Numbers and Functions

1. What is a Number?

Calculus relies on a solid understanding of different types of numbers and their properties. This section introduces the main classes of numbers used in calculus and their significance.

• Positive Integers: The simplest numbers, e.g., 1, 2, 3, ...

• Zero: The integer 0, which is neither positive nor negative.

• Negative Integers: ..., -3, -2, -1

• Rational Numbers: Numbers that can be written as a fraction of two integers, , where .

• Irrational Numbers: Numbers that cannot be written as a fraction of two integers (e.g., , ).

• Real Numbers: The set of all rational and irrational numbers. Real numbers can be represented as points on the number line.

Decimal Expansions: Rational numbers have decimal expansions that either terminate or repeat. Irrational numbers have non-terminating, non-repeating decimals.

• Example: (repeating), (non-repeating)

Distance on the Number Line: The distance between two numbers and is .

2. Intervals and Set Notation

Intervals are used to describe sets of real numbers between two endpoints. Set notation is a concise way to describe collections of numbers.

• Closed Interval:

• Open Interval:

• Half-Open Intervals: or

• Set-builder Notation:

Example: The set consists of all real numbers such that , i.e., or .

3. Functions

Functions are fundamental objects in calculus, describing how one quantity depends on another.

• Definition: A function assigns to each element in a set (the domain) exactly one element in another set (the range).

• Notation: ,

• Domain: The set of all for which is defined.

• Range: The set of all possible values can take.

Example: has domain (all real numbers), range .

3.1. Graphing a Function

The graph of a function is the set of all points in the plane. The domain is the set of -values for which the function is defined, and the range is the set of -values the function attains.

• Example: The graph of is a curve in the -plane.

3.2. Linear Functions

A linear function has the form , where is the slope and is the -intercept.

• Graph: A straight line with slope and -intercept .

• Equation of a line through with slope :

3.3. Domain and Range from Formulas

To find the domain of a function given by a formula, determine all for which the formula makes sense (e.g., avoid division by zero, square roots of negative numbers).

• Example: has domain (all real numbers except 0).

• Example: has domain .

3.4. The Vertical Line Test

A curve in the plane is the graph of a function if and only if no vertical line intersects the curve more than once. This is called the vertical line test.

• Example: The graph of passes the vertical line test; the graph of a circle does not.

3.5. Functions in Real Life

Functions are used to model relationships between quantities in science, engineering, and everyday life. For example, the distance traveled as a function of time, or the temperature as a function of location.

4. Summary Table: Types of Numbers

Additional info: These notes are based on the first chapter of a standard college calculus course, introducing foundational concepts necessary for further study in calculus, such as limits, derivatives, and integrals.