AP Calculus

Special Focus: The Fundamental
Theorem of Calculus
Since there are no x-values where the derivative does not exist, the only
critical points are x = 2, 0, –2. The domain of H is all real numbers, and
H is continuous everywhere (its derivative exists everywhere). The Extreme
Value Theorem does not apply because the domain is not a closed interval.
We are not guaranteed the existence of any absolute extrema.
Consider a sign chart:
H has a local minimum at x = –2 and at x = 2 because H′
( x) changes from
negative to positive at x = –2 and at x = 2. H has a local maximum at x = 0
because H′
( x) changes from positive to negative at x = 0.
H is increasing on [ −2, 0] ∪ [ 2, ∞)
.
H is decreasing on ( −∞, −2] ∪ [ 0, 2]
.
−2 5
H( − 2) = H( 2) = 60 − 100e
= −7.
032, H( 0)
= 0, and H( − 4) = H( 4)
> 0.
−2 5
The absolute minimum value of H is H( − 2) = H( 2) = 60 − 100e
= −7.
032.
H has no absolute maximum value.
(d) Using technology, here is a graph of y = H( x):
AP® Calculus: 2006–2007 Workshop Materials 123