SECTION 5.5
(
In Exercises 61-66, find the integral.
61.
J 3-' dx
62.
J
f
63.
367
Bases Other Than e and Applications
-
.
75. The table of values below was obtained by evaluating a
function. Determine which of the statements may be true
and which must be false, and explain why.
5-' dx
(a) y is an exponential
x(5-")
function of x.
(b)
dx
(c) .r is an exponential function of y.
64. J(3 -
dx
X)7(3-xl'
(d) Y is a linear function of x.
ITEEJ
32x
65.
J
I
J
66.
+
32x dx
~
2'in x cas x dx
76. Consider the function f(x)
In Exercises 67-70, evaluate
the integral.
(a) What is the domain of
68.
69.
70.
f2
f
f
(c) If x is a real number between 1000 and 10,000, determine the interval in which j(x) will be found.
4,/2 dx
(d) Determine the interval in which x will be found if f(x)
is negati ve.
(5-' - 3') dx
(e) If f(x) is increased by one unit, x must have been
increased by what factor?
(6-'" - 2') dx
(f) Find the ratio of x, to
f(xz) = 11.
Area
In Exercises 71 and 72, find the area
bounded by the graphs of the equations.
of the region
cOSX
ftt
sin .r, y = 0, x = 0, x =
ix
= 0.4'/3,
t)
(0,
y
y
.•... /-
/////
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////
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I
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i
_-
+//////--
III
///_-I
_-
T~;~~~;;4
/ j
-4
dy
.
74. dx = eSln, cas x,
///
'//--
given that f(x,)
= 3n and
= log,
X,
g(x) = x"
hex) =
X2,
and k(x) = 2'
from the one with the greatest rate of growth to the one with
the smallest rate of growth for large values of x.
7T
Slope Fields In Exercises 73 and 74, a differential equation, a
point, and a slope field are given. (a) Sketch two approximate
solutions of the differential equation on the slope field, one of
which passes through the given point. (b) Use integration to find
the particular solution of the differential equation and use a
graphing utility to graph the solution. Compare the result with
the sketches in part (a). To print an enlarged copy of the graph,
go to the website www.mathgraphs.com,
73.
"'2
77. Order the functions
f(x)
71. y = 3-', y = 0, x = 0, x = 3
72. Y = 3
j?
(b) Find j :'.
J2 2' dx
-,
67.
= log,ox.
(7T,2)
78. Find the derivative
constant.
of each function,
(a) y = x"
(b) y = aX
(c) y = x-'
(d) y = a"
given that a is
79. Inflation
If the annual rate of inflation averages 5% over the
next 10 years, the approximate cost C of goods or services
during any year in that decade is
C(t) = P(1.05)t
where t is the time in years and P is the present cost.
(a) The price of an oil change for your car is presently $24.95.
Estimate the price 10 years from now.
(b) Find the rates of change of C with respect to t when t =
and t = 8.
(c) Verify that the rate of change of C is proportional to C.
What is the constant of proportionality?
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