596
● Chapter 9
Derivatives
Farm workers The percent of U.S. workers in farm
occupations during certain years is shown in the table.
f(t)
80
Year
Percent
Year
Percent
Year
Percent
60
1820
1850
1870
1900
1920
71.8
63.7
53
37.5
27
1930
1940
1950
1960
1970
21.2
17.4
11.6
6.1
3.6
1980
1985
1990
1994
2002
2.7
2.8
2.4
2.5
2.5
40
Source: Bureau of Labor Statistics
Assume that the percent of U.S. workers in farm occupations can be modeled with the function
f(t) ! 1000 "
#7.4812t $ 1560.2
1.2882t2 # 122.18t $ 21,483
20
t
50
65. (a)
(b)
(c)
66. (a)
(b)
(c)
100 150 200
Find lim f(t), if it exists.
tS210
What does this limit predict?
Is the equation accurate as t S 210? Explain.
Find lim f(t), if it exists.
tS140
What does this limit predict?
Is the equation accurate as t S 140? Explain.
where t is the number of years past 1800. (A graph of
f (t) along with the data in the table is shown in the
accompanying figure.) Use the table and equation in
Problems 65 and 66.
OBJECTIVES
●
●
●
9.2
To determine whether a
function is continuous or
discontinuous at a point
To determine where a function
is discontinuous
To find limits at infinity and
horizontal asymptotes
Continuous Functions; Limits at Infinity
] Application Preview
Suppose that a friend of yours and her husband have a taxable income of $117,250, and she tells
you that she doesn’t want to make any more money because that would put them in a higher tax
bracket. She makes this statement because the tax rate schedule for married taxpayers filing a
joint return (shown in the table) appears to have a jump in taxes for taxable income at $117,250.
Schedule Y-1—If your filing status is
Married filing jointly or Qualifying widow(er)
If your taxable
income is:
Over ––
The tax is:
of the
amount
over ––
But not
over ––
$0
$14,300
10%
$0
14,300
58,100
$1,430.00 + 15%
14,300
58,100
117,250
117,250
178,650
8,000.00 + 25%
22,787.50 + 28%
58,100
117,250
178,650
319,100
319,100
39,979.50 + 33%
86,328.00 + 35%
178,650
319,100
Source: Internal Revenue Service, 2004, Form 1040 Instructions
To see whether the couple’s taxes would jump to some higher level, we will write the function that gives income tax for married taxpayers as a function of taxable income and show that
the function is continuous (see Example 3). That is, we will see that the tax paid does not jump
at $117,250 even though the tax on income above $117,250 is collected at a higher rate. In this
section, we will show how to determine whether a function is continuous, and we will investigate
some different types of discontinuous functions.
9.2 Continuous Functions; Limits at Infinity ●
597
We have found that f (c) is the same as the limit as x S c for any polynomial function f (x)
and any real number c. Any function for which this special property holds is called a continuous function. The graphs of such functions can be drawn without lifting the pencil
from the paper, and graphs of others may have holes, vertical asymptotes, or jumps that
make it impossible to draw them without lifting the pencil. Even though a function may
not be continuous everywhere, it is likely to have some points where the limit of the function as x S c is the same as f (c). In general, we define continuity of a function at the value
x " c as follows.
Continuity at a Point
The function f is continuous at x ! c if all of the following conditions are satisfied.
2. lim f(x) exists
1. f(c) exists
y
xSc
3. lim f(x) " f(c)
xSc
The figure at the left illustrates these three conditions.
If one or more of the conditions above do not hold, we say the function is discontinuous at x ! c.
y = f (x)
f (c)
Figure 9.11 shows graphs of some functions that are discontinuous at x " 2.
c
x
y
y
8
40
6
20
4
x
-1
-20
1
2
3
2
4
x
-2 -1
-2
-40
1
x#2
lim f(x) and f(2) do not exist.
(a) f(x) "
xS2
(b) f(x) "
100
5
80
4
40
2
20
1
1
2
3
4
x
4 # x if x ! 2
4
if x " 2
lim f(x) " 2 ! 4 " f(2)
xS2
x3 # 2x2 # x $ 2
x#2
60
3
Figure 9.11
3
y
6
(c) f(x) " b
2
f(2) does not exist.
y
-2 -1
1
-1
1
2
3
4
x
1!(x # 2)2 if x ! 2
0
if x " 2
lim f(x) does not exist.
