Optimization, related rates, and curve sketching via first and second derivatives. Addresses real-world scenarios in physics and economics using the Mean Value Theorem and L'Hôpital's Rule.
A comprehensive two-page calculus worksheet on differentiation rules, tangent lines, and instantaneous rates of change with dedicated student workspace and analytical problem solving.
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.
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 graduate-level exploration of the Calculus of Variations, focusing on optimizing functionals. Students derive the Euler-Lagrange equation and apply it to physics and geometry problems like the Brachistochrone and Isoperimetric challenges.
A graduate-level sequence on constrained optimization, covering Lagrange Multipliers, KKT conditions, and sensitivity analysis for economics and engineering applications.
A comprehensive graduate-level exploration of numerical optimization algorithms, moving from first-order gradient descent to second-order Newton methods and computationally efficient Quasi-Newton approaches. Students analyze convergence rates, stability, and strategies for navigating complex, non-convex landscapes.
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.
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.
Students explore the application of derivatives in real-world contexts beyond physics, specifically focusing on instantaneous rates of change in biology, finance, and social media growth using the limit definition.
An 11th-grade honors lesson connecting the limit definition of the derivative to instantaneous velocity through rocket launch simulations. Students will analyze height functions to determine peak altitude and impact force.
This lesson introduces 11th-grade students to the distinction between average and instantaneous rates of change. Students analyze real-world COVID-19 data and explore a quadratic function by 'shrinking the interval' to discover the concept of a tangent line.
