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Interpret marginal rates and local sensitivities in supplied continuous cost, revenue and response models. Requires derivative meaning, logarithmic derivatives and linearization. Distinguish marginal rates from averages and exact finite changes, and use units to estimate small input or output adjustments.
Oct 27, 2026
Use differentials to estimate the effect of one small measurement error, retaining signs, units and a nonzero reference for relative error. Requires linearization. Compare first-order estimates with exact changes and recognize when a claimed tolerance has not been proved.
Oct 27, 2026
Build a tangent-line model from a known value and derivative, choose a useful base point, and estimate nearby outputs. Requires tangent lines and the chain rule. Distinguish local predictions from exact values and unsupported distant extrapolations.
Oct 27, 2026
Check your choice of implicit, inverse and logarithmic differentiation methods, including branch, domain and joining conditions. Requires the six Chapter 06 skills. Twelve independent questions diagnose method choice and local validity; no new theory is introduced.
Oct 27, 2026
Match values and finite one-sided slopes at a piecewise join. Solve for parameters, distinguish no solution from nonunique solutions, and verify the result with the original difference quotients. Prerequisites: continuity, elementary derivatives and differentiability failures. Cover: [Piecewise_smooth.svg](https://commons.wikimedia.org/wiki/File:Piecewise_smooth.svg) by Shim'on; [Public domain](https://commons.wikimedia.org/wiki/File:Piecewise_smooth.svg). Rasterized and padded for display.
Oct 27, 2026
Choose logarithmic differentiation for multiplicative expressions and positive-base variable powers. Keep track of nonzero intervals and convert relative rate into the actual derivative. Prerequisites: logarithmic, exponential and implicit differentiation. Cover: [Logarithmic_functions.svg](https://commons.wikimedia.org/wiki/File:Logarithmic_functions.svg) by Christoffer Stausland; [CC BY-SA 3.0](https://creativecommons.org/licenses/by-sa/3.0/). Rasterized and padded for display. The adapted cover is also licensed CC BY-SA 3.0.
Oct 27, 2026
Differentiate principal arcsine, arccosine and arctangent compositions in radians. Use branch signs, chain factors and real domains to interpret angle recovery. Prerequisites: inverse derivatives and elementary trigonometric derivatives. Cover: [Arcsin.svg](https://commons.wikimedia.org/wiki/File:Arcsin.svg) by Keytotime; curve refined by Geek3; [Public domain](https://commons.wikimedia.org/wiki/File:Arcsin.svg). Rasterized and padded for display.
Oct 27, 2026
Recover an input rate from an output rate using the correct inverse-function point. Practice reciprocal slopes, table lookup, local branch conditions and calibration sensitivity. Prerequisites: function inverses and the chain rule. Cover: [Inverse.svg](https://commons.wikimedia.org/wiki/File:Inverse.svg) by Markus Schmaus; [Public domain](https://commons.wikimedia.org/wiki/File:Inverse.svg). Rasterized and padded for display.
Oct 27, 2026
Find implicit-curve tangents and distinguish finite, vertical, and unresolved first-order cases. Requires introductory implicit differentiation and tangent-line meaning. Check curve membership, continuous first partial derivatives, and nonzero coefficients before classifying a point.
Oct 27, 2026
Differentiate a constraint while treating the dependent coordinate as a local function. Requires product and chain rules. Track every dependent-rate factor, check a point on the original relation, and isolate a rate only when its coefficient is nonzero.
Oct 27, 2026
Check your choice and interpretation of differentiation rules across algebraic, trigonometric, exponential, and logarithmic models. Requires the Chapter 05 rules and local-rate meaning. Use symbolic expressions, exact data, domain checks, and a short mixed model; review errors by skill after the attempt.
Oct 27, 2026
Practice successive derivatives, rate units, and the distinction between a quantity and its changing rate. Bring the basic derivative rules, chain rule, and signed-motion interpretation. Use short computations and rate data to explain how a decreasing quantity can have an increasing rate.
Oct 27, 2026