(d) f(x) " b
xS2
In the previous section, we saw that if f is a polynomial function, then lim f(x) " f(c) for
xSc
f(x)
every real number c, and also that lim h(x) " h(c) if h(x) "
is a rational function and
xSc
g(x)
g(c) ! 0. Thus, by definition, we have the following.
598
● Chapter 9
Derivatives
Every polynomial function is continuous for all real numbers.
Every rational function is continuous at all values of x except those that make the
denominator 0.
● EXAMPLE 1 Discontinuous Functions
For what values of x, if any, are the following functions continuous?
3x $ 2
4x # 6
x2 # x # 2
(b) f(x) "
x2 # 4
(a) h(x) "
Solution
(a) This is a rational function, so it is continuous for all values of x except for those that
make the denominator, 4x # 6, equal to 0. Because 4x # 6 " 0 at x " 3!2, h(x) is
continuous for all real numbers except x " 3!2. Figure 9.12(a) shows a vertical
asymptote at x " 3!2.
(b) This is a rational function, so it is continuous everywhere except where the denominator is 0. To find the zeros of the denominator, we factor x2 # 4.
f(x) "
x2 # x # 2
x2 # x # 2
"
2
x #4
(x # 2)(x $ 2)
Because the denominator is 0 for x " 2 and for x " #2, f(2) and f(#2) do not exist
(recall that division by 0 is undefined). Thus the function is discontinuous at x " 2
and x " #2. The graph of this function (see Figure 9.12(b)) shows a hole at x " 2
and a vertical asymptote at x " #2.
y
y
y = 3x + 2
4x – 6
5
5
4
4
3
3
2
2
1
1
x
-3
-2
-1
1
-1
2
3
-3
x=
-4
Checkpoint
-2
-1
-1
1
2
3
3
2
-3
-4
-5
-5
●
x
-3
-2
-2
Figure 9.12
x2 − x − 2
x2 − 4
f (x) =
(a)
(b)
1. Find any x-values where the following functions are discontinuous.
x3 # 1
(a) f(x) " x3 # 3x $ 1
(b) g(x) "
(x # 1)(x $ 2)
If the pieces of a piecewise defined function are polynomials, the only values of x where
the function might be discontinuous are those at which the definition of the function
changes.
9.2 Continuous Functions; Limits at Infinity ●
599
● EXAMPLE 2 Piecewise Defined Functions
Determine the values of x, if any, for which the following functions are discontinuous.
(a) g(x) " b
(x $ 2)3 $ 1
3
if x ( #1
if x & #1
(b) f(x) " b
4 # x2
x#2
if x ' 2
if x % 2
Solution
(a) g(x) is a piecewise defined function in which each part is a polynomial. Thus, to see
whether a discontinuity exists, we need only check the value of x for which the definition of the function changes—that is, at x " #1. Note that x " #1 satisfies
x ( #1, so g(#1) " (#1 $ 2)3 $ 1 " 2. Because g(x) is defined differently for
x ' #1 and x & #1, we use left- and right-hand limits. For x S #1#, we know that
x ' #1, so g(x) " (x $ 2)3 $ 1:
lim g(x) " lim # 3 (x $ 2)3 $ 14 " (#1 $ 2)3 $ 1 " 2
xS#1#
xS#1
Similarly, for x S #1$, we know that x & #1, so g(x) " 3:
lim g(x) " lim $ 3 " 3
xS#1$
xS#1
Because the left- and right-hand limits differ, lim g(x) does not exist, so g(x) is
xS#1
discontinuous at x " #1. This result is confirmed by examining the graph of g,
shown in Figure 9.13.
