AP Calculus

Special Focus: The Fundamental
Theorem of Calculus
These values look like multiples of π ! By hand, we investigate when Si′ ( x) = 0.
sin( x)
Si′ ( x)
= 0 ⇒ = 0 ⇒ sin( x)
= 0 and x ≠ 0 ⇒ x = ± kπ for some
x
integer k ≠ 0
x = ± π, ± 2π, ± 3 π, ...
(b) Consider these zeros and the graph of y = Si′( x) to find local maximum and
local minimum values.
For x >0:
Local maximum values:
Si( π ) =1.
8594, Si( 3π ) = 1.
6748, Si( 5π ) = 1.
6340, Si( 7π ) = 1.
6161,
Si( 9π ) = 1.
6061
Local minimum values:
Si( 2π ) = 1.
4182, Si( 4π ) = 1.
4922, Si( 6π ) = 1.
5180, Si( 8π ) = 1.
5311,
Si( 10π ) = 1.
5390
The local maximum values are getting smaller as x increases, and the local
minimum values are getting bigger as x increases. Although it’s hard to tell,
even using technology, these values are converging, and the graph of y = Si( x)
π
has a horizontal asymptote at y = ≈
2 1. 571.
For x