LAB MANUAL: FUNCTIONS AND CALCULUS IN PYTHON

Module: Data Science Language: Python Programming Style: Functional Programming Libraries Used: math (only where necessary)

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PROJECT OVERVIEW

This lab manual demonstrates the implementation of fundamental calculus concepts using pure Python without relying on any machine learning or scientific libraries.

The objective is to build a strong conceptual foundation in:

• Mathematical functions
• Numerical differentiation
• Numerical integration
• Optimization using gradient descent
• Probability interpretation using area under curves
• Linear regression from scratch

All implementations follow a functional programming approach with simple, readable, and testable functions.

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OBJECTIVES

• Understand how mathematical functions are represented in Python
• Approximate derivatives using the limit definition
• Approximate definite integrals using Riemann sums
• Implement gradient descent for optimization
• Interpret probability as area under a curve
• Build a simple linear regression model without ML libraries

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PROJECT STRUCTURE

Part 1: Understanding Functions in Python

• Definition and evaluation of mathematical functions
• Example: f(x) = x²
• Challenge: g(x) = 3x + 2

Part 2: Numerical Derivative Approximation

• Derivative using limit definition
• Approximation using small step size (h)
• Examples:
  • f(x) = x²
  • f(x) = x³
• Challenge:
  • Approximate derivative of sin(x) at x = π/4
  • Compare with exact value cos(π/4)

Part 3: Numerical Integration (Riemann Sum)

• Left Riemann sum method
• Approximation of definite integrals
• Examples:
  • ∫ x² dx from 0 to 1
  • ∫ 2x dx from 0 to 3
• Challenge:
  • ∫ (1/x) dx from 1 to 2
  • Compare with exact value ln(2)

Part 4: Gradient Descent (Optimization)

• Implementation of gradient descent algorithm
• Numerical gradient using derivative approximation
• Example:
  • Minimize f(x) = (x − 3)²
• Challenge:
  • Minimize f(x) = x⁴ − 5x² + 6
  • Analyze effect of different learning rates

Part 5: Area Under Curve for Probability Density

• Bell-shaped (normal-like) function
• Interpretation of area as probability
• Normalization using √π
• Challenge:
  • Modify bell curve to f(x) = exp(-(x−1)²)
  • Compute probability from x = 0 to x = 2

Final Project: Linear Regression from Scratch

• Linear model: y = mx + c
• Loss function: Mean Squared Error (MSE)
• Manual computation of gradients:
  • ∂Loss/∂m
  • ∂Loss/∂c
• Optimization using gradient descent
• Dataset: x = [1, 2, 3, 4] y = [2, 4, 6, 8]
• Verification with analytical solution

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KEY CONCEPTS COVERED

• Functions as first-class objects in Python
• Numerical calculus techniques
• Optimization using gradient descent
• Probability density and normalization
• Core ideas behind machine learning training
• Relationship between calculus and data science

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REQUIREMENTS

• Python 3.x
• No external libraries
• math module allowed where necessary

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HOW TO RUN

1. Ensure Python 3 is installed.

2. Open terminal or command prompt.

3. Navigate to the project directory.

4. Run individual Python files using:

   python filename.py

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EXPECTED LEARNING OUTCOMES

After completing this lab, students will be able to:

• Implement calculus concepts programmatically
• Understand optimization techniques used in ML
• Build intuition for probability distributions
• Develop models without relying on libraries
• Transition smoothly into machine learning concepts

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ACADEMIC INTEGRITY

This work is intended for educational purposes. Students are encouraged to understand each function and test with different inputs for better learning.

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END OF README