g(x) =
(x + 2) 3 + 1 if x ≤ −1
3
if x > −1
y
4
2
1
x
-4
-2
1
-1
2
-1
-2
Figure 9.13
(b) As with g(x), f(x) is continuous everywhere except perhaps at x " 2, where the definition of f(x) changes. Because x " 2 satisfies x % 2, f(2) " 2 # 2 " 0. The leftand right-hand limits are
lim f(x) " lim#(4 # x2) " 4 # 22 " 0
xS2#
xS2
and
lim f(x) " lim$(x # 2) " 2 # 2 " 0
xS2$
xS2
Because the right- and left-hand limits are equal, we conclude that lim f(x) " 0.
xS2
The limit is equal to the functional value
lim f(x) " f(2)
xS2
so we conclude that f is continuous at x " 2 and thus f is continuous for all values
of x. This result is confirmed by the graph of f, shown in Figure 9.14.
600
● Chapter 9
Derivatives
4 − x 2 if x < 2
x − 2 if x ≥ 2
f (x) =
y
4
3
2
1
x
-4 -3
-1
-1
1
2
3
4
-2
-3
-4
Figure 9.14
] EXAMPLE 3 Taxes (Application Preview)
The tax rate schedule for married taxpayers filing a joint return (shown in the table) appears
to have a jump in taxes for taxable income at $117,250.
Schedule Y-1—If your filing status is
Married filing jointly or Qualifying widow(er)
If your taxable
income is:
Over ––
The tax is:
of the
amount
over ––
But not
over ––
$0
14,300
$14,300
58,100
10%
$1,430.00 + 15%
$0
14,300
58,100
117,250
117,250
178,650
8,000.00 + 25%
22,787.50 + 28%
58,100
117,250
178,650
319,100
319,100
39,979.50 + 33%
86,328.00 + 35%
178,650
319,100
Source: Internal Revenue Service, 2004, Form 1040 Instructions
(a) Use the table and write the function that gives income tax for married taxpayers as a
function of taxable income, x.
(b) Is the function in part (a) continuous at x " 117,250?
(c) A married friend of yours and her husband have a taxable income of $117,250, and she
tells you that she doesn’t want to make any more money because doing so would put
her in a higher tax bracket. What would you tell her to do if she is offered a raise?
Solution
(a) The function that gives the tax due for married taxpayers is
0.10x
1430 $ 0.15(x # 14,300)
8000 $ 0.25(x # 58,100)
T(x) " f
22,787.50 $ 0.28(x # 117,250)
39,979.50 $ 0.33(x # 178,650)
86,328 $ 0.35(x # 319,100)
if
if
if
if
if
if
0 ( x ( 14,300
14,300 ' x ( 58,100
58,100 ' x ( 117,250
117,250 ' x ( 178,650
178,650 ' x ( 319,100
x & 319,100
9.2 Continuous Functions; Limits at Infinity ●
(b) We examine the three conditions for continuity at x " 117,250.
(i) T(117,250) " 22,787.50, so T(117,250) exists.
(ii) Because the function is piecewise defined near 117,250, we evaluate
by evaluating one-sided limits:
From the left:
lim
xS117,250#
From the right:
T(x) "
lim
T(x) "
xS117,250$
lim
xS117,250#
lim
lim
xS117,250
601
T(x)
3 8000 $ 0.25(x # 58,100)4 " 22,787.50
xS117,250$
3 22,787.50 $ 0.28(x # 117,250)4 " 22,787.50
Because these one-sided limits agree, the limit exists and is
lim
xS117,250
(iii) Because
lim
xS117,250
T(x) " 22,787.50
T(x) " T(117,250) " 22,787.50, the function is continuous at
117,250.
(c) If your friend earned more than $117,250, she and her husband would pay taxes at a
higher rate on the money earned above the $117,250, but it would not increase the
tax rate on any income up to $117,250. Thus she should take any raise that’s offered.
●
2. If f (x) and g(x) are polynomials, h(x) " b
Checkpoint
f(x)
g(x)
if x ( a
is continuous everywhere
if x & a
except perhaps at _______.
