Numbers and Functions

1.1 Different Kinds of Numbers

Calculus begins with understanding the types of numbers used in mathematics. The main sets are:

• Positive Integers:

• Negative Integers:

• 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 together.

Example: is irrational because it cannot be expressed as a fraction of two integers.

1.2 The Real Number Line and Intervals

The real number line is a geometric representation of all real numbers. Intervals are subsets of the real line:

• Open Interval: contains all such that .

• Closed Interval: contains all such that .

• Half-Open Interval: or .

Example: The interval includes all real numbers between 0 and 1, including the endpoints.

1.3 Sets and Set Operations

Sets are collections of numbers. Operations include:

• Union: is the set of elements in or .

• Intersection: is the set of elements in both and .

2. Functions

A function is a rule that assigns to each element in a set (domain) a unique element in a set (range).

• Domain: The set of all possible input values.

• Range: The set of all possible output values.

Example: has domain and range .

2.1 Graphs of Functions

The graph of a function is the set of points in the plane. Important properties include:

• Vertical Line Test: A graph represents a function if no vertical line intersects it more than once.

Example: The graph of passes the vertical line test.

2.2 Linear Functions

A linear function has the form , where is the slope and is the y-intercept.

Example: is a linear function.

2.3 Inverse Functions and Implicit Functions

An inverse function reverses the effect of the original function. If , then .

Implicit functions are defined by equations not solved for one variable explicitly.

Example: The equation defines implicitly as a function of .

Limits and Continuity

3.1 Informal Definition of Limits

The limit of as approaches is the value that gets closer to as gets closer to .

Notation:

Example:

3.2 Formal Definition of Limits

The formal (epsilon-delta) definition states: if for every , there exists such that whenever , .

3.3 Properties of Limits

• Sum Rule:

• Product Rule:

• Quotient Rule: (if denominator not zero)

3.4 Left and Right Limits

Left limit: is the value as approaches from the left.

Right limit: is the value as approaches from the right.

3.5 Continuity

A function is continuous at if .

• Polynomials are continuous everywhere.

• Rational functions are continuous where the denominator is not zero.

3.6 The Sandwich (Squeeze) Theorem

If for all near , and , then .

Derivatives

4.1 The Tangent Line and Rate of Change

The derivative of a function at a point measures the slope of the tangent line to its graph at that point. It represents the instantaneous rate of change.

Definition: The derivative of at is

Example: For , .

4.2 Differentiability Implies Continuity

If is differentiable at , then is continuous at .

4.3 The Differentiation Rules

4.4 The Power Rule for Rational Exponents

If where is rational, then .

4.5 Non-Differentiable Functions

Some functions are not differentiable at certain points, such as those with corners or cusps.

Example: is not differentiable at .

Applications and Examples

5.1 Instantaneous Velocity

The derivative of position with respect to time gives the instantaneous velocity:

5.2 Acceleration

The derivative of velocity with respect to time gives acceleration:

5.3 Rate of Change in General

For any function , the rate of change over an interval is:

The instantaneous rate of change is the derivative.

Summary Table: Differentiation Rules

Key Concepts and Examples

• Function:

• Limit:

• Derivative:

• Product Rule:

• Quotient Rule:

Additional info: These notes are based on the first chapters of a college Calculus I course, covering foundational concepts in numbers, functions, limits, continuity, and introductory differentiation. The content is suitable for exam preparation and provides a self-contained overview of the main ideas and formulas.