then find a value of δ>0 such that a < x < b whenever 0 < |x−c| < δ.


a=4/9, b=4/7, c=1/2

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First, understand the problem: We need to find a value of δ such that the interval (a, b) is centered around the point c, and for any x in this interval, the condition 0 < |x - c| < δ holds true.
Identify the given values: a = 4/9, b = 4/7, and c = 1/2. These values represent the endpoints of the interval and the center point, respectively.
Calculate the distance from c to each endpoint of the interval. This involves finding |c - a| and |c - b|. These distances will help determine the maximum allowable δ.
Compute |c - a|: Convert the fractions to a common denominator if necessary, and find the absolute difference between c and a.
Compute |c - b|: Similarly, convert the fractions to a common denominator if necessary, and find the absolute difference between c and b. The value of δ will be the smaller of these two distances, ensuring that the interval (a, b) is centered around c with the condition 0 < |x - c| < δ.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Intervals

An interval is a set of real numbers that contains all numbers between any two numbers in the set. In this context, the interval (a, b) represents all values x such that a < x < b. Understanding how to represent and visualize intervals on the number line is crucial for determining the relationship between the point c and the interval endpoints.
추천 영상:
02:59
Finding Area Between Curves that Cross on the Interval

Delta (δ) in Limits

In calculus, δ (delta) is used to define the precision of a limit. Specifically, it represents a distance from a point c such that if x is within δ of c (i.e., |x - c| < δ), then x is also within the interval (a, b). Finding an appropriate δ ensures that the values of x remain within the specified interval, which is essential for limit definitions and continuity.
추천 영상:
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One-Sided Limits

Absolute Value

The absolute value of a number measures its distance from zero on the number line, disregarding direction. In the expression |x - c|, it indicates how far x is from the point c. This concept is fundamental in determining the conditions under which x remains within the interval (a, b) when considering values close to c.
추천 영상:
06:37
Average Value of a Function
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