Mathematical Principles & Theorems
Calculus investigates rates of change and accumulation. Derivative \(f'(x) = \lim_{h \to 0}\frac{f(x+h)-f(x)}{h}\) represents tangent slope. Definite Riemann integral \(\int_a^b f(x)dx = \lim_{n \to \infty}\sum_{i=1}^n f(x_i^*)\Delta x\) represents net signed area. The Fundamental Theorem of Calculus links differentiation and integration: \(\frac{d}{dx}\int_a^x f(t)dt = f(x)\) and \(\int_a^b f(x)dx = F(b) - F(a)\).
Operating Instructions
- Input an algebraic function \(f(x)\) (e.g., \(x^3 - 3x\), \(\sin(x)\), \(e^{-x^2}\)).
- Choose mode: Symbolic Differentiation, Numerical Derivative, or Riemann Integration.
- Configure bounds \([a, b]\) and partition count \(n\) for Riemann sums (Left, Right, Midpoint, Trapezoidal).
- Interact with the 2D plot to inspect tangent lines, roots, extrema, and shaded integral partitions.