BackCalculus I Study Notes: Numbers, Functions, and Graphs
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Numbers and Functions
1. What is a Number?
Calculus begins with understanding the types of numbers used in mathematics, especially real numbers. This section introduces the basic kinds of numbers and their properties.
Positive Integers: The set {1, 2, 3, ...}.
Zero: The number 0, which is neither positive nor negative.
Negative Integers: The set {..., -3, -2, -1}.
Rational Numbers: Numbers that can be written as a fraction , where and are integers and .
Irrational Numbers: Numbers that cannot be written as a fraction, such as or .
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. Set notation is a concise way to specify collections of numbers.
Closed Interval: includes all such that .
Open Interval: includes all such that .
Half-Open Interval: or .
Set Notation:
Example: is the set of all positive real numbers.
3. Functions
A function is a rule that assigns to each input (from its domain) exactly one output (in its range). Functions are central to calculus and are often represented by formulas, graphs, or tables.
Definition: A function from a set to a set assigns to each in a unique in .
Domain: The set of all possible input values for which the function is defined.
Range: The set of all possible output values.
Piecewise Functions: Functions defined by different formulas on different intervals.
Example: The function has domain and range .
4. Graphing Functions
The graph of a function is a visual representation showing how each input is mapped to an output. The graph consists of all points for in the domain.
Linear Functions: is a straight line with slope and -intercept .
Vertical Line Test: A graph represents a function if any vertical line crosses it at most once.
Example: The graph of is a parabola. The graph of is a straight line.
5. Domain and Range Examples
Finding the domain and range of a function is a key skill in calculus.
Example: For , the domain is and the range is .
Example: For , the domain is and the range is .
6. Functions in Real Life
Functions can model real-world relationships, such as distance, speed, and growth. For example, the distance between two points on a line can be described by a function.
Example: The function gives the distance between and .
7. Summary Table: Types of Numbers
Type | Definition | Examples |
|---|---|---|
Positive Integers | Whole numbers greater than zero | 1, 2, 3, ... |
Negative Integers | Whole numbers less than zero | -1, -2, -3, ... |
Rational Numbers | Numbers expressible as , | , , 5 |
Irrational Numbers | Numbers not expressible as | , |
Real Numbers | All rational and irrational numbers | 2, , , |
8. Key Formulas
Distance on the number line:
Linear function:
Domain of : The set of for which is defined
Additional info: These notes cover foundational concepts for Calculus I, including types of numbers, set notation, functions, and graphing. Later chapters (as seen in the table of contents) will cover limits, derivatives, integrals, and applications, which are essential for a full calculus course.
