Derivatives, Extrema, and Rates of Change
Understand tangent slopes and instantaneous change, then use them to find extrema and solve optimization problems in science and engineering.
Learning Objectives
Interpret the derivative as instantaneous rate and tangent slope.
Use first/second derivative tests and closed-interval method.
Model related rates and design optimization solutions with constraints.
Communicate results with units, feasibility, and sensitivity checks.
Lessons
Lesson 6-1
Derivatives – Concept & Geometric Meaning
Derivative as instantaneous rate, tangent line slope, and foundational differentiation rules.
Lesson 6-2
Derivatives – Extrema and Optimization
First/second derivative tests, closed-interval extrema, and classic optimization models.
Lesson 6-3
Rates of Change & Advanced Optimization
Velocity/acceleration, marginal analysis, and multi-constraint problems via variable reduction.
Why This Unit Matters
Geometric Insight
Derivatives link algebra and geometry through tangent slopes.
Optimization Skills
Find best designs under constraints across STEM fields.
Modeling Practice
Related rates connect math to motion, fluids, and economics.