9.2 Continuous Functions; Limits at Infinitydas/Teaching/1100sum13/section 9.2.pdf9.2 Continuous...

596 ! Chapter 9 Derivatives

Farm workers The percent of U.S. workers in farmoccupations during certain years is shown in the table.

Year Percent Year Percent Year Percent

1820 71.8 1930 21.2 1980 2.71850 63.7 1940 17.4 1985 2.81870 53 1950 11.6 1990 2.41900 37.5 1960 6.1 1994 2.51920 27 1970 3.6 2002 2.5

Source: Bureau of Labor Statistics

Assume that the percent of U.S. workers in farm occu-pations can be modeled with the function

where t is the number of years past 1800. (A graph off (t) along with the data in the table is shown in theaccompanying figure.) Use the table and equation inProblems 65 and 66.

f(t) ! 1000 "#7.4812t $ 1560.2

1.2882t2 # 122.18t $ 21,483

65. (a) Find if it exists.

(b) What does this limit predict?(c) Is the equation accurate as ? Explain.

66. (a) Find if it exists.

(b) What does this limit predict?(c) Is the equation accurate as Explain.t S 140?

limtS140

f(t),t S 210

limtS210

f(t),

50 100 150 200

20

40

60

80

t

f(t)

9.2OBJECTIVES

! To determine whether afunction is continuous ordiscontinuous at a point

! To determine where a functionis discontinuous

! To find limits at infinity andhorizontal asymptotes

Continuous Functions; Limits at Infinity

] Application PreviewSuppose that a friend of yours and her husband have a taxable income of $117,250, and she tellsyou that she doesn’t want to make any more money because that would put them in a higher taxbracket. She makes this statement because the tax rate schedule for married taxpayers filing ajoint 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 isMarried filing jointly or Qualifying widow(er)

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 func-tion that gives income tax for married taxpayers as a function of taxable income and show thatthe function is continuous (see Example 3). That is, we will see that the tax paid does not jumpat $117,250 even though the tax on income above $117,250 is collected at a higher rate. In thissection, we will show how to determine whether a function is continuous, and we will investigatesome different types of discontinuous functions.

If your taxableincome is:

The tax is:

Over––But notover––

$014,30058,100

117,250178,650319,100

$14,30058,100

117,250178,650319,100

of theamountover––

$014,30058,100

117,250178,650319,100

10%$1,430.00 + 15%

8,000.00 + 25%22,787.50 + 28%39,979.50 + 33%86,328.00 + 35%

Continuity at a Point

9.2 Continuous Functions; Limits at Infinity ! 597

We have found that f (c) is the same as the limit as for any polynomial function f (x)and any real number c. Any function for which this special property holds is called a con-tinuous function. The graphs of such functions can be drawn without lifting the pencilfrom the paper, and graphs of others may have holes, vertical asymptotes, or jumps thatmake it impossible to draw them without lifting the pencil. Even though a function maynot be continuous everywhere, it is likely to have some points where the limit of the func-tion as is the same as f (c). In general, we define continuity of a function at the value

as follows.

The function f is continuous at if all of the following conditions are satisfied.

1. f(c) exists 2. exists 3.

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 discontinu-ous at

Figure 9.11 shows graphs of some functions that are discontinuous at

(a) (b)

and f(2) do not exist. f(2) does not exist.

(c) (d)

does not exist.

In the previous section, we saw that if f is a polynomial function, then for

every real number c, and also that if is a rational function and

Thus, by definition, we have the following.g(c) ! 0.

h(x) "f(x)g(x)

limxSc

h(x) " h(c)

limxSc

f(x) " f(c)

limxS2

f(x)limxS2

f(x) " 2 ! 4 " f(2)

f(x) " b1!(x # 2)2 if x ! 20 if x " 2

f(x) " b4 # x if x ! 24 if x " 2

-1 1 2 3 4

20

40

60

80

100

x

y

-2 -1 1 2 3 4

1

2

3

4

5

6

x

y

limxS2

f(x)

f (x) "x3 # 2x2 # x $ 2

x # 2f(x) "

1x # 2

-2 -1 1 2 3-2

2

4

6

8

x

y

-40

-20

20

40

x

y

-1 1 3 42

x " 2.

x ! c.

limxSc

f(x) " f(c)limxSc

f(x)

x ! c

x " cx S c

x S c

x

y

y = f (x)

f (c)

c

Figure 9.11


Every polynomial function is continuous for all real numbers.

