Calculus with Python Lecture Notes

Overview

• Instructor: Ed Powski
• Focus: Implementation of calculus concepts using Python
• Audience: Suitable for all, especially those interested in computer science

Course Benefits

• Python Integration: Learn to write code for calculus concepts
  • Saves time compared to manual calculations
  • Enables focus on applications

Course Structure

• Google Collab Notebook Setup
  • Organized table of contents
  • Easy problem-solving workflow

Key Calculus Concepts

Limits

• Introduction: Transition from algebraic to calculus thinking
• Graphical Representation: Understanding holes in graphs due to limits
• Example: Function with a hole where x cannot be 2
  • Cancel terms in numerator and denominator
  • Graph as y = 3x with a hole at x=2

Python Code for Limits

• Equation Setup: Use of variable x and small increment h
• Limit Calculation: Checking left and right-hand limits
• Output: Print values to confirm limit existence

Asymptotes and Limits

• Vertical Asymptotes: Occur when denominator is zero
• Graphical Insights: Dotted lines indicate x values where function is undefined

Limits in Python

• Denominator Zero Check: Solve equations in symbolic form
• Approach from Right/Left: Using small values of h
• Output: Determine limit behavior as approaches infinity

Applications and Code Implementation

Calculating Limits with Code

• SymPy Utilization: Simplify and solve equations
• Graphing with Matplotlib: Visual representation of limits

Building a Python Calculator for Calculus

• Calculator Setup: Organize code for repeated use
• Function Import and Execution: Efficient setup for solving limits, derivatives, and integration

Advanced Concepts

Derivatives

• Differentiation Basics: Power rule, product rule, quotient rule
• Chain Rule: Important for composite functions
• Python Implementation: SymPy to calculate and evaluate derivatives

Trigonometric Derivatives

• Sine and Cosine: Key derivatives and their applications
• Python Code Execution: Simplifying trigonometric expressions

Implicit Differentiation

• Implicit vs. Explicit: Handling equations where y is not isolated

Integral Calculus

• Reimann Sums: Approximating area under curves
• Definite Integrals: Calculating exact area under curve
• Fundamental Theorem of Calculus: Relationship between integrals and derivatives

Special Integrals

• Normal Distribution: Calculating probabilities and critical values in statistics
• Python Code for Integrals: Utilize SymPy for complex integral calculations

Applications in Physics and Economics

• Projectile Motion: Using calculus to determine trajectory
• Revenue Maximization: Calculus for finding optimal pricing

Conclusion

• Comprehensive Tools: Python as a tool to handle complex calculus problems
• Future Applications: Prepare for advanced calculus and real-world applications