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Classify stationary points with the second derivative test when its hypotheses hold, and switch to a first derivative sign chart when it is inconclusive. Requires critical numbers and concavity. Practice checking stationarity before drawing an extremum conclusion.
Oct 27, 2026
Interpret concavity through changing slopes, check actual changes of concavity, and judge tangent-line estimates on a stated interval. Requires higher derivatives, derivative signs, and linearization. Distinguish increasing output from an increasing rate.
Oct 27, 2026
Read derivative signs over valid domain intervals and use sign changes to classify local extrema. Requires critical numbers and the mean value theorem. Connect factored derivatives, rate graphs, and nearby function behavior without finding antiderivatives. Cover: “Extrema example” by KSmrq, Wikimedia Commons, https://commons.wikimedia.org/wiki/File:Extrema_example.svg, CC BY-SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0/). Rasterized and fitted to a white canvas; adapted image remains CC BY-SA 3.0.
Oct 27, 2026
Use derivative bounds to constrain total change and separate root existence from uniqueness. Requires the mean value theorem and intermediate value theorem. Practice interval assumptions, strict inequalities, and checks on rate measurements.
Oct 27, 2026
Analyze positive-base indeterminate powers through natural logarithms and recover the original power limit. Requires logarithmic laws, variable powers and quotient-limit methods. Check real domains, exponent rates and finite or infinite logarithmic outcomes. Cover: [Xth root of x.svg](https://commons.wikimedia.org/wiki/File:Xth_root_of_x.svg) by Jalanpalmer; latest SVG by Rubber Duck; [Public domain](https://commons.wikimedia.org/wiki/File:Xth_root_of_x.svg). Rasterized or resized and padded; no data changed.
Oct 27, 2026
Convert indeterminate products and differences into equivalent quotients using reciprocals, common denominators and conjugates. Requires algebraic limit methods and the hypotheses of L’Hôpital’s rule. Choose a useful rewrite before taking derivatives. Cover: [X-lbx.pdf](https://commons.wikimedia.org/wiki/File:X-lbx.pdf) by Jochen Burghardt; [CC BY-SA 4.0](https://creativecommons.org/licenses/by-sa/4.0/). Rasterized or resized and padded; no data changed. Adapted cover also CC BY-SA 4.0.
Oct 27, 2026
Practice rechecking every application of L’Hôpital’s rule, handling one-sided domains and preserving signs. Requires the rule’s local hypotheses and higher derivatives. Learn when to stop and when an inconclusive derivative ratio calls for another limit method. Cover: [Mplwp ln.svg](https://commons.wikimedia.org/wiki/File:Mplwp_ln.svg) by Geek3; [CC BY 3.0](https://creativecommons.org/licenses/by/3.0/). Rasterized or resized and padded; no data changed.
Oct 27, 2026
Learn to recognize indeterminate quotients and justify L’Hôpital’s rule on a suitable nearby interval. Requires limit laws, established derivative rules and mean value reasoning. Practice choosing algebra or differentiation and separating a limit form from its value. Cover: [Hopital sin x by -0.5x.png](https://commons.wikimedia.org/wiki/File:Hopital_sin_x_by_-0.5x.png) by Jochen Burghardt; [CC BY-SA 4.0](https://creativecommons.org/licenses/by-sa/4.0/). Rasterized or resized and padded; no data changed. Adapted cover also CC BY-SA 4.0.
Oct 27, 2026
Use independently established exponential limit facts to justify exponential and logarithmic derivative rules. Prerequisites: limit laws, exponential/logarithm structure and the chain rule. Distinguish fixed-base exponentials from powers and retain real logarithm domains.
Oct 27, 2026
Derive and apply tangent, cotangent, secant and cosecant derivatives while retaining their denominator restrictions. Prerequisites: sine/cosine derivatives and the quotient rule. Use equivalent trig forms and interpret simple viewing-angle sensitivities.
Oct 27, 2026
Connect radian sine and cosine derivatives to local graph slopes, phase and angular frequency. Prerequisites: foundational trigonometric limits, radian measure and the chain rule. Includes one explicitly converted degree input and short periodic-rate interpretations.
Oct 27, 2026
Practice tracing nested dependencies and selecting product, quotient and chain rules. Prerequisites: the quotient rule and a single composition. Learn to retain every inner rate and the original real domain while choosing an efficient representation.
Oct 27, 2026