Every rational function is continuous at all values of x except those that make thedenominator 0.

! EXAMPLE 1 Discontinuous Functions

For what values of x, if any, are the following functions continuous?

(a)

(b)

Solution(a) This is a rational function, so it is continuous for all values of x except for those that

make the denominator, equal to 0. Because at h(x) iscontinuous for all real numbers except Figure 9.12(a) shows a verticalasymptote at

(b) This is a rational function, so it is continuous everywhere except where the denomi-nator is 0. To find the zeros of the denominator, we factor

Because the denominator is 0 for and for f(2) and do not exist(recall that division by 0 is undefined). Thus the function is discontinuous at and The graph of this function (see Figure 9.12(b)) shows a hole at and a vertical asymptote at x " #2.

x " 2x " #2.x " 2

f(#2)x " #2,x " 2

f(x) "x2 # x # 2

x2 # 4"

x2 # x # 2(x # 2)(x $ 2)

x2 # 4.

x " 3!2.x " 3!2.

x " 3!2,4x # 6 " 04x # 6,

f(x) "x2 # x # 2

x2 # 4

h(x) "3x $ 24x # 6

! Checkpoint 1. Find any x-values where the following functions are discontinuous.

(a) (b)

If the pieces of a piecewise defined function are polynomials, the only values of x wherethe function might be discontinuous are those at which the definition of the functionchanges.

g(x) "x3 # 1

(x # 1)(x $ 2)f(x) " x3 # 3x $ 1

-3 -2 -1 1 2 3

-4

-2

-3

-5

-1

1

3

5

2

4

x

y

y = 3x + 24x – 6

x = 32

(a)

-5

-4

-3

-2

-1

1

2

3

4

5

-3 -2 -1 1 2 3

f (x) = x2 ! x ! 2x2 ! 4

x

y

(b)Figure 9.12


! EXAMPLE 2 Piecewise Defined Functions

Determine the values of x, if any, for which the following functions are discontinuous.

(a) (b)

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 defi-nition of the function changes—that is, at Note that satisfies

so Because g(x) is defined differently forand we use left- and right-hand limits. For we know thatso

Similarly, for we know that so

Because the left- and right-hand limits differ, does not exist, so g(x) is

discontinuous at This result is confirmed by examining the graph of g,shown in Figure 9.13.

(b) As with g(x), f(x) is continuous everywhere except perhaps at where the defi-nition of f(x) changes. Because satisfies The left-and right-hand limits are

and

Because the right- and left-hand limits are equal, we conclude that The limit is equal to the functional value

so we conclude that f is continuous at and thus f is continuous for all valuesof x. This result is confirmed by the graph of f, shown in Figure 9.14.

x " 2

limxS2

f(x) " f(2)

limxS2

f(x) " 0.

limxS2$

f(x) " limxS2$

(x # 2) " 2 # 2 " 0

limxS2#

f(x) " limxS2#

(4 # x2) " 4 # 22 " 0

x % 2, f(2) " 2 # 2 " 0.x " 2x " 2,

-4 -2 -1 1 2

-2

-1

1

2

4

x

y

g(x) = (x + 2)3 + 1 if x " !13 if x > !1

x " #1.

limxS#1

g(x)

limxS#1$

g(x) " limxS#1$

3 " 3

g(x) " 3:x & #1,x S #1$,

limxS#1#

g(x) " limxS#1#

3 (x $ 2)3 $ 1 4 " (#1 $ 2)3 $ 1 " 2

g(x) " (x $ 2)3 $ 1:x ' #1,x S #1#,x & #1,x ' #1

g(#1) " (#1 $ 2)3 $ 1 " 2.x ( #1,x " #1x " #1.

f (x) " b4 # x2 if x ' 2x # 2 if x % 2

g(x) " b(x $ 2)3 $ 1 if x ( #13 if x & #1

Figure 9.13


] EXAMPLE 3 Taxes (Application Preview)

The tax rate schedule for married taxpayers filing a joint return (shown in the table) appearsto have a jump in taxes for taxable income at $117,250.


