AP Calculus

Special Focus: The Fundamental
Theorem of Calculus
(c) Consider these zeros, the graph of y = S′( x), and/or a sign chart to find the
intervals on which S(x) is increasing and decreasing.
Increasing: ...,[ − 10, − 8], [ − 6, − 4], [ − 2, 2], [ 4, 6], [ 8, 10 ],...
Decreasing: ...,[ − 8, − 6], [ − 4, − 2], [ 2, 4], [ 6, 8 ],...
⎛
(d) Find the second derivative using the chain rule: S′′ ( x) = π xcos⎜
⎝
⎛ π x
Set S′′ ( x) = 0 and solve for x: π xcos⎜
⎝ 2
⎛
We conclude that x = 0 or cos π x2 ⎞
⎜ ⎟ = 0.
⎝ 2 ⎠
2
⎞
⎟ = 0.
⎠
π x
2
2
Since the cosine is zero when its argument is equal to π 2 + k π , for some
integer k, π 2
x π kπ
2 = 2
+ for some integer k. In other words, x 2
1
= +
2 2
k for
2
some integer k, and consequently, x = 1+ 2k
for some integer k. This implies
that x = ± 1+
2 k for some integer k. Finally, we conclude that S′′ ( x) = 0
when x = 0, ± 1, ± 3, ± 5, ....
Consider these values and where the graph of y = S′( x) is increasing/decreasing to
find the intervals on which the graph of y = S( x) is concave up/concave down.
Concave up: ...,( − 7, − 5), ( − 3, − 1), ( 0, 1), ( 3, 5), ( 7, 9 ),...
Concave down: ...,( − 9, − 7), ( − 5, − 3), ( −1, 0), ( 1, 3), ( 5, 7 ),...
⎞
⎟ .
⎠
AP® Calculus: 2006–2007 Workshop Materials 117