Start from slope on the graph itself, use one constrained rectangle bench to make a real maximum visible, keep limit and continuity behavior available on one target point, and then widen into signed area and accumulation so rate and total change stay connected on one visual branch.
This first calculus page stays on an honest graphing bench instead of splitting slope, limits, optimization, and accumulation into detached notation too early. Derivative as Slope / Local Rate of Change keeps the moving point, tangent line, secant line, and derivative graph linked so local steepness never drifts into symbols alone, Limits and Continuity / Approaching a Value keeps one-sided approach and continuity visible at a target point, Optimization / Maxima, Minima, and Constraints turns that local-slope language into one fixed-perimeter rectangle bench where the objective curve peaks at the square, and Integral as Accumulation / Area keeps the same graph-first language while the running total grows from signed area.
Best first concepts
The topic page keeps these starts in their own compact row so the first screen is about orientation and next action, not stacked feature cards.
Slide a point along one curve, tighten a secant into a tangent, and connect local steepness to the derivative graph without leaving the same live bench.
Tangent and local change
Strong first stop for getting into this topic without scanning the whole library.
Approach one target point from the left and right, compare the limiting height with the actual function value, and contrast continuous, removable, jump, and blow-up behavior on one honest graph.
Limits and visible continuity
Strong first stop for getting into this topic without scanning the whole library.
Move one rectangle width under a fixed perimeter, watch the area curve peak, and use the local slope to see why the square is the best constrained shape.
Constrained maxima and minima
Strong first stop for getting into this topic without scanning the whole library.
Move one upper bound across a source curve and watch signed area build into a running total so accumulation stays visual instead of symbolic.
Accumulation and area
Strong first stop for getting into this topic without scanning the whole library.
Specific learning goals
These goal cards stay authored and transparent. They reuse the current topic page, starter tracks, guided collections, concept bundles, and progress cues instead of adding a separate recommendation system on top of this branch.
Use the functions topic route, the new lesson set, the compact math starter track, and the calculus topic route so graph moves, rational asymptotes, exponential change, local slope, and accumulation stay on one coherent bench.
Primary move
Open topic route
No saved progress yet inside Functions.
Entry diagnostic
Start from the opening step
No saved diagnostic checks are available yet, so the opening step is still the best entry into the collection.
Reuses the guided collection entry for Functions and Change Lesson Set, with 0 of 3 probes already ready.
No saved progress yet inside Functions.
Open the functions topic route is the next guided collection step.
Graph Transformations opens this track and sets up the rest of the path.
No saved progress yet inside Calculus.
Grouped concept overview
Each group is authored in the topic catalog, but the actual concepts, progress badges, and track cues still come from the canonical concept metadata and shared progress model.
Group 01
Keep one graph-first branch in view while the secant line collapses into the tangent, one fixed-perimeter rectangle turns that local-rate language into a real maximum, one target point keeps limit and continuity behavior visible, and signed area finally builds a running total on the same branch.
Slide a point along one curve, tighten a secant into a tangent, and connect local steepness to the derivative graph without leaving the same live bench.
Strong first stop for getting into this topic without scanning the whole library.
從左與右逼近同一個目標點，比較極限高度與函數實際值，並在同一張圖上對照連續、可去間斷、跳躍與發散行為。
Strong first stop for getting into this topic without scanning the whole library.
在固定周長下改變矩形寬度，觀察面積曲線如何達到峰值，並用局部斜率看出正方形為何是最佳受限形狀。
Strong first stop for getting into this topic without scanning the whole library.
把上限沿著來源曲線移動，看有號面積如何累積成為運行總量，讓累積保持可視化而不是只剩符號。
Strong first stop for getting into this topic without scanning the whole library.