Limits at Infinity
6
1
y=x
-6
6
We noted in Section 2.4, “Special Functions and Their Graphs,” that the graph of y " 1!x
has a vertical asymptote at x " 0 (shown in Figure 9.15(a)). By graphing y " 1!x and evaluating the function for very large x values we can see that y " 1!x never becomes negative for positive x-values regardless of how large the x-value is. Although no value of x
makes 1!x equal to 0, it is easy to see that 1!x approaches 0 as x gets very large. This is
denoted by
1
lim
"0
xS$) x
and means that the line y " 0 (the x-axis) is a horizontal asymptote for y " 1!x. We also
see that y " 1!x approaches 0 as x decreases without bound, and we denote this by
-6
(a)
lim
30
xS#)
y=
-3
(b)
Figure 9.15
These limits for f(x) " 1!x can also be established with numerical tables.
2
x2
x
3
0
1
"0
x
100
100,000
100,000,000
f(x) " 1!x
0.01
0.00001
0.00000001
T
x
#100
#100,000
#100,000,000
T
T
0
$)
lim
xS$)
f(x) " 1!x
#0.01
#0.00001
#0.00000001
T
0
#)
1
"0
x
lim
xS#)
1
"0
x
We can use the graph of y " 2!x2 in Figure 9.15(b) to see that the x-axis (y " 0) is a
horizontal asymptote and that
lim
xS$)
2
"0
x2
and
lim
xS#)
2
"0
x2
602
● Chapter 9
Derivatives
By using graphs and/or tables of values, we can generalize the results for the functions
shown in Figure 9.15 and conclude the following.
Limits at Infinity
If c is any constant, then
and
lim c " c.
1. lim c " c
xS$)
xS#)
c
2. lim p " 0, where p & 0.
xS$) x
c
3. lim n " 0, where n & 0 is any integer.
xS#) x
In order to use these properties for finding the limits of rational functions as x
approaches $) or #), we first divide each term of the numerator and denominator by
the highest power of x present and then determine the limit of the resulting expression.
● EXAMPLE 4 Limits at Infinity
Find each of the following limits, if they exist.
(a) lim
xS$)
2x # 1
x$2
(b) lim
xS#)
x2 $ 3
1#x
Solution
(a) The highest power of x present is x1, so we divide each term in the numerator and
denominator by x and then use the properties for limits at infinity.
lim
xS $ )
2x # 1
" lim
xS $ )
x$2
"
1
2x
1
#
2#
x
x
x
" lim
xS $ )
2
x
2
$
1$
x
x
x
2#0
" 2 (by Properties 1 and 2)
1$0
Figure 9.16(a) shows the graph of this function with the y-coordinates of the graph
approaching 2 as x approaches $) and as x approaches #). That is, y " 2 is a
horizontal asymptote. Note also that there is a discontinuity (vertical asymptote)
where x " #2.
(b) We divide each term in the numerator and denominator by x2 and then use the
properties.
lim
xS#)
x2
3
3
$ 2
1$ 2
2
$3
x
x
x
" lim
" lim
xS#) 1
xS#) 1
1#x
x
1
# 2
#
x
x2
x
x2
x2
This limit does not exist because the numerator approaches 1 and the denominator
approaches 0 through positive values. Thus
x2 $ 3
S $) as x S #)
1#x
The graph of this function, shown in Figure 9.16(b), has y-coordinates that increase
without bound as x approaches #) and that decrease without bound as x approaches
$). (There is no horizontal asymptote.) Note also that there is a vertical asymptote
at x " 1.
9.2 Continuous Functions; Limits at Infinity ●
x2 + 3
→ + ) as x → − )
1−x
y
y
6
8
6
4
2
4
lim 2x − 1 = 2
x→+ ) x + 2
−6
−4
−2
2
−4
Figure 9.16
x
y=2
−8 −6 −4 −2
−2
x
−8
603
4
f (x) =
2x − 1
f (x) =
x+2
(a)
x2 + 3
1−x
2
4
6
8
−4
−6
−8
−10
(b)
In our work with limits at infinity, we have mentioned horizontal asymptotes several
times. The connection between these concepts follows.
Limits at Infinity and
Horizontal Asymptotes
●
Checkpoint
Calculator
Note
If lim f(x) " b or lim f(x) " b, where b is a constant, then the line y " b is a
xS)
xS#)
horizontal asymptote for the graph of y " f(x). Otherwise, y " f(x) has no horizontal
asymptotes.
x2 # 4
.
xS$) 2x2 # 7
(b) What does part (a) say about horizontal asymptotes for f(x) " (x2 # 4)!(2x2 # 7)?
