Numbers and Functions

1.1 What is a Number?

Calculus is fundamentally concerned with functions of real numbers. Understanding the types of numbers used in calculus is essential for further study.

• Positive Integers:

• Negative Integers:

• Zero: $0$

• Rational Numbers: Numbers that can be written as , where and are integers and .

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

• Real Numbers: The set of all rational and irrational numbers. Every real number can be represented by a (possibly infinite) decimal expansion.

Example: (repeating decimal), (non-repeating, non-terminating decimal).

1.2 The Real Number Line and Intervals

The real number line is a geometric representation of all real numbers as points on a line. Intervals are subsets of the real line defined by inequalities.

• Closed Interval: (includes endpoints)

• Open Interval: (excludes endpoints)

• Half-Open Intervals: or

• Distance on the Real Line: The distance between and is .

Example: The interval includes all real numbers between $0, including $0.

1.3 Set Notation

Sets are collections of objects, often numbers. In calculus, sets are used to describe domains and ranges.

• Set-builder notation:

• Union: (elements in or )

• Intersection: (elements in both and )

• Example:

1.4 Functions

A function is a rule that assigns to each element in a set called the domain exactly one element in a set called the range. Functions are the central objects of study in calculus.

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

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

• Range: The set of all possible values can take as varies over the domain.

• Example: has domain and range .

1.5 Graphing a Function

The graph of a function is the set of all points in the plane such that . The graph visually represents the relationship between and .

• 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.

• Example: The graph of is a parabola opening upwards.

1.6 Linear Functions

A linear function is a function of the form , where and are constants. The graph of a linear function is a straight line.

• Slope: represents the rate of change of the function.

• Y-intercept: is the value of when .

• Equation of a line through two points:

Example: has slope $2.

1.7 Domain and Range (Finding and Expressing)

To determine the domain of a function, identify all -values for which the formula makes sense (e.g., avoid division by zero, square roots of negative numbers). The range is the set of all possible output values.

• Example: For , the domain is .

• Example: For , the domain is .

1.8 Functions in Real Life

Functions are used to model relationships in the real world, such as distance, velocity, and population growth. For example, the distance traveled at constant speed over time is .

• Example: The height of a ball thrown upward as a function of time can be modeled by .

1.9 Piecewise-defined Functions

Some functions are defined by different formulas in different parts of their domain. These are called piecewise-defined functions.

• Example:

1.10 Exercises and Practice

Practice problems often involve identifying domains, ranges, and graphing functions. Understanding these foundational concepts is crucial for success in calculus.

Tables

Table: Types of Numbers

Table: Interval Notation

Additional info: These notes are based on the first chapter of a standard college calculus course, covering foundational concepts necessary for understanding limits, derivatives, and integrals in later chapters.