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 fractions of integers, and real numbers include all rational and irrational numbers (such as ).

• Rational numbers: Numbers that can be written as , where and are integers and .

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

• Real numbers: All rational and irrational numbers, represented on the number line.

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

1.2 Sets and Intervals

Sets are collections of numbers, and intervals are subsets of the real line. For example, the interval includes all real numbers such that .

• Closed interval: includes endpoints and .

• Open interval: excludes endpoints.

• Union and intersection: The union is all elements in or ; intersection is all elements in both.

1.3 Functions

A function is a rule that assigns to each input in a set a unique output in a set . The domain is the set of all possible inputs, and the range is the set of all possible outputs.

• Notation: means maps to .

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

• Linear function: is a straight line.

Example: has domain and range .

1.4 Inverse and Implicit Functions

Some functions can be reversed, giving an inverse function . An implicit function is defined by an equation involving and , not solved for explicitly.

• Inverse function: If , then .

• Implicit function: Defined by equations like .

Limits and Continuous Functions

2.1 Informal Definition of Limits

The limit of as approaches is the value that gets close to as gets close to . Notation: .

• Example: .

• Substitution: Sometimes plugging in values suggests the limit, but not always.

2.2 Formal Definition of Limits

The formal definition uses (epsilon) and (delta): For every , there exists such that if , then .

• Left and right limits: and .

• Properties: Limits of sums, products, and quotients.

Example: .

2.3 Where Limits Fail to Exist

Limits may not exist if the function jumps, oscillates, or diverges. Examples include the sign function and functions with infinite oscillations.

• Jump discontinuity: The function has different left and right limits.

• Infinite oscillation: The function does not settle to a value.

2.4 Continuity

A function is continuous at if . Polynomials are continuous everywhere; rational functions are continuous where their denominators are nonzero.

• Discontinuous function: Has jumps, holes, or asymptotes.

Derivatives

3.1 The Tangent and Rate of Change

The derivative of at measures the rate of change or the slope of the tangent to the graph at . The tangent line approximates the graph near a point.

• Definition:

• Geometric meaning: Slope of the tangent line at .

Example: For , .

3.2 Instantaneous Velocity

The derivative also represents instantaneous velocity in physics: the rate at which position changes with respect to time.

• Average velocity:

• Instantaneous velocity:

3.3 Differentiation Rules

There are several rules for computing derivatives:

3.4 Differentiability and Continuity

If a function is differentiable at a point, it is also continuous there. However, not all continuous functions are differentiable (e.g., functions with corners or cusps).

• Non-differentiable point: The graph has a sharp corner or vertical tangent.

3.5 Examples and Applications

Derivatives are used to find slopes, rates of change, and to solve problems in physics and engineering.

• Example: The derivative of is .

• Application: Finding maximum and minimum values of functions.

Summary Table: Differentiation Rules

Key Concepts and Definitions

• Function: A rule assigning each input to a unique output.

• Limit: The value a function approaches as the input approaches a point.

• Continuity: No jumps or breaks in the graph of a function.

• Derivative: The rate of change or slope of a function at a point.

Examples

• Find the limit:

• Find the derivative:

• Application: The velocity of an object at time is the derivative of its position function .

Additional info: These notes cover the foundational topics in Calculus I, including functions, limits, continuity, and introductory differentiation, as outlined in the course contents. The structure and examples are expanded for clarity and completeness.