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AP Calculus · Unit 4

Contextual Applications of Differentiation

Rates in context, motion, related rates, linear approximation and L'Hôpital's rule.

Topic 4.1Explain a derivative in context

  1. Identify the quantity and its rateDistinguish a function's amount from its derivative's rate of change.
  2. Interpret a derivative's value and signSelect an accurate statement about how the quantity is changing at the specified input.
  3. Give the derivative's unitsExpress output units per input unit, including conversions when needed.
  4. Interpret a changing rateDistinguish an increasing quantity from an increasing rate of change.

Topic 4.2Motion on a line

  1. Connect position, velocity, and accelerationDifferentiate the appropriate function and evaluate the requested quantity.
  2. Find rest times and direction of motionUse the sign and zeros of velocity, including intervals.
  3. Identify a change of directionCheck a velocity sign change rather than relying on a zero alone.
  4. Decide when speed increases or decreasesCompare the signs of velocity and acceleration.
  5. Distinguish speed from velocityUse the magnitude of velocity and interpret signs correctly.

Topic 4.3Rates outside motion

  1. Choose the quantity whose rate is neededIdentify the dependent and independent variables in a non-motion context.
  2. Calculate the contextual derivativeDifferentiate the model, including products or quotients of changing quantities.
  3. Interpret a signed rate with unitsSelect a statement that names the quantity, instant, sign, and units correctly.
  4. Separate amount, average rate, and instantaneous rateDo not interpret a derivative as a total or as a whole-interval average.

Topic 4.4Set up related rates

  1. Identify the changing quantitiesDistinguish variables that change with time from fixed lengths or constants.
  2. Write the relationship between themSelect or construct the geometric or algebraic constraint.
  3. Differentiate with respect to timeInclude a rate for each changing quantity, using chain and product rules as needed.
  4. Substitute information at the correct stageDifferentiate the general relationship before inserting instantaneous measurements.

Topic 4.5Solve related rates

  1. Build and differentiate the rate relationshipTranslate the situation into an equation linking the changing quantities and their rates.
  2. Find any missing instantaneous measurementUse the constraint to determine a length, angle, area, or other needed value.
  3. Solve for the requested rateSubstitute the supplied values and isolate the unknown rate.
  4. Interpret the resultGive the correct sign and units and distinguish related but different rates.

Topic 4.6Approximate with a tangent line

  1. Build a tangent-line approximationUse the function value and derivative at a suitable nearby input.
  2. Estimate a nearby function valueSubstitute the target input into the linear model.
  3. Predict whether the estimate is high or lowUse concavity on the relevant interval rather than guessing from the derivative's sign.
  4. Use supplied values or a tableForm the approximation without needing an explicit formula for the function.

Topic 4.7Use L’Hôpital’s rule carefully

  1. Check the form and conditions for L'Hôpital's ruleRecognize zero over zero or infinity over infinity, check the required differentiability and denominator-derivative conditions near the limit, and reject an unjustified application.
  2. Differentiate numerator and denominator separatelyApply the rule to the quotient limit without using the quotient rule.
  3. Check the form again after each stepRepeat the rule only when its conditions remain appropriate.
  4. Rewrite other indeterminate formsConvert a product, difference, or variable power into a suitable quotient when possible.