Foundations and limits
Change you can see.
Explore limits, derivatives, and accumulation through motion, measurement, and design.
Follow the real situation, derive the mathematics, and test it with interactive graphs and worked practice.
Start exploring ↗Reading your saved completion. Completion is saved on this browser.
Foundations and limits
Functions describe a changing system.
A cart's position depends on time. Choose an input and watch its position respond; the graph records that relationship rather than the shape of its path.
Explore the idea ↗Interactive lessonUndo a measurement with an inverse.
A sensor reports a squared displacement. To recover displacement, you must know which sign of the original input is allowed.
Explore the idea ↗Interactive lessonFollow a value toward a missing point.
A measurement formula can have a hole at one input while its nearby predictions approach a perfectly finite value.
Explore the idea ↗Interactive lessonA limit and a value must agree.
A sensor calibration should not produce an isolated jump in output. Move the assigned value and see whether it joins the surrounding response.
Explore the idea ↗Interactive lessonA continuous response must cross intermediate levels.
If a continuous temperature or sensor response changes from below a target to above it, some intermediate input reaches that target.
Explore the idea ↗Interactive lessonA tiny angle connects arc length and height.
A small rotation changes a point's vertical coordinate almost in proportion to the rotation angle. The unit chosen for that angle matters.
Explore the idea ↗Interactive lessonA response can level off without ever reaching its limit.
Increasing the input can produce diminishing changes. A rational response illustrates how a horizontal asymptote emerges.
Explore the idea ↗Interactive lessonChoose an input tolerance that guarantees an output tolerance.
A calibration engineer specifies an allowed output error. The slope determines how tightly the input must be controlled.
Explore the idea ↗