Numbers and Functions

1. What is a Number?

Calculus is built on the concept of functions of one real variable. Understanding the types of numbers used in calculus is essential for grasping its foundations.

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

• Zero: The number 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, e.g., , .

• Real Numbers: All rational and irrational numbers together.

Decimal Expansions: Rational numbers have either terminating or repeating decimals. Irrational numbers have non-repeating, non-terminating decimals.

• Example: (repeating)

• Example: (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.

• Closed Interval: includes both endpoints and .

• Open Interval: excludes both endpoints.

• Half-Open Interval: or includes one endpoint.

Set Notation: Used to describe collections of numbers satisfying certain properties.

• Example: is the set of all such that .

• Example: is the set of all positive real numbers.

3. Functions

A function assigns to each input (from its domain) exactly one output (in its range). Functions are central to calculus.

• Definition: A function from a set to a set is a rule that assigns to each in a unique in .

• Domain: The set of all possible inputs for which the function is defined.

• Range: The set of all possible outputs.

Graphing a Function: The graph of is the set of points for all in the domain.

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

Linear Functions: Functions of the form .

• Slope (): Measures the rate of change.

• Intercept (): The value where the graph crosses the -axis.

• Example:

Piecewise Functions: Functions defined by different formulas on different intervals.

4. Domain and Range

Finding the domain and range is a key skill in calculus.

• Domain: All for which is defined.

• Range: All possible values of .

• Example: For , domain is , range is .

5. The Vertical Line Test

The vertical line test is used to determine if a graph represents a function.

• If any vertical line crosses the graph more than once, it is not a function.

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

6. Functions in Real Life

Functions model relationships in science, engineering, and everyday life.

• Distance Function: measures the distance between two points.

• Temperature, population growth, and speed can all be modeled as functions.

7. Exercises and Applications

Practice problems help reinforce understanding of numbers, intervals, and functions.

• Find the domain and range of .

• Determine if the graph of represents a function.

• Classify numbers as rational or irrational.

8. Table: Types of Numbers

The following table summarizes the main types of numbers used in calculus:

Additional info: These notes cover foundational concepts for calculus, including types of numbers, intervals, set notation, and the definition and properties of functions. Understanding these topics is essential for success in calculus and its applications.