AP Calculus

Special Focus: The Fundamental
Theorem of Calculus
Examples to Reinforce Concepts That Are Connected to the
Fundamental Theorem of Calculus
Steve Kokoska
Bloomsburg University
Bloomsburg, Pennsylvania
When the Fundamental Theorem of Calculus is introduced, most students have plenty of
straightforward practice evaluating definite integrals and using the inverse relationship
between the process of differentiation and of integration:
d
dx
x
∫ f ( t) dt = f ( x).
a
The following examples focus on applications that involve the FTC and usually appear
later in a calculus, physics, probability, or statistics course.
x ⎛ πt
⎞
1. The Fresnel function defined by S( x) = ∫
2
sin ⎜ ⎟ dt can be used to measure the
0
⎝ 2 ⎠
amount of reflection versus refraction. Even for dark, dense material, some reflection
takes place, especially at sharp angles, and S( x) can also be used to determine the
color reflected off metal.
(a) Use the Fundamental Theorem of Calculus to find the derivative S'(x)
of the Fresnel function.
(b) Sketch a graph of y = S′( x) on [ − 4, 4 ]. This is a little tricky by hand because
of the t 2 in the argument of the sine. Since the derivative involves a common
trigonometric function, you can be pretty sure the graph oscillates. However,
the graph does not have a constant period. Use technology to help construct
this graph and find the pattern for the zeros of the derivative—the values of x
for which the derivative crosses the x-axis.
(c) Use the derivative, S'(x), and its graph to find the find the intervals on which
the graph of y = S(x) is increasing and the intervals on which the graph is
decreasing. Even though the function S(x) is an accumulation function, its
derivative still provides the usual information about the graph of y = S(x).
The graph and pattern of zeros in part (b) lead to an interesting pattern of
intervals here.
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AP® Calculus: 2006–2007 Workshop Materials