Skip to content

FTC Connection Visualization

Run the FTC Connection Visualization Fullscreen

Edit with the p5.js Editor

About This MicroSim

This visualization shows the deep connection between the two parts of the Fundamental Theorem:

  • Panel 1: f(x) - the integrand (what we're integrating)
  • Panel 2: F(x) - the antiderivative (whose derivative is f)
  • Panel 3: The connection - the slope of F equals f(x)

The arrows show how integration and differentiation are inverse operations.

Iframe Code

1
<iframe src="https://dmccreary.github.io/calculus/sims/ftc-connection/main.html" height="602px" scrolling="no" style="width: 100%;"></iframe>

Lesson Plan

Learning Objectives

Students will be able to:

  1. Analyze the relationship between differentiation and integration
  2. Explain how FTC Part 1 and Part 2 express the same underlying relationship
  3. Verify that the derivative of the antiderivative returns the original function

Activities

  1. Visual Exploration (5 min): Move x and observe the tangent line slope matching f(x)
  2. Part 1 Focus (5 min): Verify that d/dx[∫f(t)dt] = f(x) visually
  3. Part 2 Focus (5 min): Connect F(b) - F(a) to the shaded area
  4. Synthesis (5 min): Explain in your own words why these two theorems are really one

The Grand Connection

FTC Part 1: Differentiation undoes integration \(\(\frac{d}{dx}\left[\int_a^x f(t)\,dt\right] = f(x)\)\)

FTC Part 2: Integration "undoes" differentiation \(\(\int_a^b F'(x)\,dx = F(b) - F(a)\)\)

Both say the same thing: differentiation and integration are inverse operations.

References