notebook/notes/calculus/intervals.md

title	TARGET DECK	FILE TAGS	tags
Intervals	Obsidian::STEM	calculus::intervals	calculus

Overview

An interval corresponds to a continuous segment of the real number line. There are a few different types. For all a, b \in \mathbb{R} satisfying a < b:

• [a, b] denotes a closed interval, all x satisfying a \leq x \leq b;
• (a, b) denotes an open interval, all x satisfying a < x < b;
• (a, b] denotes a half-open interval, all x satisfying a < x \leq b;
• [a, b) denotes a half-open interval, all x satisfying a \leq x < b.

%%ANKI Basic Let a, b \in \mathbb{R} and consider interval [a, b]. How is a and b assumed to relate? Back: a < b Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Cloze Let a, b \in \mathbb{R} s.t. a < b. Then {[a, b]} denotes a {closed} interval. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic Let a, b \in \mathbb{R} s.t. a < b. How is [a, b] expressed as a chain of inequalities? Back: As all x satisfying a \leq x \leq b. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Cloze Let a, b \in \mathbb{R} s.t. a < b. Then {(a, b)} denotes an {open} interval. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic Let a, b \in \mathbb{R} s.t. a < b. How is (a, b) expressed as a chain of inequalities? Back: As all x satisfying a < x < b. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Cloze Let a, b \in \mathbb{R} s.t. a < b. Then {(a, b] and [a, b)} denote {half-open} intervals. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic Let a, b \in \mathbb{R} s.t. a < b. How is (a, b] expressed as a chain of inequalities? Back: As all x satisfying a < x \leq b. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic Let a, b \in \mathbb{R} s.t. a < b. How is [a, b) expressed as a chain of inequalities? Back: As all x satisfying a \leq x < b. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Cloze Interval {(a, b)} is called the {interior} of interval [a, b]. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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Partitions

Let a, b \in \mathbb{R} such that a < b. A partition P of interval [a, b] is a set of points x_0 = a, x_1, \ldots, x_{n-1}, x_n = b satisfying $x_0 < x_1 < \cdots < x_{n-1} < x_n.$

We use the symbol P = \{x_0, x_1, \ldots, x_n\} to designate this partition.

%%ANKI Basic Let a, b \in \mathbb{R} s.t. a < b. What is a partition of interval [a, b]? Back: A set of points x_0 = a, x_1, \ldots, x_{n-1}, x_n = b satisfying $x_0 < x_1 < \cdots < x_{n-1} < x_n.$ Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic Let a, b \in \mathbb{R} s.t. a < b. How is a partition of interval [a, b] denoted? Back: As \{a, x_1, \ldots, x_{n-1}, b\} where a < x_1 < \cdots < x_{n-1} < b. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic Let a, b \in \mathbb{R} s.t. a < b and P = \{a, x_1, \ldots, x_{n-1}, b\} be a partition of [a, b]. What are P's closed subintervals? Back: [a, x_1], [x_1, x_2], \ldots, [x_{n-1}, b] Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic Let a, b \in \mathbb{R} s.t. a < b and P = \{a, x_1, \ldots, x_{n-1}, b\} be a partition of [a, b]. What are P's open subintervals? Back: (a, x_1), (x_1, x_2), \ldots, (x_{n-1}, b) Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic Let a, b \in \mathbb{R} s.t. a < b and P = \{a, x_1, \ldots, x_{n-1}, b\} be a partition of [a, b]. How are the members of P assumed to relate? Back: a < x_1 < \cdots < x_{n-1} < b Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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A refinement P' of some partition P is created by adjoining more subdivision points to those of P. P', also a partition, is said to be finer than P.

%%ANKI Basic Let P be a partition of [a, b]. What is a refinement of P? Back: A partition created by adjoining one or more subdivision points to those of P. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic Let P' be a refinement on partition P of [a, b]. How do P and P' relate? Back: P \subseteq P' Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic Let P' be a refinement on partition P of [a, b]. Which of P or P' have more subdivision points? Back: P' Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Cloze Let P' be a refinement on partition P of [a, b]. Then P' is said to be {finer} than P. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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Given two partitions P_1 and P_2, the common refinement of P_1 and P_2 is the partition formed by adjoining the subdivision points of P_1 and P_2 together.

%%ANKI Basic How is the common refinement of partitions P_1 and P_2 defined? Back: The partition formed by adjoining the subdivision points of P_1 and P_2 together. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic Consider partitions P_1 = \{a, x_1, b\} and P_2 = \{a, x_2, b\}. What are the possible common refinements of P_1 and P_2? Back: \{a, x_1 = x_2, b\}, \{a, x_1, x_2, b\}, or \{a, x_2, x_1, b\}. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic Partition P' = \{0, 1, 2, 3, 4\} is the common refinement of P = \{0, 1, 4\} and what other (minimal) partition? Back: \{0, 2, 3, 4\} Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Cloze Partition P' = \{0, 1, 2, 3, 4\} is a {refinement} of partition P = \{0, 1, 4\}. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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Step Functions

A function s, whose domain is a closed interval [a, b], is called a step function if and only if there exists a #Partitions P = \{a, x_1, \ldots, x_{n-1}, b\} of [a, b] such that s is constant on each open subinterval of P.

At each of the endpoints x_{k-1} and x_k, the function must have some well-defined value.

Step functions are also called piecewise constant functions.

%%ANKI Basic With maximum specificity, what is the domain of a step function? Back: A closed interval. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic Suppose s is a step function with domain [a, b]. This implies existence of what? Back: A partition of [a, b] such that s is constant on each open subinterval. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic Let s be a step function on [a, b]. What of its corresponding partition must be constant? Back: The value along each open subinterval. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic Let s be a step function on [a, b] with partition P. What criteria must the endpoints of P's open subintervals satisfy? Back: They must be well-defined. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Cloze {Step} functions are also called {piecewise constant} functions. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic The following is an example of what kind of function? ! Back: A step function. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic What minimally refined partition is associated with the following step function? ! Back: \{0, 1, 2, 3, 4\} Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic What maximally refined partition is associated with the following step function? ! Back: N/A. There is no maximally refined partition. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic How is the sum of two real functions f and g denoted? Back: f + g Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic Let f and g be real functions with a common domain. What kind of mathematical entity is f + g? Back: A function. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic How is the product of two real functions f and g denoted? Back: f \cdot g Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic Let f and g be real functions with a common domain. What kind of mathematical entity is f \cdot g? Back: A function. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic How is the quotient of two real functions f and g denoted? Back: f / g Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic Let f and g be real functions with a common domain. What kind of mathematical entity is f / g? Back: A function, provided g(x) \neq 0 for all x in g's domain. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic When is the sum of two step functions also a step function? Back: Always. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic Why is the sum of two step functions also a step function? Back: The partition of the sum is the common refinement of the partitions of the summands. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic When is the multiplication of two step functions also a step function? Back: Always. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic Why is the product of two step functions also a step function? Back: The partition of the product is the common refinement of the partitions of the factors. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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%%ANKI Basic When is the quotient of two step functions also a step function? Back: When the divisor has nonzero value along its domain. Reference: Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).

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Bibliography

• Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).