AP Calculus AB Lesson

FTC Part 1: Accumulation Functions and the Rate-Distance Connection

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11th-12th GradeThin-SlicingAP Calculus AB Learning Objective FUN-5.A, FUN-5.B50 min

Learning Objective

Students will be able to define and evaluate accumulation functions of the form $F (x) = \int_{a}^{x} f (t) d t$ , interpret them in context, and apply FTC Part 1 — including the chain rule extension — to differentiate accumulation functions.

Lesson Overview

Students build intuition for accumulation functions using a car velocity context, then formalize FTC Part 1 by discovering that differentiating an integral returns the integrand. Tasks progress from estimating area under a velocity curve to differentiating complex accumulation functions with chain rule.

Materials

Vertical whiteboards or chart paper
Dry-erase markers
Colored markers (2 colors per group)
Random grouping tool
Printed velocity graph reference (optional)

Scaffolded Task Progression

1Task 1

A car's velocity (in miles per hour) is given by the graph below, which shows

v (t) = 20

(a constant) for

0 \leq t \leq 4

hours. [ Interactive graph / diagram — available in the full lesson ] Define the accumulation function

F (x) = \int_{0}^{x} 20 d t

for

0 \leq x \leq 4

.(a) Evaluate

F (1)

F (2)

F (3)

, and

F (4)

. What does each value represent in context?(b) Plot these four points and describe the shape of

F (x)

2Task 2

Now the car's velocity is

v (t) = 2 t

(mph), starting from

t = 0

. [ Interactive graph / diagram — available in the full lesson ] Define

F (x) = \int_{0}^{x} 2 t d t

for

0 \leq x \leq 4

.
(a) Use the area of a triangle to find

F (1)

F (2)

F (3)

F (4)

.(b) What kind of function is

F (x)

? Write a formula for

F (x)

.(c) Find F'(x). What do you notice?

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