Domain: Humanities and Social Sciences — Economics
Target concept: Characteristics of the Average Cost Curve: U-shaped distribution around the Marginal Cost Curve

Video generation prompt: A coordinate axis is displayed on the screen, with the horizontal axis representing quantity and the vertical axis representing cost. Data points indicating how cost changes with increasing quantity are marked on the coordinate axis, and these points, when connected, form the average cost curve. In addition, plot a marginal cost curve on the same coordinate axis.

Evaluation guide:
Target concept: The average cost (AC) curve is U-shaped *because* marginal cost (MC) lies below it during falling AC (pulling it down) and above it during rising AC (pushing it up), intersecting exactly at AC's minimum - a necessary and sufficient condition grounded in calculus (d(AC)/dq = (MC - AC)/q). Minimum mechanistic assumption: AC(q) = TC(q)/q, so d(AC)/dq = (MC - AC)/q, meaning the sign of the slope is determined solely by MC-AC. Phased storyline: (1) Low-q regime: plotted points descend -> AC curve slopes downward; MC curve drawn visibly beneath -> confirms causal pull; (2) Mid-q: points flatten -> AC reaches minimum; MC intersects AC at that exact coordinate -> visual anchor; (3) High-q: points ascend -> AC slopes upward; MC now lies clearly above -> confirms causal push. Visibility constraints: All elements must be front-facing, orthographic (no perspective), high-contrast, and labeled unambiguously; no occlusion; the intersection point must be clearly and unambiguously marked and highlighted (e.g., a pulse or ring) so it is easy to verify by eye within 10 seconds.

Manual:
I. Basic Demonstration Information
Demonstration Title: Average Cost Curve U-Shape Around Marginal Cost Curve Demonstration
Subject: Economics - Microeconomics - Production Costs
Demonstration Purpose: To visually demonstrate that the average cost (AC) curve is U-shaped and positioned such that the marginal cost (MC) curve intersects it at its minimum point - with MC lying below AC on the left (causing AC to fall) and above AC on the right (causing AC to rise); this illustrates the core cost behavior principle underlying short-run production decisions.

II. Demonstration Carrier/Equipment
Core Carrier: Dynamic 2D coordinate visualization (digital screen rendering)
Auxiliary Equipment: None (fully software-generated); background: neutral white or light gray; font: sans-serif, legible at 16pt; color scheme: AC curve = solid black (#000000), MC curve = dashed red (#CC0000), data points = filled blue circles (#0066CC), axes = thin gray (#999999).

III. Demonstration Content Description
Layout: Full-screen Cartesian plane with origin at bottom-left; x-axis labeled 'Quantity' (0 to 10 units, evenly spaced ticks); y-axis labeled 'Cost' ($0 to $100, linear scale). Data points plotted at (1,90), (2,70), (3,55), (4,45), (5,40), (6,42), (7,48), (8,57), (9,68), (10,80). AC curve connects points smoothly. MC curve (calculated as discrete slope approximations) passes through (5,40) and extends as a continuous dashed line crossing AC only at its minimum.

IV. Observation of Core Phenomena
Expected Phenomenon: A smooth, symmetric U-shaped AC curve with clear minimum at (5,40); MC curve intersects AC *exactly* at that minimum; MC lies visibly lower than AC for x < 5 and higher for x > 5 - forming the defining 'U-shaped distribution *around* MC'.
Expected Screen: Clean, uncluttered coordinate plane. No extraneous labels, legends, or animations beyond essential elements. All visual features (axis labels, points, curves) remain stable and legible throughout.
Visual Verification Features: (1) X-axis must read 'Quantity'; (2) Y-axis must read 'Cost'; (3) At least 7 distinct blue data points trace a concave-up path; (4) AC curve is a single unbroken black line connecting them; (5) MC curve is a red dashed line intersecting AC at its lowest vertex; (6) Vertical gap between MC and AC is observable and directionally consistent left/right of intersection.

V. Summary of Key Knowledge Points and Principles
Core Concept Definition: In the short run, average total cost first decreases due to spreading fixed costs and increasing efficiency (economies of scale), reaches a minimum where MC = AC, then increases due to diminishing marginal returns (diseconomies). The MC curve's position relative to AC determines AC's slope: when MC < AC -> AC falls; when MC > AC -> AC rises.
Principle of Visual Recognition: The U-shape is not merely aesthetic - it is *defined operationally* by the intersection and relative positioning of two distinct cost curves. Recognition requires verifying both geometric shape (U) and relational topology (MC bisects AC's minimum and straddles it).

VI. Demonstration Precautions
- No verbal narration or text explanations - pure visual demonstration.
- All curves must be mathematically consistent: MC values are derived from adjacent total-cost differences, where total cost TC(q) = AC(q) x q (e.g., MC between q=4 and q=5 is approximately TC(5) - TC(4)), not from raw AC differences; no arbitrary drawing.
- Maintain constant aspect ratio; avoid zooming, panning, or rotating.
- Animation duration <= 10 seconds: points plot sequentially (1s each for first 5, accelerated for rest), AC draws continuously, MC reveals fully at t=7s with intersection highlight at t=8s.

