AP Calculus

Special Focus: The Fundamental
Theorem of Calculus
The graph of y = erf ( x) is concave up on the interval ( −∞, 0 ) and concave
down on the interval ( 0, ∞ ).
2 0 2
−t
(e) erf ( 0)
= ∫ e dt = 0.
π
0
Using technology, here is a graph of y = erf ( x). Notice that this graph confirms
the results obtained in parts (a) through (d).
4. (a) h(x) is defined as the composition of two functions. Although we do not have a
specific rule for the function f, we can still simplify the derivative of h:
2 .
h( x) = ( f g)( x) = f ( g( x)) = f ( x )
We use the chain rule to obtain that h′ ( x ) d
= = ′
dx f ( x 2 ) f ( x 2
)( 2 x)
.
(b) Let G( x) = x
2 x −t
10
and F( x) = ∫ ( t − 4)
e dt.
0
x
Then ( F G)( x) = F( G( x)) = F( x 2 2
−
) = ( t − 4) e t 10 dt = H(
x) .
∫
0
2
2 2 − 10
As in part (a), H′ ( x) = ( 2x) F′ ( x ) = ( 2x)( x − 4)
e x
2
2 −x
10
(c) H '( x) = 0⇔( 2x)( x − 4)
e = 0 ⇔ 2x
= 0 or x 2 − 4 = 0 or e x
Thus we conclude that H′ ( x) = 0 ⇔ x = 0 or x = ± 2 (since e x
never zero).
.
− 2 10
=
− 2 10 is
0.
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AP® Calculus: 2006–2007 Workshop Materials