Definitions and Extrema

Local and Absolute Extrema

In calculus, understanding the behavior of functions at their maximum and minimum values is fundamental. These points are called extrema and are classified as local (relative) or absolute (global) maxima and minima.

• Local Maximum at c: f has a local maximum at c if there is an open interval (a, b) containing c such that for all in .

• Local Minimum at c: f has a local minimum at c if there is an open interval (a, b) containing c such that for all in .

• Absolute Maximum: f has an absolute maximum if there exists a point c in the domain such that for all in the domain.

• Absolute Minimum: f has an absolute minimum if there exists a point c in the domain such that for all in the domain.

The absolute maximum and minimum, if they exist, are called the extreme values of the function.

Critical Values

• A critical value of f is a value c in its domain such that either or does not exist.

Increasing and Decreasing Functions

• Increasing: f is increasing if for all and in the domain.

• Decreasing: f is decreasing if for all and in the domain.

Concavity and Inflection Points

Concavity

• Concave Up: f is concave up on an interval if for all in the interval.

• Concave Down: f is concave down on an interval if for all in the interval.

Inflection Points

• A point c is an inflection point if the concavity of f changes at c.

• If and the sign of changes at c, then c is an inflection point.

• If and the concavity changes, but the tangent is horizontal, it is called a saddle point.

Theorems and Tests

First Derivative Test

• If changes from positive to negative at , then has a local maximum at .

• If changes from negative to positive at , then has a local minimum at .

Second Derivative Test

• If and , then has a local minimum at .

• If and , then has a local maximum at .

• If , the test is inconclusive.

Extreme Value Theorem

• If is continuous on , then attains both a maximum and minimum value at either a critical point or at the endpoints.

Fundamental Theorem of Calculus (FTC)

• Suppose is continuous on .

• Part 1: .

• Part 2: , where is any anti-derivative of .

Reference Formulas

Derivative Rules

Trigonometric Derivatives

Inverse Trigonometric Derivatives

Logarithmic Differentiation

Logarithm Rules

Exponent Rules

Algebraic Identities

Anti-derivative Rules

• (for )

Properties of Integrals

• Linearity:

• Definite Integral as Area: gives the net area between and the -axis from to .

• Sum of Areas:

Example: Area Under |x|

Trigonometric Reference

Graphs and Properties

• sin(x): Domain: , Range: , Periodicity:

• cos(x): Domain: , Range: , Periodicity:

• tan(x): Domain: , Range: , Periodicity:

• sec(x): Domain: , Range: , Periodicity:

Trigonometric Identities

Unit Circle and Special Angles

• Values of and for , etc.

• Periodicity: , for

Exponential and Logarithmic Functions

Properties of and

• Exponential: is defined for all real , , ,

• Logarithm: is defined for , , ,

Vertical Asymptotes of Rational Functions

• For , ,

• For , ,

Additional Algebraic and Trigonometric Facts

• Square root properties:

• Examples of evaluating roots, exponents, and trigonometric values at special points.

Summary Table: Trigonometric Functions

Additional info: Some explanations and context have been expanded for clarity and completeness, including the summary table and explicit statements of theorems and properties.