3. (a) Evaluate lim
We can use the graphing and table features of a graphing calculator to help locate and investigate discontinuities and limits at infinity (horizontal asymptotes). A graphing calculator
can be used to focus our attention on a possible discontinuity and to support or suggest
■
appropriate algebraic calculations.
● EXAMPLE 5 Limits with Technology
Use a graphing utility to investigate the continuity of the following functions.
(a) f(x) "
(c) h(x) "
x2 $ 1
x$1
"x $ 1"
x$1
x2 # 2x # 3
x2 # 1
#x2
# 2x if x ( #1
2
(d) k(x) " d
x
$2
if x & #1
2
(b) g(x) "
Solution
(a) Figure 9.17(a) shows that f(x) has a discontinuity (vertical asymptote) near x " #1.
Because f (#1) DNE, we know that f(x) is not continuous at x " #1.
(b) Figure 9.17(b) shows that g(x) is discontinuous (vertical asymptote) near x " 1, and
this looks like the only discontinuity. However, the denominator of g(x) is zero at
x " 1 and x " #1, so g(x) must have discontinuities at both of these x-values. Evaluating or using the table feature confirms that x " #1 is a discontinuity (a hole, or
missing point). The figure also shows a horizontal asymptote; evaluation of lim g(x)
xS)
confirms this is the line y " 1.
604
● Chapter 9
Derivatives
(c) Figure 9.17(c) shows a discontinuity (jump) at x " #1. We also see that h(#1) DNE,
which confirms the observations from the graph.
(d) The graph in Figure 9.17(d) appears to be continuous. The only “suspicious” x-value
is x " #1, where the formula for k(x) changes. Evaluating k(#1) and examining a
table near x " #1 indicates that k(x) is continuous there. Algebraic evaluations of the
two one-sided limits confirm this.
8
8
y = g(x)
y = f (x)
-10
10
-8
8
-8
-8
(a)
(b)
8
5
y = k(x)
y = h(x)
-10
10
-5
3
-8
Figure 9.17
(c)
-3
(d)
Summary
The following information is useful in discussing continuity of functions.
A. A polynomial function is continuous everywhere.
f(x)
, where f(x) and g(x) are polynomials.
B. A rational function is a function of the form
g(x)
1. If g(x) ! 0 at any value of x, the function is continuous everywhere.
2. If g(c) " 0, the function is discontinuous at x " c.
(a) If g(c) " 0 and f (c) ! 0, then there is a vertical asymptote at x " c.
f(x)
" L, then the graph has a missing point at (c, L).
(b) If g(c) " 0 and lim
xSc g(x)
C. A piecewise defined function may have a discontinuity at any x-value where the function changes its formula. One-sided limits must be used to see whether the limit exists.
The following steps are useful when we are evaluating limits at infinity for a rational function f(x) " p(x)!q(x).
1. Divide both p(x) and q(x) by the highest power of x found in either polynomial.
2. Use the properties of limits at infinity to complete the evaluation.
●
Checkpoint Solutions
1. (a) This is a polynomial function, so it is continuous at all values of x (discontinuous at
none).
(b) This is a rational function. It is discontinuous at x " 1 and x " #2 because these
values make its denominator 0.
2. x " a.
9.2 Continuous Functions; Limits at Infinity ●
605
4
1
#4
x2
1#0
3. (a) lim
" lim
"
"
2
xS$) 2x # 7
xS$)
7
2#0
2
2# 2
x
(b) The line y " 1!2 is a horizontal asymptote.
x2
1#
9.2 Exercises
Problems 1 and 2 refer to the following figure. For each
given x-value, use the figure to determine whether the
function is continuous or discontinuous at that x-value.
