Introduction to Calculus
Calculus is the branch of mathematics that studies change and accumulation. It is built on two main ideas:
- Derivatives - Measuring rates of change
- Integrals - Measuring accumulation
The derivative of a function \(f(x)\) at a point \(x\) is defined as: \[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \] \[ \textbf{Example:} \] \[ f(x) = x^2 \quad \Rightarrow \quad f'(x) = 2x \] This means the slope of the curve \(y = x^2\) at any point \(x\) is \(2x\).
Integrals:
The integral of a function represents the area under its curve: \[ \int_a^b f(x)\,dx \] Example: \[ \int_0^1 x^2\,dx = \left[ \frac{x^3}{3} \right]_0^1 = \frac{1}{3} \] This means the area under \(y = x^2\) from \(x=0\) to \(x=1\) is \(\tfrac{1}{3}\).
Fundamental Theorem of Calculus
Differentiation and integration are connected:
\[ \frac{d}{dx} \left( \int_a^x f(t)\,dt \right) = f(x) \] Applications- Physics: velocity and acceleration
- Economics: cost and revenue
- Geometry: areas and volumes
