📗 Chapter 2 – Numerical Integration and Differentiation

In many scientific and engineering problems, derivatives and integrals of functions are not known explicitly or are difficult to compute analytically. The need to approximate these quantities has existed for centuries, from the early development of calculus by Newton and Leibniz, to modern applications in signal processing, control systems, and telecommunications. Numerical methods provide practical and reliable ways to compute these approximations with controlled accuracy.

This chapter introduces fundamental techniques for numerical differentiation and numerical integration. Students will learn how to estimate derivatives and integrals when analytical solutions are impractical, how to evaluate the accuracy of these approximations, and how to apply them in real-world engineering contexts, such as analyzing dynamic systems, computing power and energy, or processing signals.

By the end of this chapter, students will be able to:

• Approximate derivatives and integrals of functions using basic numerical methods.
• Apply standard numerical integration techniques:
  • Rectangle method
  • Trapezoidal rule
  • Simpson’s rule
• Use two- and three-point differentiation formulas to estimate derivatives.
• Evaluate the order of accuracy of different numerical methods.
• Estimate and analyze approximation errors to improve reliability.
• Understand the historical development and practical significance of numerical integration and differentiation in science and engineering.