Engneering Mathematics

1. Fundamentals to Calculus

Definition and History of Calculus

Calculus is the mathematical study of continuous change. It provides a framework for understanding how quantities change over time or space and is divided into two main branches:

Differential Calculus – Deals with rates of change and slopes of curves.
Integral Calculus – Focuses on accumulation of quantities and areas under curves.

Historical Background:

The foundations of calculus were independently developed in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz.
Newton used calculus to describe motion and gravity in his work Principia Mathematica.
Leibniz introduced the modern notation for calculus, including the integral (∫) and derivative (d/dx) symbols.
Over time, mathematicians such as Euler, Cauchy, and Riemann refined the concepts, making calculus more rigorous and applicable to various scientific fields.

Key Concepts of Calculus

Limits
- A limit describes the value that a function approaches as the input (x) gets closer to a particular point.
- It is essential in defining derivatives and integrals.
- Example: $\lim_{{x \to 2}} (3x + 1) = 7$ $x \to 2 lim (3 x + 1) = 7$ This means that as $x$ x approaches 2, the function $3x + 1$ $3 x + 1$ approaches 7.
Continuity
- A function is continuous at a point if its graph has no breaks, holes, or jumps.
- Mathematically, a function $f(x)$ f(x) is continuous at $x = a$ $x = a$ if: $\lim_{{x \to a}} f(x) = f(a)$ $x \to a lim f (x) = f (a)$
- Example: The function $f(x) = x^2$ $f (x) = x 2$ is continuous everywhere, but $f(x) = \frac{1}{x}$ $f (x) = x 1$ is discontinuous at $x = 0$ $x = 0$ .
Functions
- A function is a relationship between an input (x) and an output where each input has exactly one output.
- Types of functions commonly used in calculus:
  - Polynomial functions (e.g., $f(x) = x^3 - 2x + 5$ $f (x) = x 3 - 2 x + 5$ )
  - Exponential functions (e.g., $f(x) = e^x$ $f (x) = e x$ )
  - Trigonometric functions (e.g., $f(x) = \sin x$ $f (x) = sin x$ )