Using derivatives
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.
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Using derivatives
A tangent is a local prediction tool.
A small measurement change can be translated into an approximate output change using sensitivity at a reference point.
Explore the idea ↗Interactive lessonFind critical points, then inspect the interval.
Choosing a maximum response requires more than finding where a derivative is zero. Endpoints may win.
Explore the idea ↗Interactive lessonAn average rate must occur somewhere instantaneously.
A trip with a known average velocity must match that velocity at some instant under the theorem's smoothness conditions.
Explore the idea ↗Interactive lessonRead shape from first and second derivatives.
A response can be increasing while flattening out. Separate slope sign from the direction in which slope changes.
Explore the idea ↗Interactive lessonCheck the indeterminate form before changing the quotient.
An apparently undefined small-input response can have a finite limit. A valid limit tool should be applied only after checking its hypotheses.
Explore the idea ↗Interactive lessonA constraint turns a design question into one variable.
A river supplies one side of an enclosure. Decide how to distribute a fixed amount of fencing to maximize useful area.
Explore the idea ↗Interactive lessonA tangent can guide a root search.
A numerical solver predicts where a local tangent hits zero, then repeats the prediction from that new location.
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