The Bridge Between Change and Area
The Fundamental Theorem of Calculus (FTC) is the "glue" that holds calculus together. Part I states that differentiation and integration are inverse processes. If you define a function as an integral, its derivative is the original function:
\[\frac{d}{dx} \int_a^x f(t) dt = f(x)\]
Worked Examples
Example 1: Differentiating an Integral
Find the derivative of \(g(x) = \int_1^x \sin(t^2) dt\).
- According to FTC Part I, we just replace \(t\) with \(x\).
- Result: \(g'(x) = \sin(x^2)\).
Example 2: Using the Chain Rule
Find the derivative of \(g(x) = \int_1^{x^3} \cos(t) dt\).
- This requires the Chain Rule: \(g'(x) = \cos(x^3) \cdot \frac{d}{dx}(x^3)\).
- Result: \(3x^2 \cos(x^3)\).
The Bridge to Quantum Mechanics
In Quantum Mechanics, we often define the Potential Energy \(V(x)\) as the negative integral of the force. The FTC Part I tells us that the force is the negative derivative of the potential: \(F(x) = -\frac{dV}{dx}\). This allows us to switch back and forth between "Force" (Newton's view) and "Energy" (Schrödinger's view) seamlessly. Most of Quantum Mechanics is written in terms of Energy, but the FTC ensures we never lose touch with the classical concept of Force.