Plan of Course: Differentiation in Mathematics
Course Introduction
Welcome to the course on Differentiation in Mathematics designed for Year 11 students. This course explores the fundamental concepts and applications of differentiation, a cornerstone of calculus that allows us to understand the behavior of functions and their rates of change. Throughout the course, students will engage in both theoretical understanding and practical application, preparing them for further studies in mathematics and its many branches.
Course Goals
- To develop a strong conceptual understanding of differentiation.
- To apply differentiation techniques to solve a variety of mathematical problems.
- To analyze graphs of functions using derivatives.
- To prepare students for higher-level mathematics and related fields.
Course Aims
- To introduce the concept of limits as a precursor to differentiation.
- To teach the basic rules and techniques of differentiation, including the power rule, product rule, quotient rule, and chain rule.
- To explore higher-order derivatives and their applications.
- To investigate the graphical interpretation of derivatives.
- To solve real-world problems using differentiation.
Course Outline
Module 1: Understanding Limits (Lessons 1-5)
- Lesson 1: Introduction to Limits
- Lesson 2: Calculating Limits Algebraically
- Lesson 3: One-Sided Limits and Continuity
- Lesson 4: The Limit Laws
- Lesson 5: The Squeeze Theorem
Module 2: Introduction to Differentiation (Lessons 6-10)
- Lesson 6: Definition of the Derivative
- Lesson 7: The Derivative as a Limit
- Lesson 8: Instantaneous Rate of Change
- Lesson 9: Power Rule of Differentiation
- Lesson 10: Differentiation of Polynomial Functions
Module 3: Advanced Differentiation Techniques (Lessons 11-20)
- Lesson 11: The Product Rule
- Lesson 12: The Quotient Rule
- Lesson 13: The Chain Rule
- Lesson 14: Implicit Differentiation
- Lesson 15: Higher-Order Derivatives
- Lesson 16: Applications of Derivatives: The Tangent Line Problem
- Lesson 17: Related Rates Problems
- Lesson 18: Derivatives of Trigonometric Functions
- Lesson 19: Derivatives of Exponential and Logarithmic Functions
- Lesson 20: Derivatives of Composite Functions
Module 4: Graphical Interpretation of Derivatives (Lessons 21-25)
- Lesson 21: Analyzing the Slope of a Function
- Lesson 22: Critical Points and Local Extrema
- Lesson 23: The First Derivative Test for Increasing and Decreasing Functions
- Lesson 24: The Second Derivative Test for Concavity
- Lesson 25: Inflection Points and Curve Sketching
Module 5: Practical Applications of Differentiation (Lessons 26-30)
- Lesson 26: Optimization Problems in Real Life
- Lesson 27: Position, Velocity, and Acceleration
- Lesson 28: Linear Approximation and Differentials
- Lesson 29: Economic Applications: Cost and Revenue Functions
- Lesson 30: Review and Comprehensive Final Assessment
References
- Stewart, J. (2016). Calculus: Early Transcendentals. Cengage Learning.
- Thomas, G. B., Weir, M. D., & Hass, J. (2014). Thomas' Calculus. Pearson.
- Larson, R., & Edwards, B. H. (2013). Calculus. Cengage Learning.
- Anton, H., Bivens, I., & Davis, S. (2014). Calculus. Wiley.
- Zill, D. G., & Wright, W. S. (2016). Calculus: Early Transcendentals. Jones & Bartlett Learning.
- Desmos Graphing Calculator - Desmos
- Khan Academy - Differentiation Resources
This course structure provides a balanced approach to learning differentiation, addressing both conceptual understanding and practical experience that students will require as they progress in their mathematical education.