AP Calculus

Special Focus: The Fundamental
Theorem of Calculus
(d) Use the information in parts (a) through (c) to sketch the graph of y = Si( x) .
Then, use technology, if available, to confirm the behavior seen in the sketch.
2 x 2
−t
3. Consider the error function erf ( x) = ∫ e dt. The integrand is the Gauss curve,
π
0
and the error function is used extensively in probability and statistics calculations.
(a) Complete the following table of values for erf ( x).
x 1 2 3 4 5 6
erf(x)
x –1 –2 –3 –4 –5 –6
erf(x)
(b) Using the table in part (a), describe the behavior of erf ( x) as | x | gets large
What does this suggest for the values of lim erf ( x) and lim erf ( x) ? What
x→∞
x→−∞
characteristics of the graph of y = erf ( x) does this suggest?
(c) Use the Fundamental Theorem of Calculus to find the derivative, erf ′( x),
of the error function.
(d) Sketch a graph of the derivative of the error function and use this graph to
determine the intervals on which the graph of y = erf ( x) is increasing and
decreasing, and where the graph is concave up and concave down.
(e) Find erf ( 0 ) and use the information about increasing, decreasing, and
concavity to sketch a graph of y = erf ( x).
4. The purpose of this problem is to combine the ideas of the Fundamental Theorem of
Calculus and the chain rule.
(a) Consider an arbitrary function f and the function g( x) = x
2 . Let
h( x) = ( f g)( x)
and find h′
( x).
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AP® Calculus: 2006–2007 Workshop Materials