Fundamental Theorem of calculus, Part 1

Definition

The Fundamental Theorem of calculus (FTC) Part 1 states that differentiation and integration are inverse operations. It defines an “accumulation function” and proves that its rate of change is simply the original function.

Why It Matters

This theorem guarantees that every continuous process has a predictable ‘running total’; it is the mathematical foundation that allows us to define and use complex functions (like those in statistics or signal processing) that are otherwise impossible to write down using simple algebra.

Core Concepts

• Accumulation Function: $F(x) = \int_a^x f(t) dt$ represents the total area under $f$ from a fixed point $a$ to a variable point $x$.
  • How to read: “The function capital F of x equals the integral from a to x of f of t with respect to t.”
  • Meaning: $F(x)$ is the running total of $f$ from $a$ up to $x$ — area accumulated as the upper limit slides.
• The Derivative Rule: $\frac{d}{dx} \int_a^x f(t) dt = f(x)$.
  • How to read: “The derivative with respect to x of the integral from a to x of f of t with respect to t equals f of x.”
  • Meaning: Differentiating an accumulation function recovers the original rate function — integration and differentiation are inverse operations.
• Existence of Antiderivatives: This theorem guarantees that every continuous function has an antiderivative (the accumulation function itself).

Connected Concepts

• Fundamental Theorem of calculus, Part 2
• Fundamental Theorem of Line Integrals
• Graphs of Functions: Properties