5.2. THE SECOND FUNDAMENTAL THEOREM OF CALCULUS
285
(Hint: Let F(x) =
x
4
sin(t 2 ) dt and observe that this problem is asking you to
evaluate
d
dx
F(x 3 )]
.
⊳
Summary
In this section, we encountered the following important ideas:
• For a continuous function f , the integral function A(x) =
x
1
f (t) dt defines an antiderivative of f .
• The Second Fundamental Theorem of Calculus is the formal, more general statement
of the preceding fact: if f is a continuous function and c is any constant, then
A(x) =
x
c
f (t) dt is the unique antiderivative of f that satisfies A(c) = 0.
• Together, the First and Second FTC enable us to formally see how differentiation and
integration are almost inverse processes through the observations that
x
c
d
dt
[ f (t)] dt = f (x) − f (c)
and
d
dx
x
c
f (t) dt
= f (x).
Exercises
1. Let g be the function pictured at left in Figure 5.13, and let F be defined by F(x) =
x
2
g(t) dt. Assume that the shaded areas have values A 1 = 4.29, A 2 = 12.75, A 3 = 0.36,
and A 4 = 1.79. Assume further that the portion of A 2 that lies between x = 0.5 and
x = 2 is 6.06.
Sketch a carefully labeled graph of F on the axes provided, and include a written
analysis of how you know where F is zero, increasing, decreasing, CCU, and CCD.
2. The tide removes sand from the beach at a small ocean park at a rate modeled by the
function
R(t) = 2 + 5 sin
4πt
25
A pumping station adds sand to the beach at rate modeled by the function
S(t) =
15t
1 + 3t
Both R(t) and S(t) are measured in cubic yards of sand per hour, t is measured in
hours, and the valid times are 0 ≤ t ≤ 6. At time t = 0, the beach holds 2500 cubic
yards of sand.
285
(Hint: Let F(x) =
x
4
sin(t 2 ) dt and observe that this problem is asking you to
evaluate
d
dx
F(x 3 )]
.
⊳
Summary
In this section, we encountered the following important ideas:
• For a continuous function f , the integral function A(x) =
x
1
f (t) dt defines an antiderivative of f .
• The Second Fundamental Theorem of Calculus is the formal, more general statement
of the preceding fact: if f is a continuous function and c is any constant, then
A(x) =
x
c
f (t) dt is the unique antiderivative of f that satisfies A(c) = 0.
• Together, the First and Second FTC enable us to formally see how differentiation and
integration are almost inverse processes through the observations that
x
c
d
dt
[ f (t)] dt = f (x) − f (c)
and
d
dx
x
c
f (t) dt
= f (x).
Exercises
1. Let g be the function pictured at left in Figure 5.13, and let F be defined by F(x) =
x
2
g(t) dt. Assume that the shaded areas have values A 1 = 4.29, A 2 = 12.75, A 3 = 0.36,
and A 4 = 1.79. Assume further that the portion of A 2 that lies between x = 0.5 and
x = 2 is 6.06.
Sketch a carefully labeled graph of F on the axes provided, and include a written
analysis of how you know where F is zero, increasing, decreasing, CCU, and CCD.
2. The tide removes sand from the beach at a small ocean park at a rate modeled by the
function
R(t) = 2 + 5 sin
4πt
25
A pumping station adds sand to the beach at rate modeled by the function
S(t) =
15t
1 + 3t
Both R(t) and S(t) are measured in cubic yards of sand per hour, t is measured in
hours, and the valid times are 0 ≤ t ≤ 6. At time t = 0, the beach holds 2500 cubic
yards of sand.