Numbers and Functions

1.1 Different Kinds of Numbers

Calculus begins with understanding the types of numbers used in mathematics. The positive integers are 1, 2, 3, ..., and the negative integers are ..., -3, -2, -1. Together, these form the integers. Rational numbers are those that can be written as fractions of integers, while irrational numbers (such as ) cannot be expressed as a ratio of integers.

• Rational numbers: where are integers and

• Irrational numbers: Numbers that cannot be written as a fraction, e.g.,

• Real numbers: All rational and irrational numbers

Intervals are used to describe sets of real numbers between two endpoints. For example, includes all such that .

1.2 Sets and Intervals

Sets are collections of numbers. Intervals are specific sets that include all numbers between two endpoints, possibly including or excluding the endpoints.

• Closed interval: includes and

• Open interval: excludes and

• Half-open interval: or includes one endpoint

1.3 Functions

A function is a rule that assigns to each input (from the domain) exactly one output (in the range). Functions can be represented by formulas, graphs, or tables.

• Domain: The set of all possible input values

• Range: The set of all possible output values

• Graph of a function: The set of points in the plane

Linear functions have the form and their graphs are straight lines.

1.4 Inverse Functions and Implicit Functions

An inverse function reverses the effect of the original function. If maps to , then maps back to . Implicit functions are defined by equations that relate and without explicitly solving for one variable.

• Example: defines implicitly as

Limits and Continuous Functions

2.1 Informal Definition of Limits

The limit of a function as approaches is the value that gets closer to as gets closer to . This is written as .

• Example:

2.2 Formal Definition of Limits

The formal definition uses (epsilon) and (delta) to describe how close gets to as gets close to :

• such that

2.3 Properties of Limits

• Sum:

• Product:

• Quotient: (if denominator not zero)

2.4 Left and Right Limits

The left limit and right limit describe the behavior of as approaches from the left or right, respectively.

2.5 Continuity

A function is continuous at if . Polynomials and many other functions are continuous everywhere in their domains.

2.6 The Squeeze (Sandwich) Theorem

If for all near , and , then .

Derivatives

3.1 The Tangent Line and Rate of Change

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

• Definition:

• Geometric interpretation: Slope of the tangent line at

• Physical interpretation: Instantaneous velocity if is position as a function of time

3.2 Examples of Derivatives

• Power rule:

• Constant rule:

• Sum rule:

• Product rule:

• Quotient rule:

3.3 Differentiability and Continuity

If a function is differentiable at a point, it is also continuous at that point. However, not all continuous functions are differentiable.

• Example: is continuous everywhere but not differentiable at

3.4 Applications of Derivatives

• Velocity: The derivative of position with respect to time

• Acceleration: The derivative of velocity with respect to time

• Rate of change: Used in physics, economics, biology, and other fields

3.5 Table: Differentiation Rules

Examples and Exercises

• Find the domain and range of

• Compute

• Find the derivative of

• Determine if is differentiable at

Additional info:

• These notes are based on the first chapters of a standard Calculus I college textbook, covering foundational topics in functions, limits, and derivatives.

• Further chapters would include techniques of differentiation, applications, and integration, as indicated in the table of contents.