Numbers and Functions

1.1 Different Kinds of Numbers

Understanding calculus begins with a solid foundation in the types of numbers used in mathematics.

• Positive Integers:

• Negative Integers:

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

• Irrational Numbers: Numbers that cannot be written as fractions, such as and .

• Real Numbers: The set of 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

Intervals are used to describe subsets of real numbers.

• Closed Interval: includes all such that .

• Open Interval: includes all such that .

• Half-Open Intervals: or include one endpoint but not the other.

Example: includes all real numbers with .

1.3 Functions

A function is a rule that assigns to each element in a set exactly one element in a set .

• Domain: The set of all possible input values ().

• Range: The set of all possible output values ().

Example: has domain and range .

1.4 Linear and Nonlinear Functions

• Linear Function: (graph is a straight line).

• Nonlinear Function: Any function whose graph is not a straight line (e.g., ).

1.5 Inverse, Implicit, and Piecewise Functions

• Inverse Function: If is one-to-one, the inverse satisfies .

• Implicit Function: Defined by an equation involving both and , not solved explicitly for .

• Piecewise Function: Defined by different expressions for different intervals of the domain.

Example:

Limits and Continuous Functions

2.1 Informal Definition of Limit

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

• Notation:

Example:

2.2 Formal Definition of Limit

For every , there exists such that if , then .

2.3 Properties of Limits

• Sum:

• Product:

• Quotient: , if

2.4 One-Sided Limits

• Right-hand limit:

• Left-hand limit:

2.5 Continuity

A function is continuous at if .

• Polynomials and rational functions are continuous on their domains.

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.

• Definition:

• Geometric Interpretation: The derivative is the slope of the tangent line to the curve at .

Example: For , .

3.2 Physical Interpretation: Velocity and Acceleration

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

• Acceleration: The derivative of velocity with respect to time (second derivative of position).

Example: If , then velocity and acceleration .

3.3 Differentiation Rules

3.4 Differentiability and Continuity

• If is differentiable at , then is continuous at .

• Not all continuous functions are differentiable (e.g., at ).

3.5 Examples and Applications

• Find the derivative of :

• Find the equation of the tangent line to at :

Slope: ; Point: ; Equation:

Additional info:

• This summary covers the first chapters of a standard Calculus I course, including numbers, functions, limits, continuity, and the basics of derivatives. The content is based on lecture notes for MATH 221: First Semester Calculus.