Calculus 1 — 2.5: Even and Odd Functions скачать в хорошем качестве

Calculus 1 — 2.5: Even and Odd Functions

Learn how to classify functions as even, odd, or neither using a simple algebraic test. This video connects the visual symmetry of function graphs — mirror symmetry across the y-axis and 180-degree rotational symmetry about the origin — to the formal definitions f(-x) = f(x) and f(-x) = -f(x), then walks through complete worked examples. Key concepts covered: • Even functions: mirror symmetry across the y-axis, where f(-x) = f(x) • Odd functions: rotational symmetry about the origin, where f(-x) = -f(x) • The four-step algebraic test: substitute -x, simplify, and compare to the original • Worked example: testing f(x) = x⁴ - x² + 1 (even) • Worked example: testing g(x) = x³ - x (odd) • Worked example: testing h(x) = x² + x (neither) • Power patterns in polynomials: all even powers, all odd powers, or mixed • Practical applications: halving graphing work and the calculus shortcut where integrals of odd functions over symmetric intervals equal zero • Challenge problem: verifying f(x) = x⁵ - 3x³ + 2x is odd by factoring out -1 ━━━━━━━━━━━━━━━━━━━━━━━━ SOURCE MATERIALS The source materials for this video are from    • Calculus 1 Lecture 0.2:  Introduction to F...