If the function is discontinuous, state which of the three
conditions that define continuity is not satisfied.
y
y = f(x)
4
2
−6
x
−4
2
4
6
−2
−4
1. (a) x " #5
2. (a) x " 2
(b) x " 1
(b) x " #4
(c) x " 3
(c) x " #2
(d) x " 0
(d) x " 5
In Problems 3–8, determine whether each function is
continuous or discontinuous at the given x-value. Examine the three conditions in the definition of continuity.
x2 # 4
x " #2
,
3. f(x) "
x#2
x2 # 9
,
x"3
4. y "
x$3
x2 # 9
,
x " #3
5. y "
x$3
2
x #4
,
x"2
6. f(x) "
x#2
x#3
if x ( 2
,
x"2
7. f(x) " b
4x # 7 if x & 2
x2 $ 1
if x ( 1
,
x"1
8. f(x) " b 2
2x # 1 if x & 1
In Problems 9–16, determine whether the given function
is continuous. If it is not, identify where it is discontinuous and which condition fails to hold. You can verify
your conclusions by graphing each function with a
graphing utility, if one is available.
9. f(x) " 4x2 # 1
10. y " 5x2 # 2x
4x2 $ 4x $ 1
4x2 $ 3x $ 2
11. g(x) "
12. y "
x $ 1!2
x$2
2x # 1
x
13. y " 2
14. y " 2
x $3
x $1
3
if x ( 1
15. f(x) " b 2
x $ 2 if x & 1
x3 $ 1 if x ( 1
16. f(x) " b
2
if x & 1
In Problems 17–20, use the trace and table features of a
graphing utility to investigate whether each of the following functions has any discontinuities.
x2 # 5x $ 4
x2 # 5x # 6
17. y "
18. y "
x$1
x#4
x#4
if x ( 3
19. f(x) " b 2
x # 8 if x & 3
x2 $ 4 if x ! 1
20. f(x) " b
8
if x " 1
Each of Problems 21–24 contains a function and its
graph. For each problem, answer parts (a) and (b).
(a) Use the graph to determine, as well as you can,
(i) vertical asymptotes, (ii) lim f(x),
xS $%
(iii) lim f(x),
(iv) horizontal asymptotes.
xS #%
(b) Check your conclusions in (a) by using the functions
to determine items (i)–(iv) analytically.
y
8
21. f(x) "
x$2
10
y = f (x)
5
x
-10
5
10
606
● Chapter 9
22. f(x) "
Derivatives
x#3
x#2
y
10
y = f (x)
5
x
-10 -5
5
10
-5
-10
23. f(x) "
y
2(x $ 1)3(x $ 5)
(x # 3)2(x $ 2)2
25
20
y = f (x)
15
10
5
x
-20 -10
10
20
-10
24. f(x) "
x2
4x2
# 4x $ 4
y
35
30
25
20
15
10
5
y = f (x)
10 20 30 40
In Problems 25–32, complete (a) and (b).
(a) Use analytic methods to evaluate each limit.
(b) What does the result from (a) tell you about horizontal asymptotes?
You can verify your conclusions by graphing the functions with a graphing utility, if one is available.
3
4
25. lim
26. lim 2
xS$) x $ 1
xS#) x # 2x
x3 # 1
3x2 $ 2
27. lim 3
28. lim
xS$) x $ 4
xS#) x2 # 4
5x3 # 4x
4x2 $ 5x
29. lim
30. lim
xS#) 3x3 # 2
xS$) x2 # 4x
3x2 $ 5x
5x3 # 8
31. lim
32. lim
xS$) 6x $ 1
xS#) 4x2 $ 5x
In Problems 33 and 34, use a graphing utility to complete (a) and (b).
(a) Graph each function using a window with 0 ! x
! 300 and #2 ! y ! 2. What does the graph indicate about lim f(x)?
xS $%
For Problems 37 and 38, let
anxn $ an#1xn#1 $ . . . $ a1x $ a0
f(x) !
bmxm $ bm#1xm#1 $ . . . $ b1x $ b0
be a rational function.
an
37. If m " n, show that lim f(x) " , and hence that
xS)
bn
an
y"
is a horizontal asymptote.
bn
38. (a) If m & n, show that lim f(x) " 0, and hence that
xS)
y " 0 is a horizontal asymptote.
(b) If m ' n, find lim f(x). What does this say about
xS)
horizontal asymptotes?