(a) Use the table and write the function that gives income tax for married taxpayers as afunction of taxable income, x.

(b) Is the function in part (a) continuous at (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 puther 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

T(x) " f 0.10x if 0 ( x ( 14,300 1430 $ 0.15(x # 14,300) if 14,300 ' x ( 58,100 8000 $ 0.25(x # 58,100) if 58,100 ' x ( 117,25022,787.50 $ 0.28(x # 117,250) if 117,250 ' x ( 178,65039,979.50 $ 0.33(x # 178,650) if 178,650 ' x ( 319,10086,328 $ 0.35(x # 319,100) if x & 319,100

x " 117,250?


The tax is:


$014,30058,100

117,250178,650319,100

$14,30058,100

117,250178,650319,100


$014,30058,100

117,250178,650319,100

10%$1,430.00 + 15%

8,000.00 + 25%22,787.50 + 28%39,979.50 + 33%86,328.00 + 35%

-4 -3 -1 1 2 3 4

-4

-3

-2

-1

1

2

3

4

x

y

f (x) = 4 ! x2 if x < 2x ! 2 if x # 2

Figure 9.14



(b) We examine the three conditions for continuity at (i) 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:

From the right:

Because these one-sided limits agree, the limit exists and is

(iii) Because 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 thetax 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, is continuous everywhere

except perhaps at _______.

We noted in Section 2.4, “Special Functions and Their Graphs,” that the graph of has a vertical asymptote at (shown in Figure 9.15(a)). By graphing and eval-uating the function for very large x values we can see that never becomes nega-tive for positive x-values regardless of how large the x-value is. Although no value of xmakes equal to 0, it is easy to see that approaches 0 as x gets very large. This isdenoted by

and means that the line (the x-axis) is a horizontal asymptote for We alsosee that approaches 0 as x decreases without bound, and we denote this by

These limits for can also be established with numerical tables.

x x100 0.01

100,000 0.00001100,000,000 0.00000001

0 0

We can use the graph of in Figure 9.15(b) to see that the x-axis is ahorizontal asymptote and that

limxS$)

2x2

" 0 and limxS#)

2x2

" 0

(y " 0)y " 2!x2

limxS#)

1x

" 0limxS$)

1x

" 0

#)$)TTTT

#0.00000001#100,000,000#0.00001#100,000#0.01#100

f(x) " 1!xf(x) " 1!x

f(x) " 1!x

limxS#)

1x

" 0

y " 1!xy " 1!x.y " 0

limxS$)

1x

" 0

1!x1!x

y " 1!xy " 1!xx " 0

y " 1!x

h(x) " b f(x) if x ( ag(x) if x & a

limxS117,250

T(x) " T(117,250) " 22,787.50,

limxS117,250

T(x) " 22,787.50

limxS117,250$

T(x) " limxS117,250$

322,787.50 $ 0.28(x # 117,250) 4 " 22,787.50

limxS117,250#

T(x) " limxS117,250#

38000 $ 0.25(x # 58,100) 4 " 22,787.50

limxS117,250

T(x)T(117,250) " 22,787.50,

x " 117,250.

Figure 9.15

! Checkpoint

Limits at Infinity

-6 6

-6

6

y = 1x

(a)

(b)

-3 30

30

y = 2x2

Limits at Infinity


By using graphs and/or tables of values, we can generalize the results for the functionsshown in Figure 9.15 and conclude the following.

If c is any constant, then

1.

2.

3.

In order to use these properties for finding the limits of rational functions as xapproaches or we first divide each term of the numerator and denominator bythe 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) (b)

Solution(a) The highest power of x present is so we divide each term in the numerator and

denominator by x and then use the properties for limits at infinity.

