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Calculus BC · AP · Contextual Applications of Differentiation

CED focus

Rates of Change in Applied Contexts Other Than Motion

This topic is assessed on the AP Exam.

Multiple-choice tips

  • Identify whether the stem tests a definition, rule, or application.
  • Identify the independent variable (often time) in related-rates MC.
  • Eliminate L’Hospital uses when the limit is not indeterminate.
  • Check linearization: is the estimate over or under?
  • Match units on related-rates distractors.

Free-response tips

  • Label f′(a) meaning in context; show the rule that justifies each step.
  • Draw and label a related-rates diagram before differentiating.
  • Show dV/dt (or similar) setup with correct chain rule.
  • State L’Hospital hypotheses before applying.
  • For linearization, show f(a) + f′(a)(x − a) with a labeled tangent point.

Common traps

  • Mixing average rate over an interval with instantaneous rate at a point
  • Plugging values before differentiating in related rates
  • Using L’Hospital on a non-indeterminate form
  • Linearizing at the wrong point a

Quick checklist

  • [ ] Can set up a related-rates equation with labels
  • [ ] Can state when L’Hospital applies
  • [ ] Can write a local linearization formula

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