Numbers and Functions

1.1 Different Kinds of Numbers

Calculus begins with understanding the types of numbers used in mathematics. The most common are the positive integers (1, 2, 3, ...), negative integers (-1, -2, -3, ...), and rational numbers (fractions of integers). The 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 simple fraction, e.g., .

• Real numbers: The set of all rational and irrational numbers.

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

1.2 The Real Number Line and Intervals

The real numbers can be represented on a number line. Intervals are subsets of the real line, such as (all numbers between and , including endpoints) or (excluding endpoints).

• Closed interval: includes and .

• Open interval: excludes and .

• Half-open interval: includes but not .

1.3 Sets and Subsets

Sets are collections of numbers. The intersection of two sets and is the set of elements belonging to both.

• Notation:

Functions

2.1 Definition of a Function

A function assigns to each element in a set (domain) a unique element in a set (range). The graph of a function is the set of points .

• Domain: The set of all possible input values .

• Range: The set of all possible output values .

• Function notation:

Example: has domain and range .

2.2 Linear Functions

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

• Slope: measures the rate of change of with respect to .

• Intercept: is the value of when .

2.3 Domain and Range of Functions

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

• Example: For , domain is , range is .

2.4 Inverse Functions and Implicit Functions

An inverse function reverses the effect of . If , then . Implicit functions are defined by equations involving both and , not solved explicitly for .

• Example: defines and .

Limits and Continuous Functions

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 definition uses and to describe how close gets to as approaches .

• For every , there exists such that if , then .

3.3 Properties of Limits

• Sum:

• Product:

• Quotient: (if denominator not zero)

3.4 Left and Right Limits

• Left limit:

• Right limit:

3.5 Continuity

A function is continuous at if .

• Polynomials: Continuous everywhere.

• Rational functions: Continuous except where denominator is zero.

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:

• Geometric meaning: Slope of the tangent line at .

4.2 Examples of Derivatives

• For :

• For :

4.3 Instantaneous Velocity

The derivative can represent physical quantities such as velocity. If is position at time , then velocity is .

• Average velocity:

• Instantaneous velocity:

4.4 Acceleration

Acceleration is the derivative of velocity: .

Techniques of Differentiation

5.1 Differentiation Rules

5.2 Differentiating Powers and Exponentials

• Power Rule for Negative and Rational Exponents: Applies to for any real .

• Exponential Functions:

5.3 Differentiability Implies Continuity

If a function is differentiable at a point, it is also continuous at that point.

Additional info:

• This summary covers the first chapters of a standard Calculus I course, including numbers, functions, limits, continuity, and introductory differentiation. Later chapters (not shown here) would include further techniques, applications, and integration.