5.1. CONSTRUCTING ACCURATE GRAPHS OF ANTIDERIVATIVES
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(c) At what point(s) does A have a relative minimum? a relative maximum?
(d) Use the given information to determine the exact values of A(0), A(1), A(2),
A(3), A(4), A(5), and A(6).
(e) Based on your responses to all of the preceding questions, sketch a complete
and accurate graph of y = A(x) on the axes provided, being sure to indicate
the behavior of A for x < 0 and x > 6.
(f) How does the graph of B compare to A if B is instead defined by B(x) =
x
0
g(t) dt?
⊳
Summary
In this section, we encountered the following important ideas:
• Given the graph of a function f , we can construct the graph of its antiderivative F
provided that (a) we know a starting value of F, say F(a), and (b) we can evaluate
the integral
b
a
f (x) dx exactly for relevant choices of a and b. For instance, if we
wish to know F(3), we can compute F(3) = F(a) +
3
a
f (x) dx. When we combine this
information about the function values of F together with our understanding of how the
behavior of F ′ = f affects the overall shape of F, we can develop a completely accurate
graph of the antiderivative F.
• Because the derivative of a constant is zero, if F is an antiderivative of f , it follows that
G(x) = F(x) + C will also be an antiderivative of f . Moreover, any two antiderivatives
of a function f differ precisely by a constant. Thus, any function with at least one
antiderivative in fact has infinitely many, and the graphs of any two antiderivatives will
differ only by a vertical translation.
• Given a function f , the rule A(x) =
x
a
f (t) dt defines a new function A that measures
the net-signed area bounded by f on the interval [a, x]. We call the function A the
integral function corresponding to f .
Exercises
1. A moving particle has its velocity given by the quadratic function v pictured in
Figure 5.6. In addition, it is given that A 1 =
7
6 and A 2 =
8
3 , as well as that for the
corresponding position function s, s(0) = 0.5.
(a) Use the given information to determine s(1), s(3), s(5), and s(6).
(b) On what interval(s) is s increasing? On what interval(s) is s decreasing?
(c) On what interval(s) is s concave up? On what interval(s) is s concave down?
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(c) At what point(s) does A have a relative minimum? a relative maximum?
(d) Use the given information to determine the exact values of A(0), A(1), A(2),
A(3), A(4), A(5), and A(6).
(e) Based on your responses to all of the preceding questions, sketch a complete
and accurate graph of y = A(x) on the axes provided, being sure to indicate
the behavior of A for x < 0 and x > 6.
(f) How does the graph of B compare to A if B is instead defined by B(x) =
x
0
g(t) dt?
⊳
Summary
In this section, we encountered the following important ideas:
• Given the graph of a function f , we can construct the graph of its antiderivative F
provided that (a) we know a starting value of F, say F(a), and (b) we can evaluate
the integral
b
a
f (x) dx exactly for relevant choices of a and b. For instance, if we
wish to know F(3), we can compute F(3) = F(a) +
3
a
f (x) dx. When we combine this
information about the function values of F together with our understanding of how the
behavior of F ′ = f affects the overall shape of F, we can develop a completely accurate
graph of the antiderivative F.
• Because the derivative of a constant is zero, if F is an antiderivative of f , it follows that
G(x) = F(x) + C will also be an antiderivative of f . Moreover, any two antiderivatives
of a function f differ precisely by a constant. Thus, any function with at least one
antiderivative in fact has infinitely many, and the graphs of any two antiderivatives will
differ only by a vertical translation.
• Given a function f , the rule A(x) =
x
a
f (t) dt defines a new function A that measures
the net-signed area bounded by f on the interval [a, x]. We call the function A the
integral function corresponding to f .
Exercises
1. A moving particle has its velocity given by the quadratic function v pictured in
Figure 5.6. In addition, it is given that A 1 =
7
6 and A 2 =
8
3 , as well as that for the
corresponding position function s, s(0) = 0.5.
(a) Use the given information to determine s(1), s(3), s(5), and s(6).
(b) On what interval(s) is s increasing? On what interval(s) is s decreasing?
(c) On what interval(s) is s concave up? On what interval(s) is s concave down?