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, ...; the negative integers are ..., -3, -2, -1. Together, these form the integers. Rational numbers are fractions of integers, and real numbers include all rational and irrational numbers (such as and ).

• 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, represented on the number line.

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

1.2 The Real Number Line and Intervals

The real numbers can be visualized as points on a line. Intervals are subsets of real numbers between two endpoints.

• Closed interval: includes both endpoints and .

• Open interval: excludes both endpoints.

• Half-open intervals: or include only one endpoint.

Example: is the set of all real numbers such that .

1.3 Sets and Set Operations

Sets are collections of objects (numbers). The union of two sets and is the set of elements in either or . The intersection is the set of elements in both and .

• Union:

• Intersection:

Functions

3.1 Definition of a Function

A function assigns to each element in a set exactly one element in a set . The set is the domain, and the set of all possible is the range.

• Notation:

• Example: with domain and range .

3.2 Linear Functions

A linear function has the form , where is the slope and is the y-intercept. Its graph is a straight line.

• Example:

3.3 Domain and Range

The domain of a function is the set of all input values for which the function is defined. The range is the set of all possible output values.

• Example: For , the domain is and the range is .

3.4 Inverse Functions

An inverse function reverses the effect of , so that and for all in the domain of .

• Example: If , then .

Limits and Continuity

2.1 Informal Definition of Limit

The limit of as approaches is if gets arbitrarily close to as gets arbitrarily close to .

• Notation:

• Example:

2.2 Formal Definition of Limit (Epsilon-Delta)

For every , there exists such that if , then .

2.3 Properties of Limits

• Sum:

• Product:

• Quotient: , provided

2.4 Left and Right Limits

• Left-hand limit:

• Right-hand limit:

• The limit exists if and only if both one-sided limits exist and are equal.

2.5 Continuity

A function is continuous at if .

• Polynomials and rational functions are continuous on their domains.

Derivatives

3.1 Definition of the Derivative

The derivative of at is the limit:

• It represents the instantaneous rate of change of at .

• The derivative is the slope of the tangent line to the graph of at .

3.2 Notation

• Lagrange:

• Leibniz:

3.3 Differentiation Rules

3.4 Differentiability and Continuity

If a function is differentiable at a point, it is also continuous at that point. However, continuity does not guarantee differentiability.

• Example: is continuous everywhere but not differentiable at .

3.5 Applications of the Derivative

• Instantaneous velocity: The derivative of position with respect to time.

• Rate of change: The derivative measures how a quantity changes as its input changes.

Additional info:

• These notes cover the foundational topics in Calculus I, including numbers, functions, limits, continuity, and the basics of differentiation, as outlined in the course contents.