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AP Calculus Lesson

Toll Road Speed Trap: From Average to Instantaneous Rate of Change

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11th-12th GradeThin-SlicingCHA-2.A, CHA-2.B50 min
Learning Objective

Students will be able to define the derivative of a function at a point as the limit of the average rate of change over a shrinking interval, set up and evaluate the limit definition of the derivative, and interpret the result as an instantaneous rate of change in context (CHA-2.A, CHA-2.B).

Lesson Overview

Students progress from computing average rates of change over intervals to discovering that shrinking the interval toward zero produces the instantaneous rate of change — the derivative — using the limit definition. The toll-road context grounds every step in physical intuition.

Materials

  • Vertical whiteboards or large chart paper
  • Dry-erase markers (multiple colors)
  • Random grouping cards or spinner
  • Four-Quadrant Note-Taking handout (one per student)
  • Graphing calculators (optional)

Scaffolded Task Progression

1Task 1
A driver's position (in miles) along a toll road is given by s(t) = 60t, where t is in hours. The car enters at t = 0 and exits at t = 1.a) What is the car's position at t = 0? At t = 1?b) Compute the average speed over [0, 1].c) Is the driver speeding if the limit is 65 mph? [ Interactive graph / diagram — available in the full lesson ]
2Task 2
Now the driver's position is s(t) = 75t (miles, t in hours). The car enters at t = 0 and exits at t = 1.
a) Compute the average speed over [0, 1].b) Is the driver speeding?c) Can we be sure they were speeding at every single moment, or just on average? [ Interactive graph / diagram — available in the full lesson ]

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