Numbers and Functions

1.1 Different Kinds of Numbers

Calculus begins with a review of the types of numbers used in mathematics, which are foundational for understanding functions and their properties.

• Positive Integers: The set {1, 2, 3, ...}.

• Negative Integers: The set {..., -3, -2, -1}.

• Rational Numbers: Numbers that can be expressed as a fraction , where and are integers and .

• Irrational Numbers: Numbers that cannot be expressed as a simple fraction, such as or .

• Real Numbers: The set of 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 Sets and Intervals

Sets and intervals are used to describe collections of numbers, especially when defining domains and ranges of functions.

• Interval Notation: denotes all real numbers such that .

• Open Interval: denotes all such that .

• Union and Intersection: The union is the set of elements in or ; the intersection is the set of elements in both and .

Functions

3.1 Definition of a Function

A function is a rule that assigns to each element in a set (the domain) exactly one element in a set (the range).

• Domain: The set of all possible input values () for which the function is defined.

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

Example: has domain and range .

3.2 Graphs of Functions

The graph of a function is the set of all points in the plane. The vertical line test is used to determine if a graph represents a function.

• If any vertical line intersects the graph more than once, it is not a function.

3.3 Linear Functions

A linear function has the form , where is the slope and is the y-intercept.

• The graph is a straight line.

3.4 Inverse Functions and Implicit Functions

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

• Implicit Functions: Sometimes functions are defined by equations not solved for ; these are called implicit functions.

Example: The equation defines implicitly as a function of (for ).

Limits and Continuity

1. Informal Definition of Limits

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

• Notation:

2. Formal (Epsilon-Delta) Definition of Limits

The formal definition uses (epsilon) and (delta) to rigorously define limits:

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

3. Properties of Limits

• Limits of sums:

• Limits of products:

• Limits of quotients: , provided

4. Continuity

A function is continuous at if .

• Polynomials and rational functions are continuous on their domains.

• Discontinuities can be classified as removable, jump, or infinite.

Derivatives

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, representing the instantaneous rate of change.

• Definition:

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

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, or the second derivative of position.

Example: If is the position at time , then velocity and acceleration .

3. Differentiation Rules

Several rules simplify the computation of derivatives:

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.

• Non-differentiable points: Corners, cusps, and vertical tangents.

5. Examples and Applications

• Find the derivative of : .

• Find the equation of the tangent line to at : Slope is $2y = 2(x-1) + 1$.

Additional info:

• This summary covers the first chapters of a standard Calculus I course, including numbers, functions, limits, continuity, and the basics of derivatives. Later chapters (not included here) would cover advanced differentiation techniques, applications, and integration.