A P P L I C AT I O N S
x
-20-10
In Problems 35 and 36, complete (a)–(c). Use analytic
methods to find (a) any points of discontinuity and (b)
limits as x S $% and x S #%. (c) Then explain why, for
these functions, a graphing utility is better as a support
tool for the analytic methods than as the primary tool for
investigation.
1000x # 1000
3000x
35. f(x) "
36. f(x) "
x $ 1000
4350 # 2x
(b) Use the table feature with x-values larger than 10,000
to investigate lim f(x). Does the table support your
xS $%
conclusions in part (a)?
x2 # 4
5x3 # 7x
33. f(x) "
34. f(x) "
2
3 $ 2x
1 # 3x3
39. Sales volume Suppose that the weekly sales volume
(in thousands of units) for a product is given by
y"
32
(p $ 8)2!5
where p is the price in dollars per unit. Is this function
continuous
(a) for all values of p?
(b) at p " 24?
(c) for all p % 0?
(d) What is the domain for this application?
40. Worker productivity Suppose that the average number
of minutes M that it takes a new employee to assemble
one unit of a product is given by
M"
40 $ 30t
2t $ 1
where t is the number of days on the job. Is this function continuous
(a) for all values of t?
(b) at t " 14?
(c) for all t % 0?
(d) What is the domain for this application?
41. Demand Suppose that the demand for a product is
defined by the equation
p"
200,000
(q $ 1)2
where p is the price and q is the quantity demanded.
(a) Is this function discontinuous at any value of q?
What value?
9.2 Continuous Functions; Limits at Infinity ●
607
(b) Because q represents quantity, we know that q % 0.
Is this function continuous for q % 0?
Describe any discontinuities for C( p). Explain what
each discontinuity means.
42. Advertising and sales The sales volume y (in thousands of dollars) is related to advertising expenditures
x (in thousands of dollars) according to
47. Pollution The percent p of particulate pollution that
can be removed from the smokestacks of an industrial
plant by spending C dollars is given by
y"
200x
x $ 10
p"
100C
7300 $ C
(a) Is this function discontinuous at any points?
(b) Advertising expenditures x must be nonnegative. Is
this function continuous for these values of x?
Find the percent of the pollution that could be removed
if spending C were allowed to increase without bound.
Can 100% of the pollution be removed? Explain.
43. Annuities If an annuity makes an infinite series of
equal payments at the end of the interest periods, it is
called a perpetuity. If a lump sum investment of An is
needed to result in n periodic payments of R when the
interest rate per period is i, then
48. Cost-benefit The percent p of impurities that can be
removed from the waste water of a manufacturing
process at a cost of C dollars is given by
An " RB
1 # (1 $ i)#n
R
i
(a) Evaluate lim An to find a formula for the lump sum
nS)
payment for a perpetuity.
(b) Find the lump sum investment needed to make payments of $100 per month in perpetuity if interest is
12%, compounded monthly.
44. Response to adrenalin Experimental evidence suggests that the response y of the body to the concentration x of injected adrenalin is given by
x
y"
a $ bx
where a and b are experimental constants.
(a) Is this function continuous for all x?
(b) On the basis of your conclusion in (a) and the fact
that in reality x % 0 and y % 0, must a and b be
both positive, be both negative, or have opposite
signs?
45. Cost-benefit Suppose that the cost C of removing
p percent of the impurities from the waste water in a
manufacturing process is given by
9800p
C(p) "
101 # p
Is this function continuous for all those p-values for
which the problem makes sense?
46. Pollution Suppose that the cost C of removing p percent of the particulate pollution from the exhaust gases
at an industrial site is given by
C(p) "
8100p
100 # p
p"
100C
8100 $ C
Find the percent of the impurities that could be removed
if cost were no object (that is, if cost were allowed to
increase without bound). Can 100% of the impurities
be removed? Explain.
49. Federal income tax The tax owed by a married couple filing jointly and their tax rates can be found in the
following tax rate schedule.
