Curve Sketching and Analysis

A graduate-level exploration of discrete dynamical systems, moving from linear growth models to the complex, chaotic behavior of the logistic map. Students apply recursive sequences to model biological and economic phenomena, emphasizing stability analysis and bifurcation theory.

A comprehensive unit for 12th Grade Calculus students focusing on the derivation and application of derivatives in polar coordinates. Students transition from Cartesian slope to polar slope, analyze horizontal and vertical tangency, investigate behavior at the pole, and solve optimization problems involving polar curves.

A comprehensive unit on parametric equations and their applications in modeling motion. Students move from the basics of parametric curves to advanced calculus concepts like derivatives, concavity, vectors, and arc length.

This sequence explores the calculus of polar functions, focusing on differentiation techniques. Students will learn to calculate slopes of tangent lines, identify horizontal and vertical tangents, analyze behavior at the pole, and apply optimization to find maximum and minimum distances from the origin.

A sequence for undergraduate students bridging pre-calculus and calculus by focusing on the analytical properties of functions with rational exponents. Students explore graphing, algebraic rewriting, rationalizing for limits, and growth comparison.

This sequence covers the calculus of parametric curves, including first and second derivatives, tangent lines, concavity, arc length, and surface area of revolution. Designed for undergraduate calculus students, it emphasizes direct parametric differentiation and integration techniques.

This sequence establishes the rigorous mathematical underpinnings necessary for advanced optimization work, moving beyond procedural calculus to analysis-based proofs. Students explore the intersection of topology, set theory, and multivariate calculus to determine the existence and uniqueness of optimal solutions.

A comprehensive workshop series on optimization in calculus. Students master the Extreme Value Theorem, learn to translate complex word problems into mathematical models, and apply differentiation to find optimal outcomes in number theory and geometric contexts.

A comprehensive calculus sequence for undergraduate students focused on the rigorous application of derivatives to industrial, geometric, and economic optimization problems. Students progress from basic modeling to multi-constraint capstone analysis.

A graduate-level sequence exploring the gradient vector as the foundational tool for modern optimization. Students move from the geometric interpretation of multivariate derivatives to the implementation of stochastic algorithms used in machine learning.

A foundational calculus unit bridging average and instantaneous rates of change. Students move from physical motion data to geometric visualization and numerical estimation, culminating in the qualitative sketching of derivative graphs and interpretation of notation in real-world contexts.

A comprehensive 11th-grade calculus sequence that synthesizes domain, intercepts, symmetry, asymptotes, derivatives, and concavity into a systematic curve sketching algorithm. Students progress from procedural mastery to critical analysis of technological limitations and a final synthesis project.

Une leçon interactive de niveau Première générale sur la notion de nombre dérivé, s'appuyant sur des visualisations graphiques du passage de la sécante à la tangente, la lecture et le tracé de coefficients directeurs, et le lien avec le sens de variation.

This AP Calculus review lesson bridges the gap between algebraic rational function rules and formal limit notation, using visual sketching as a framework for understanding asymptotic behavior and continuity. Students translate pre-calculus 'rules' into calculus 'logic' to prepare for the formal definition of limits.

Students will learn to identify and describe intervals of increase, decrease, and constant behavior in functions using interval notation through a roller coaster design challenge. The lesson emphasizes using x-values to define these intervals and distinguishing between location and value.

A 12th-grade Pre-Calculus lesson connecting the algebraic evaluation of infinite limits and limits at infinity to the visual behavior of vertical and horizontal asymptotes. Students analyze reciprocal functions from a video to bridge the gap between algebra and calculus.

A comprehensive lesson plan and student activity connecting algebraic graphing rules to rigorous calculus limit definitions, centered around a detailed rational functions tutorial.

This lesson investigates why turning points are excluded from increasing and decreasing intervals. Students analyze the 'neutral zone' of zero slope at a vertex and debate strict monotonicity versus general definitions.

A Precalculus lesson exploring the end behavior of rational functions through graphical analysis and algebraic intuition. Students use polynomial degrees to predict horizontal asymptotes and formalize their findings using limit notation.

This lesson guides undergraduate students through a review of 'Does Not Exist' limits, focusing on the visual and algebraic differences between removable discontinuities (holes) and infinite discontinuities (vertical asymptotes) using Khan Academy's quotient rule demonstration.

A Pre-Calculus lesson on determining the end behavior of polynomial functions using the Leading Coefficient Test, featuring a kinesthetic warm-up, video analysis, and a collaborative sorting activity.