AP Calculus AB Lesson

From Average to Instantaneous: Building the Derivative Concept

Shared by Rebeca Heredia★ Featured Teacher · rated 90+ in our quality review

12th GradeBTC Style / Thinking ClassroomsAP Calculus AB Learning Objectives CHA-2.B, CHA-2.D50 min

Learning Objective

Students will be able to compute average rates of change over an interval, interpret them graphically as slope of a secant line, and use limit notation to define and evaluate the instantaneous rate of change (derivative) at a point.

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 — connecting limits (prior knowledge) to the formal definition of the derivative.

Materials

Vertical whiteboards or large paper
Dry-erase markers (multiple colors)
Pencils and notebook paper

Scaffolded Task Progression

1Task 1

A car's position (in miles) is given by

s (t) = 3 t^{2}

where

t

is in hours. Find the average rate of change of

s

from

t = 1

t = 3

. Include units and interpret your answer in one sentence.

2Task 2

Using

s (t) = 3 t^{2}

, find the average rate of change from

t = 1

t = 2

. Then from

t = 1

t = 1.5

. Record both answers. What do you notice as the interval gets smaller?

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