Calculus: Single Variable Part 2 – Differentiation Course Syllabus

Overview: This course provides a comprehensive introduction to differentiation in single-variable calculus, emphasizing both theoretical understanding and practical applications. Through six structured modules, learners will explore derivatives, differentiation rules, linearization, optimization, and differentials, with an emphasis on real-world problem-solving in engineering, physics, and economics. The course includes interactive exercises and quizzes to reinforce learning. With a total time commitment of approximately 10 hours, this course is designed for beginners seeking to build a strong foundation in calculus. Lifetime access ensures flexibility for all learners.

Module 1: A New Look at Differentiation

Estimated time: 3 hours

• Revisiting the concept of derivatives beyond geometric slopes
• Introduction to asymptotic (big-O) notation
• Understanding rates of change using asymptotics
• Rules governing derivatives and their asymptotic behavior

Module 2: Putting Derivatives to Work

Estimated time: 3 hours

• Linearization and approximation using derivatives
• Higher-order derivatives and their interpretation
• Optimization problems and critical points
• Applying asymptotic reasoning to derivative applications

Module 3: Differentials and Operators

Estimated time: 2 hours

• Concept of differentials and their geometric meaning
• Implicit differentiation and related rates
• Differentiation operators and their properties

Module 4: Differentiation Rules

Estimated time: 1 hour

• Review of basic derivative rules (sum, product, quotient)
• Chain rule and its asymptotic interpretation
• Derivatives of composite and inverse functions

Module 5: Applications in Real-World Contexts

Estimated time: 1 hour

• Solving problems in physics using derivatives
• Economic models involving marginal analysis
• Engineering applications of rate-of-change concepts

Module 6: Final Project

Estimated time: 1 hour

• Apply differentiation to model a real-world scenario
• Use optimization techniques to solve a practical problem
• Submit a short report with analysis and conclusions

Prerequisites

• Familiarity with basic algebra and functions
• Understanding of limits and continuity (from Part 1 or equivalent)
• Basic knowledge of mathematical notation and reasoning

What You'll Be Able to Do After

• Compute and interpret derivatives using various rules
• Apply linearization and higher derivatives to approximate functions
• Solve optimization problems in engineering and economics
• Analyze rates of change using asymptotic (big-O) notation
• Use differentials and implicit differentiation in practical contexts