Schedule Y-1—If your filing status is
Married filing jointly or Qualifying widow(er)
If your taxable
income is:
Over ––
The tax is:
of the
amount
over ––
But not
over ––
$0
14,300
$14,300
58,100
10%
$1,430.00 + 15%
$0
14,300
58,100
117,250
117,250
178,650
8,000.00 + 25%
22,787.50 + 28%
58,100
117,250
178,650
319,100
319,100
39,979.50 + 33%
86,328.00 + 35%
178,650
319,100
Source: Internal Revenue Service, 2004, Form 1040 Instructions
From this schedule, the tax rate R(x) is a function of
taxable income x, as follows.
0.10
0.15
0.25
R(x) " f
0.28
0.33
0.35
if
if
if
if
if
if
0 ( x ( 14,300
14,300 ' x ( 58,100
58,100 ' x ( 117,250
117,250 ' x ( 178,650
178,650 ' x ( 319,100
x & 319,100
Identify any discontinuities in R(x).
608
● Chapter 9
Derivatives
50. Calories and temperature Suppose that the number of
calories of heat required to raise 1 gram of water (or
ice) from #40°C to x°C is given by
1
x $ 20
f(x) " b 2
x $ 100
if #40 ( x ' 0
if 0 ( x
(a) What can be said about the continuity of the function f(x)?
(b) What happens to water at 0°C that accounts for the
behavior of the function at 0°C?
51. Electrical usage costs The monthly charge in dollars
for x kilowatt-hours (kWh) of electricity used by a residential consumer of Excelsior Electric Membership
Corporation from November through June is given by
the function
10 $ 0.094x
C(x) " c 19.40 $ 0.075(x # 100)
49.40 $ 0.05(x # 500)
if
if
if
0 ( x ( 100
100 ' x ( 500
x & 500
(a) What is the monthly charge if 1100 kWh of electricity is consumed in a month?
(b) Find lim C(x) and lim C(x), if the limits exist.
xS100
xS500
(c) Is C continuous at x " 100 and at x " 500?
52. Postage costs First-class postage is 37 cents for the
first ounce or part of an ounce that a letter weighs and
is an additional 23 cents for each additional ounce or
part of an ounce above 1 ounce. Use the table or graph
of the postage function, f(x), to determine the following.
(a) lim f(x)
xS2.5
(b) f(2.5)
(c) Is f(x) continuous at 2.5?
(d) lim f(x)
xS4
Weight x
0'x(1
1'x(2
2'x(3
3'x(4
4'x(5
$0.37
0.60
0.83
1.06
1.29
f (x)
Year
Interest Paid as a
Percent of
Federal Expenditures
Year
Interest Paid as a
Percent of
Federal Expenditures
1930
1940
1950
1955
1960
1965
1970
0
10.5
13.4
9.4
10.0
9.6
9.9
1975
1980
1985
1990
1995
2000
2003
9.8
12.7
18.9
21.1
22.0
20.3
14.7
Source: Bureau of Public Debt, Department of the Treasury
If t is the number of years past 1900, use the table to
complete the following.
(a) Use the data in the table to find a fourth-degree
function d(t) that models the percent of federal
expenditures devoted to payment of interest on the
public debt.
(b) Use d(t) to predict the percent of federal expenditures devoted to payment of interest in 2009.
(c) Calculate lim d(t).
tS$)
(d) Can d(t) be used to predict the percent of federal
expenditures devoted to payment of interest on the
public debt for large values of t? Explain.
(e) For what years can you guarantee that d(t) cannot
be used to predict the percent of federal expenditures devoted to payment of interest on the public
debt? Explain.
54. Students per computer By using data from Quality
Education Data Inc., Denver, CO, the number of students per computer in U.S. public schools (1983–2002)
can be modeled with the function
(e) f(4)
(f) Is f(x) continuous at 4?
Postage
f(x)
53. Modeling Public debt of the United States The
interest paid on the public debt of the United States of
America as a percent of federal expenditures for
selected years is shown in the following table.
Postage Function
f(x) "
1.29
1.06
0.83
0.60
0.37
x
1
2
3
4
5
375.5 # 14.9x
x $ 0.02
where x is the number of years past the school year ending in 1981.
(a) Is this function continuous for school years from
1981 onward?
(b) Find the long-range projection of this model by
finding lim f(x). Explain what this tells us about
xS)
the validity of the model.