AP Calculus

Special Focus: The Fundamental
Theorem of Calculus
(d) We can learn more about the shape of the graph y = S(x) by considering
the second derivative. Find S"(x) and the intervals on which the graph of
y = S(x) is concave up and the intervals on which the graph is concave down.
The second derivative also involves a trigonometric function and leads to a
similar pattern of intervals on which the Fresnel function is concave up and
concave down.
(e) Put this information (increasing, decreasing, and concavity) together to sketch
a graph of y = S(x). Find S(0) first, and be careful as you sketch the local
maximum peaks and local minimum valleys as | x | increases. What do you
suppose happens to the values of S(x) as | x | increases? Use technology to
sketch y = S(x) and to find some values of S(x) for large values of | x | .
(Even with a fast calculator, this may take a while.)
x sin( t)
2. Consider the sine integral function defined by Si( x)
= ∫ dt .
0 t
sin( t)
The integrand, f ( t)
= , is not defined at t = 0, but the limit is 1. Define
t
f(0) = 1 to make f continuous everywhere. The integrand is called the Sine Cardinal
function, Cardinal Sine function, or the sinc function. As you will see in this problem,
the graph of the integrand exhibits damped harmonic oscillation, that is, harmonic
oscillation with diminishing amplitude. The Sine Cardinal function is used in signal
processing and Fourier analysis and is related to the spherical Bessel function of the
first kind.
(a) Use the Fundamental Theorem of Calculus to find Si′( x) on [ −10, 10 ]. Use
technology to sketch a graph of y = Si′( x). Approximate several of the zeros of
the derivative. Try to find a pattern in the zeros.
(b) For x > 0, find the first five local maximum values and first five local
minimum values. You should be able to solve this problem analytically:
find the values of x such that Si′ ( x) = 0 and use a sign chart. Check your
solutions using technology. Describe the behavior of these values (the
maximum and minimum values). What characteristic of the graph of
y = Si( x) does this suggest? Do the same thing for x < 0.
(c) Unfortunately, there is no nice, inherent pattern in the inflection points.
However, we want to find at least one of these. Find the coordinates of the first
inflection point to the right of the origin.
AP® Calculus: 2006–2007 Workshop Materials 113