BackMATH 221: First Semester Calculus – Study Notes (Chapter 1: Numbers and Functions)
Study Guide - Smart Notes
Tailored notes based on your materials, expanded with key definitions, examples, and context.
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 foundational number systems used in calculus.
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. Every real number can be represented by a (possibly infinite) decimal expansion.
Example: (repeating decimal), (non-repeating, non-terminating decimal).
2. The Real Number Line and Intervals
The real number line is a geometric representation of all real numbers as points on a straight line. Intervals are subsets of the real line defined by inequalities.
Closed Interval:
Open Interval:
Half-Open Intervals: or
Distance on the Real Line: The distance between two numbers and is .
Example: The interval includes all real numbers greater than -1 and up to and including 2.
3. Set Notation
Sets are collections of numbers or objects. In calculus, set notation is used to describe intervals, domains, and ranges.
Set-builder notation: means the set of all such that .
Union: is the set of all elements in or .
Intersection: is the set of all elements in both and .
Example: is the set of all such that , i.e., or .
4. Functions
Functions are central to calculus. A function assigns to each input exactly one output.
Definition: A function is a rule that assigns to each in its domain a unique value .
Domain: The set of all for which is defined.
Range: The set of all possible values can take.
Piecewise Functions: Functions defined by different formulas on different intervals.
Example: has domain and range .
5. 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.
Linear Functions: is a straight line with slope and -intercept .
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.
6. Domain and Range: Examples
To find the domain and range of a function, determine for which the formula makes sense and what values are possible.
Example: has domain and range .
Example: has domain and range .
7. Functions in Real Life
Functions can model real-world relationships, such as distance over time, population growth, or physical laws.
Example: The distance from the origin after seconds if moving at constant speed is .
8. Summary Table: Types of Numbers
The following table summarizes the main types of numbers discussed:
Type | Symbol | Examples | Description |
|---|---|---|---|
Natural Numbers | \( \mathbb{N} \) | 1, 2, 3, ... | Counting numbers |
Integers | \( \mathbb{Z} \) | ..., -2, -1, 0, 1, 2, ... | Whole numbers, positive and negative |
Rational Numbers | \( \mathbb{Q} \) | , , | Fractions of integers |
Irrational Numbers | – | , | Non-repeating, non-terminating decimals |
Real Numbers | \( \mathbb{R} \) | All above | All points on the number line |
Additional info: These notes are based on the first chapter of a standard college calculus course, covering foundational concepts necessary for further study in calculus.
