AP Calculus

Special Focus: The Fundamental
Theorem of Calculus
(c) This question will be one of the most challenging for students because the
composite function is created by the upper limit. The domain of f is 0 ≤ x ≤ 4.
Since f is continuous on this interval, F 3 accepts all x for which 0 ≤ 2x ≤ 4.
This implies that the domain for F 3
is 0 ≤ x ≤ 2, and the graph that results is a
horizontal compression of the graph of F. It might help for students to build a
table to investigate the values F 3
( 0), F 3
( 1 2), F 3
( 1), F 3
( 3 2), and F 3
( 2) .
The graph of F 3
on [0,2]
(d) Students must next consider y = f (| t|) and must realize that it is an even
function and that its domain requires 0 ≤ t ≤ 4 or, in other words, −4 ≤ t ≤ 4.
The graph of y = f (| t|) is given below.
The graph of f
(| t|) on [–4,–4]
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AP® Calculus: 2006–2007 Workshop Materials