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Connect secant and tangent slopes under the mean value theorem, with exact interval conditions and rate units. Requires average rates and Rolle’s theorem. Calculate admissible interior points and diagnose claims about midpoints, endpoint averages and nonsmooth reports. Cover: [Photo by Gezer Amorim on Pexels](https://www.pexels.com/photo/dashboard-in-car-23533488/), [Pexels License](https://www.pexels.com/license/).
Oct 27, 2026
Check Rolle’s three hypotheses separately and interpret or calculate interior horizontal tangents. Requires continuity, differentiability and critical-number reasoning. Separate sufficient guarantees from direct conclusions when a hypothesis fails. Cover: [Photo by detait on Pexels](https://www.pexels.com/photo/gray-concrete-arch-bridge-over-river-6410671/), [Pexels License](https://www.pexels.com/license/).
Oct 27, 2026
Select all valid extremum candidates, including corners and endpoints, while excluding points outside the original domain. Requires extrema, differentiability and elementary derivative rules. Critical numbers mean interior domain points; endpoints are checked separately. No second derivative test is needed. Cover: [Photo by James Gampper on Pexels](https://www.pexels.com/photo/urban-staircase-with-modern-architecture-in-view-34495387/), [Pexels License](https://www.pexels.com/license/).
Oct 27, 2026
Distinguish local and absolute extrema, and decide when continuity on a closed bounded interval guarantees attained best and worst values. Requires continuity and basic graph reading. Only interior points count as local extrema in this quiz; endpoints can be absolute extrema. Cover: [Photo by Aydın Kiraz on Pexels](https://www.pexels.com/photo/landscape-with-mountains-and-valley-8863813/), [Pexels License](https://www.pexels.com/license/).
Oct 27, 2026
Check your choice of local estimates and simultaneous-rate models without method labels. Requires the six earlier local-change skills; interpret assumptions, signs and units and diagnose the first unjustified step in a proposed solution.
Oct 27, 2026
Convert camera angles, radar ranges and calibrated readings into physical rates. Requires chain-rule and trigonometric derivatives in radians; distinguish measured rates from object speed and interpret signs and units.
Oct 27, 2026
Apply signed related rates to moving ladders, expanding shapes and conical tanks. Requires differentiation of a constraint; supplied formulas and similarity connect dimensions to one unknown rate.
Oct 27, 2026
Connect simultaneous rates by differentiating a changing constraint before substituting data. Requires the chain rule, product rule and implicit differentiation; practice signed rates, units and consistency checks.
Oct 27, 2026
Optimize geometric area or volume under one explicit resource constraint. Requires feasible-domain modeling and first/second derivative tests. Practice separating an optimizing dimension from the optimum value, rejecting impossible stationary points, and checking boundaries for a global conclusion.
Oct 27, 2026
Translate a fixed-resource situation into a one-variable objective with a physically feasible domain. Requires elementary extrema and critical-number ideas plus algebraic substitution. Distinguish what is optimized from the constraint and explain which endpoints represent valid designs.
Oct 27, 2026
Check theorem hypotheses, derivative sign tests and short counterexamples that connect local rates to global conclusions. Requires derivative bounds, extremum tests and curve synthesis. This checkpoint introduces no new theorem; use the local retrieval map and objective-specific remediation after attempting it.
Oct 27, 2026
Combine domain, asymptotes, first-derivative signs and concavity to test a proposed curve. Requires elementary rational limits and first/second derivative sign charts. Practice selecting compatible graph features and repairing a sketch; the separate local companion develops full sketch construction.
Oct 27, 2026
