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Derivatives: The Tangent Line

Audience: UniversityCategory: Math

Description

Builds intuition for the derivative by animating the limit definition. A secant line between two points on f(x) = xΒ² is drawn and animated as h approaches 0, visually converging to the tangent line. The difference quotient formula transforms into the derivative formula.

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Derivatives: The Tangent Line

Description

Builds intuition for the derivative by animating the limit definition. A secant line between two points on f(x) = xΒ² is drawn and animated as h approaches 0, visually converging to the tangent line. The difference quotient formula transforms into the derivative formula.

Phases

# Phase Name Duration Description
1 Setup 5s Title, draw axes, plot f(x) = xΒ² in blue
2 Secant Line 7s Mark point (x, f(x)) at x=2, mark second point at x+h, draw secant
3 Difference Quotient 6s Display rise/run formula, label Ξ”x and Ξ”y on graph
4 Animate h β†’ 0 12s ValueTracker h from 1.5 down to 0.01; secant morphs to tangent
5 Limit Formula 8s Show full limit definition of derivative, highlight as h→0
6 Compute f'(2) 7s Algebraically compute derivative of xΒ², show f'(x) = 2x, f'(2) = 4
7 Summary 5s Tangent line at x=2, slope label, geometric interpretation text

Layout

+--------------------------------------------------+
|  Title: "Derivatives: The Tangent Line"          |
+--------------------------------------------------+
|                                                  |
|   Axes with f(x) = xΒ²        | Formula panel    |
|   (left 60%)                 | (right 40%)      |
|                              |                  |
|   β€’ Point P = (2, 4)         | f(x+h) - f(x)    |
|   β€’ Point Q = (2+h, (2+h)Β²)  | ─────────────    |
|   β€’ Secant line (orange)     |      h           |
|   β†’ tangent line (yellow)    |                  |
|                              | → lim as h→0     |
|   Ξ”x and Ξ”y labels           | β†’ f'(x) = 2x    |
|                              |                  |
+--------------------------------------------------+

Area Descriptions

  • Left 60%: Axes with curve, points, secant/tangent lines, Ξ”x and Ξ”y visual indicators
  • Right 40%: Evolving formula from difference quotient to derivative definition
  • Below formula: Step-by-step algebraic simplification

Assets & Dependencies

  • Fonts: LaTeX
  • Manim version: ManimCE 0.19.1

Notes

  • Use ValueTracker for h; always_redraw for secant line, Q point, and Ξ” labels
  • xβ‚€ fixed at x=2 for concreteness, f'(2)=4 is the expected tangent slope
  • Color coding: curve = BLUE, secant = ORANGE, tangent = YELLOW, Ξ”x = GREEN, Ξ”y = RED