Prompt Grounding rubric:
level_1: No coordinate system is visible; axes are unlabeled; no data points or curves are drawn; the screen shows something unrelated (e.g., text only, an unrelated graph, or a schematic without quantity/cost axes).
level_2: A coordinate system appears but its labeling is wrong: the horizontal axis is mislabeled (e.g., 'Time' or 'Price') or the vertical axis is mislabeled (e.g., 'Revenue' or 'Output'), or only one axis is present, or scattered points appear with no line connecting them into a curve.
level_3: Axes are correctly labeled (horizontal = 'Quantity', vertical = 'Cost'); a smooth curve is drawn from the plotted points, but it is not U-shaped (e.g., linear, an inverted U, or monotonically increasing/decreasing); or the curve is U-shaped but no discrete data points are marked along it; a marginal cost curve may be entirely absent at this level.
level_4: Correct axes labeled 'Quantity' (x) and 'Cost' (y); discrete data points are plotted along a clearly U-shaped average cost curve; however, either (a) the marginal cost curve is missing entirely, or (b) the marginal cost curve is present but rendered in the same color, thickness, and line style as the average cost curve so the two are hard to tell apart, or (c) the data points lack visual distinction from the curve line (e.g., same color and size).
level_5: Screen displays a clean Cartesian coordinate system with explicitly labeled x-axis 'Quantity' and y-axis 'Cost'; multiple distinct, uniformly sized circular data points are plotted at increasing quantity intervals; these points are smoothly connected into a continuous, symmetric U-shaped average cost curve rendered as a solid black or dark-blue line; a second marginal cost curve is also drawn, in a visibly distinct style (e.g., thinner, dashed, and a different color such as red) so the two curves are never confused with one another anywhere on the graph.

Scientific and Causal Correctness rubric:
level_1: No causal or structural relationship between quantity and cost is conveyed; the curve's shape contradicts basic cost theory (e.g., average cost rises monotonically with no minimum, or falls without bound); the marginal cost curve intersects the average cost curve at its maximum or not at all; the animation shows no sign of the diminishing-then-increasing-returns pattern that produces a U-shape.
level_2: The curve shows some non-monotonic behavior (a dip followed by a rise), but the intersection logic is wrong: the marginal cost curve crosses the average cost curve at a point that is not its minimum, or crosses it more than once; or the average cost minimum sits at zero quantity; or cost values decrease into unrealistic or negative territory with no visual justification.
level_3: A U-shaped average cost curve appears with plausible cost magnitudes; a marginal cost curve is drawn but does not cross the average cost curve at its lowest point (it is offset horizontally or vertically); or it does cross near the minimum but fails to stay below average cost beforehand and above it afterward - the curve shapes are right but the MC-AC positioning that explains the U-shape is only partly shown.
level_4: The average cost curve shows the textbook pattern: cost falls as quantity first increases (economies of scale), reaches a minimum, then rises (diseconomies of scale); the marginal cost curve crosses the average cost curve exactly at that minimum, staying below average cost to the left of the crossing and above it to the right. However, there is no visual emphasis (arrow, color contrast, or highlight) tying the marginal cost curve's position to the average cost curve's change in slope, so the causal link is present but not foregrounded.
level_5: The full causal picture is visible without narration: (a) as quantity rises from low values, the average cost curve declines while the marginal cost curve stays visibly below it; (b) at the average cost minimum, the two curves cross, marked with a subtle highlight (e.g., a brief pulse or a small label); (c) beyond that point, average cost rises while marginal cost stays visibly above it; (d) this unfolds within the video so a viewer can watch the marginal cost curve pull the average cost curve down and then push it up, purely through the curves' relative positions over time.

Spatiotemporal Consistency rubric:
level_1: The axes jump in position or resize mid-video; the curves flicker in and out of view; data points appear or disappear with no visible continuity; the marginal cost and average cost curves swap which one is drawn in front of the other inconsistently; frame transitions look like jarring cuts or warping rather than continuous motion.
level_2: Axis labels shift position across frames; curve thickness or color changes erratically partway through; data points vary in size or color inconsistently; the marginal cost and average cost curves visibly swap which one is above the other at the same quantity value between frames; points appear to plot out of order (e.g., a later quantity value appears before an earlier one).
level_3: There is slight jitter in point placement or in how smooth the curves look; axis labels stay in place but their font weight varies; the marginal cost and average cost curves keep the correct relative positioning overall, but the crossing point visibly shifts a little from frame to frame; points are plotted in the correct forward order (low to high quantity) but appear abruptly rather than fading or drawing in smoothly.
level_4: Axes stay stable in position, scale, and labeling throughout; the data points and both curves keep a consistent line thickness, opacity, and color; the crossing of the two curves stays at one visually stable location; points animate in left-to-right order at a steady, even pace; the rendering never flips which curve is drawn on top at their crossing point once that layering has been established for the frame.
level_5: The coordinate grid stays visually fixed with no drift; every plotted point, both curves, and the axis labels keep a clean, uncluttered appearance throughout; points appear in strict increasing-quantity order, evenly spaced and similar in size, at a steady pace; the average cost curve appears to draw itself smoothly as points are added, rather than appearing all at once; the marginal cost curve's crossing with the average cost curve shifts smoothly as new points appear, with no sudden jump in its location; there are no frame drops, stutters, or stray visual artifacts, and the whole animation reads as one continuous sequence.