FDWK_3ed_Ch05_pp262-319

Section 5.4 Fundamental Theorem of Calculus 305
Explorations
71. The Sine Integral Function The sine integral function
Six x
sin t
dt
0
t
is one of the many useful functions in engineering that are
defined as integrals. Although the notation does not show it, the
function being integrated is
{
sin t
, t 0
f t t
1, t 0,
the continuous extension of sin tt to the origin.
(a) Show that Six is an odd function of x.
(b) What is the value of Si0
(c) Find the values of x at which Six has a local extreme
value.
(d) Use NINT to graph Six.
72. Cost from Marginal Cost The marginal cost of printing a
poster when x posters have been printed is
dc 1
d x 2 x
dollars. Find
(a) c100 c1, the cost of printing posters 2 to 100. $9
(b) c400 c100, the cost of printing posters 101 to 400. $10
73. Revenue from Marginal Revenue Suppose that a
company’s marginal revenue from the manufacture and sale
of eggbeaters is
dr
2
2 ,
d x x 1 2
where r is measured in thousands of dollars and x in thousands
of units. How much money should the company expect from a
production run of x 3 thousand eggbeaters To find out,
integrate the marginal revenue from x 0 to x 3. $4500
74. Average Daily Holding Cost Solon Container receives 450
drums of plastic pellets every 30 days. The inventory function
(drums on hand as a function of days) is Ix 450 x 2 2.
(a) Find the average daily inventory (that is, the average value of
Ix for the 30-day period). 300 drums
Answers:
57. (b) H is increasing on [0, 6] where H(x) = f (x) > 0.
(c) H is concave up on (9, 12) where H(x) f (x) 0.
(d) H(12) 12
f(t) dt 0 because there is more area above the x-axis
0
than below for y f(x).
(e) x 6 since H (6) f (6) 0 and H (6) f (6) 0.
(f ) x 0 since H(x) 0 on (0, 12].
58. (c) s(3) 3 f (x) dx 1 0
2 (9) 9 2 units
(d) s(6) f (6) 0, s(6) f(6) 0 so s has its largest value
at t 6 sec.s has its largest value at t = 6 sec.
(e) s(t) f(t) 0 at t 4 sec and t 7 sec.
(f ) Particle is moving toward on (6, 9) and away on (0, 6) since
s(6) f (t) 0 on (0, 6) and then s(6) f (t) 0 on (6, 9).
(b) If the holding cost for one drum is $0.02 per day, find the
average daily holding cost (that is, the per-drum holding cost
times the average daily inventory). $6.00
75. Suppose that f has a negative derivative for all values of x and
that f 1 0. Which of the following statements must be true
of the function
hx x
f t dt
0
Give reasons for your answers.
(a) h is a twice-differentiable function of x.
True, h(x) f(x) ⇒ h(x) f(x)
(b) h and dhdx are both continuous.
True, h and h are both differentiable.
(c) The graph of h has a horizontal tangent at x 1.
h(1) f(1) 0
(d) h has a local maximum at x 1.
True, h(1) f(1) 0 and h(1) 0
(e) h has a local minimum at x 1.
False, h(1) f(1) 0
(f) The graph of h has an inflection point at x 1.
False, h(1) f(1) 0
(g) The graph of dhdx crosses the x-axis at x 1.
True, d h
f(x) 0 at x 1 and h(x) f(x) is a decreasing function.
dx
Extending the Ideas
76. Writing to Learn If f is an odd continuous function, give a
graphical argument to explain why x
f t dt is even.
0
77. Writing to Learn If f is an even continuous function, give a
graphical argument to explain why x
f t dt is odd.
0
78. Writing to Learn Explain why we can conclude from
Exercises 76 and 77 that every even continuous function is
the derivative of an odd continuous function and vice versa.
79. Give a convincing argument that the equation
x
0
sin t
dt 1
t
has exactly one solution. Give its approximate value.
59. (c) s(3) 3 f (x) dx 1 (3)(6) 9 units
0
2
(d) s(t) 0 at t 6 sec because 6 f (x) dx 0
0
(e) s(t) f(t) 0 at t 7 sec
(f) 0 t 3: s 0, s 0 ⇒ away
3 t 6: s 0, s0 ⇒ toward
t 6: s 0, s0 ⇒ away
76. Using area, x
f(t) dt 0 f(t) dt x
f(t) dt
0
x
0
77. Using area, x
f(t) dt 0 f(t) dt x
f(t) dt
0
x
0
78. f odd → x
d
f(t) dt is even, but
0
d
x
f(t) dt f(x)
x 0
so f is the derivative of an even function. Similarly for f even.
79. x 1.0648397. lim sin t
0. Not enough area
t→∞ t
between x-axis and sin x
for x 1.0648397 to get
x
x
sin t
dt back to 1.
0 t
or: Si(x) doesn’t decrease enough (for x 1.0648397) to get back to 1.