BackNumbers, Functions, and Foundations of Calculus
Study Guide - Smart Notes
Tailored notes based on your materials, expanded with key definitions, examples, and context.
Numbers and Functions
What is a Number?
Calculus begins with understanding the types of numbers used in mathematics, especially real numbers, and how they relate to functions.
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: All rational and irrational numbers; can be represented on the number line.
Decimal Expansions: Rational numbers have either terminating or repeating decimals. Irrational numbers have non-repeating, non-terminating decimals.
Example: (repeating), (non-repeating)
The Real Number Line and Intervals
The real number line is a visual representation of all real numbers. Intervals are subsets of the real line, defined by endpoints.
Closed Interval: includes endpoints and .
Open Interval: excludes endpoints and .
Distance on the Number Line: The distance between and is .
Set Notation
Sets are collections of numbers. Intervals and other sets are often described using set notation.
Example: is the set of all such that .
Intersection: is the set of elements in both and .
Functions
Definition of a Function
A function is a rule that assigns to each element in a set (the domain) exactly one element (the range).
Notation: denotes the value of the function at .
Domain: The set of all for which is defined.
Range: The set of all possible values can take.
Graphing a Function
The graph of a function is the set of all points in the plane, where is in the domain of .
Example: The graph of is a curve in the -plane.
Vertical Line Test: A curve is the graph of a function if and only if no vertical line intersects the curve more than once.
Linear Functions
A linear function has the form , where is the slope and is the -intercept.
Graph: The graph is a straight line.
Slope Formula:
Domain and Range from Formulas
To find the domain and range of a function given by a formula, determine for which the formula makes sense (e.g., no division by zero, no square roots of negative numbers).
Example: For , the domain is all real numbers except .
Example: For , the domain is .
Functions in Real Life
Functions model relationships in science, engineering, and everyday life, such as distance, speed, and growth.
Example: The distance between two points on a line:
Piecewise Defined Functions
Some functions are defined by different formulas on different intervals.
Example:
Summary Table: Types of Numbers
Type | Definition | Example |
|---|---|---|
Integer | Whole numbers (positive, negative, zero) | |
Rational | Can be written as | |
Irrational | Cannot be written as | |
Real | All rational and irrational numbers |
Summary Table: Types of Intervals
Interval Type | Notation | Includes Endpoints? |
|---|---|---|
Closed | Yes | |
Open | No | |
Half-Open | or | One endpoint |
Additional info:
These notes cover foundational concepts for Calculus, including numbers, functions, domains, ranges, and graphing, which are essential for understanding limits, derivatives, and integrals in later chapters.
