This page delves into the concept of absolute extrema and provides practical examples of function analysis. The content emphasizes the distinction between relative and absolute extrema.

Definition: An absolute maximum M is a value where f$x$ ≤ M for all x in the domain, while an absolute minimum m is where m ≤ f$x$ for all x.

Highlight: The Weierstrass Theorem guarantees that continuous functions on closed intervals [a,b] have both absolute maximum and minimum values.

Example: For f$x$ = x³ - 3x, the analysis includes finding critical points and determining intervals of monotonicity.

This page covers the theory of asymptotes and their classification. The content explains vertical, horizontal, and oblique asymptotes in detail.

Definition: An asymptote is a line that a function approaches but never reaches as x approaches infinity or a specific value.

Vocabulary: Vertical asymptotes occur where the function approaches infinity as x approaches a finite value.

Highlight: A function cannot have both horizontal and oblique asymptotes simultaneously (unless defined piecewise).

This page provides a comprehensive approach to graphing functions through systematic analysis of key features.

Example: A detailed analysis of f$x$ = x²/$x² - 4$ demonstrates the complete process of function investigation.

Highlight: The systematic approach includes: 1) Domain determination 2) Intercepts 3) Asymptotes 4) Critical points 5) Intervals of monotonicity and curvature 6) Graph construction.

Vocabulary: Critical points are values where f'$x$ = 0 or f'$x$ is undefined, potentially indicating relative extrema or inflection points.

This section provides a comprehensive approach to analyzing and graphing functions using calculus techniques.

Example: Detailed analysis of a rational function including domain, critical points, and asymptotic behavior.

Highlight: The process includes finding intersection points with axes, analyzing asymptotes, and determining intervals of monotonicity.

This page introduces fundamental concepts about function monotonicity and derivative applications. The content focuses on how derivatives determine function behavior and identify critical points.

Definition: A function f$x$ is increasing on [a,b] if x₁ < x₂ implies f(x₁) ≤ f(x₂), and decreasing if x₁ < x₂ implies f(x₁) ≥ f(x₂).

Highlight: The first derivative test reveals that if f'(x₀) > 0, the function is increasing at x₀, and if f'(x₀) < 0, the function is decreasing at x₀.

Example: For the function f$x$ = x² - 4x + 3, the derivative f'$x$ = 2x - 4 helps determine increasing intervals (x > 2) and decreasing intervals (x < 2).

Vocabulary: Relative extrema refers to local maximum and minimum points where the function value is highest or lowest compared to nearby points.