Derivative as a Geometric Limit

Overview

An animation illustrating the geometric meaning of the derivative for the function $y = x^{2}$. A point $P(2,4)$ on the parabola is used to draw a secant line with a nearby point $Q$ separated by a horizontal offset $h$. As $h\ o 0$, the secant approaches the tangent, visually demonstrating the limit definition $\displaystyle \frac{\Delta y}{\Delta x}=\lim_{h\ o0}\frac{f(2+h)-f(2)}{h}$.

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Phases

#	Phase Name	Duration	Description
1	Intro & Axes	~4s	Fade‑in a white background, draw centered Cartesian axes with equal scaling; label the origin subtly (no persistent text).
2	Plot Parabola & Point	~6s	Sketch the curve $y=x^{2}$ in a thin black line; place a white‑filled dot at $P(2,4)$ and highlight it with a brief pulse.
3	Secant with Initial $h$	~8s	Introduce a second point $Q(2+h, (2+h)^{2})$ with $h=1$. Draw the secant line $PQ$ in a contrasting color (e.g., blue). Show the fraction $\displaystyle \frac{\Delta y}{\Delta x}=\frac{(2+h)^{2}-4}{h}$ beside the line (small inline formula).
4	Shrink $h$	~30s	Animate $h$ decreasing smoothly from 1 to 0.02 in many small steps; for each step, update the position of $Q$ and the secant line. The fraction updates in real time, visually approaching a constant value.
5	Tangent Emerges	~8s	When $h$ is sufficiently small, fade the secant into the tangent line at $P$. Highlight the tangent in red and display the limit value $\displaystyle \lim_{h\ o0}\frac{\Delta y}{\Delta x}=4$ (the derivative $f'(2)=4$).
6	Zoom Out & End	~4s	Slowly zoom the camera out to show the whole parabola while keeping the tangent and point visible; then fade to white.

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Layout

┌─────────────────────────────────────────────┐
│                                             │
│                MAIN (graph)                 │
│                                             │
├─────────────────────────────────────────────┤
│  Caption (optional, small formula display)   │
└─────────────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Main	Cartesian axes, parabola $y=x^{2}$, points $P$ and $Q$, secant/tangent lines	Occupies most of the frame; background is solid white
Caption	Small inline formula showing $\Delta y/\Delta x$ or the limit value	Appears near the bottom edge; fades when not needed

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Notes

• All objects are white or black except the dynamic secant (blue) and final tangent (red) to keep contrast on the white background.
• The camera remains centered on the origin; only a gentle zoom out occurs in the final phase.
• No persistent textual labels are used except the brief formula in the caption area.
• Total runtime is approximately 60 seconds, respecting the requested duration.