Calculus: local rates and accumulated change
Connect limits, derivatives and integrals, use the chain rule and partial derivatives, and justify an optimization claim with its domain and constraints.
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Recommended preparation and ways to catch up
- Substitute into algebraic functions and interpret graphs.
- Compute interval slopes; the existing nonlinear-change lesson supplies a recap.
- Square roots and basic circle area support the related-rates example.
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Your units and lessons
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From intervals to local rates
Distinguish a function value from a limit and test whether a derivative exists.
Recognize when the rate changes
Compare equal input steps on a line and a curve without assuming a constant rate.
Nearby behavior is not the value at the point
Can a function approach a number where its own value is undefined?
Shrink an interval without dividing by zero
When does an average rate become a unique local rate?
Recover total change
Interpret an integral as signed accumulation and connect antiderivatives with rates.
Accumulation can cancel without erasing motion
Why can displacement be zero while distance traveled is positive?
Follow several changing quantities
Use chain factors and partial derivatives while stating what is held fixed.
A changing radius creates a changing area
How does a length rate become an area rate?
Change one input while holding another fixed
How does a two-input function have more than one local rate?
Follow a gradient through a computation
Connect a derivative, chain factors, partial gradients and a numerical check.
Make a constrained choice
Use existing slope and optimization experiments to compare stationary candidates, endpoints and feasible designs.
Use a slope to improve a choice
When does following the downhill direction overshoot?
Build the largest enclosure
Which rectangle gives the most space for a fixed fence?
Explain and optimize a changing system
Choose a geometric growth, motion or bounded design model. Connect a local rate to finite change, calculate accumulated change where appropriate, and justify an optimum only within a stated feasible domain. Use captures from at least two calculus mechanisms and include a counterexample to a tempting shortcut.
Build from lesson evidence, compare alternatives, write an artifact, and tackle unfamiliar numerical cases. Use the rubric with a teacher, peer or colleague to assess the quality of your reasoning.
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