Figure 9.16(a) shows the graph of this function with the y-coordinates of the graphapproaching 2 as x approaches and as x approaches That is, is ahorizontal asymptote. Note also that there is a discontinuity (vertical asymptote)where

(b) We divide each term in the numerator and denominator by and then use theproperties.

This limit does not exist because the numerator approaches 1 and the denominatorapproaches 0 through positive values. Thus

The graph of this function, shown in Figure 9.16(b), has y-coordinates that increasewithout bound as x approaches and that decrease without bound as x approaches

(There is no horizontal asymptote.) Note also that there is a vertical asymptoteat x " 1.$).

#)

x2 $ 31 # x

S $) as x S #)

limxS#)

x2 $ 31 # x

" limxS#)

x2

x2$

3x2

1x2

#xx2

" limxS#)

1 $3x2

1x2

#1x

x2x " #2.

y " 2#).$)

"2 # 01 $ 0

" 2 (by Properties 1 and 2)

limxS $)

2x # 1x $ 2

" limxS $)

2xx

#1x

xx

$2x

" limxS $)

2 #1x

1 $2x

x1,

limxS#)

x2 $ 31 # x

limxS$)

2x # 1x $ 2

#),$)

limxS#)

cxn

" 0, where n & 0 is any integer.

limxS$)

cxp

" 0, where p & 0.

limxS$)

c " c and limxS#)

c " c.

Limits at Infinity andHorizontal Asymptotes


In our work with limits at infinity, we have mentioned horizontal asymptotes severaltimes. The connection between these concepts follows.

If or where b is a constant, then the line is a

horizontal asymptote for the graph of Otherwise, has no horizontalasymptotes.

3. (a) Evaluate

(b) What does part (a) say about horizontal asymptotes for

We can use the graphing and table features of a graphing calculator to help locate and inves-tigate discontinuities and limits at infinity (horizontal asymptotes). A graphing calculatorcan be used to focus our attention on a possible discontinuity and to support or suggestappropriate algebraic calculations. "

! EXAMPLE 5 Limits with Technology

Use a graphing utility to investigate the continuity of the following functions.

(a) (b)

(c) (d)

Solution(a) Figure 9.17(a) shows that f(x) has a discontinuity (vertical asymptote) near

Because DNE, we know that f(x) is not continuous at (b) Figure 9.17(b) shows that g(x) is discontinuous (vertical asymptote) near and

this looks like the only discontinuity. However, the denominator of g(x) is zero atand so g(x) must have discontinuities at both of these x-values. Evalu-

ating or using the table feature confirms that is a discontinuity (a hole, ormissing point). The figure also shows a horizontal asymptote; evaluation of confirms this is the line y " 1.

limxS)

g(x)x " #1

x " #1,x " 1

x " 1,x " #1.f (#1)

x " #1.

k(x) " d#x2

2# 2x if x ( #1

x2

$ 2 if x & #1h(x) "

"x $ 1"x $ 1

g(x) "x2 # 2x # 3

x2 # 1f(x) "

x2 $ 1x $ 1

f(x) " (x2 # 4)!(2x2 # 7)?

limxS$)

x2 # 42x2 # 7

.

y " f(x)y " f(x).

y " blimxS#)

f(x) " b,limxS)

f(x) " b

x

y

!8 !6 !4 2 4

!4

4

6

8

2

!2

limx$+ )

2x ! 1x + 2

= 2

2x ! 1x + 2f (x) =

y = 2

(a)

x

y

!8 !6 !4 !2 2 4 6 8

!10

!8

!6

!4

!2

4

6

x2 + 31 ! x

$ + ) as x $ ! )

x2 + 31 ! xf (x) =

(b)Figure 9.16

! Checkpoint

Calculator Note


(c) Figure 9.17(c) shows a discontinuity (jump) at We also see that DNE,which confirms the observations from the graph.

(d) The graph in Figure 9.17(d) appears to be continuous. The only “suspicious” x-valueis where the formula for k(x) changes. Evaluating and examining atable near indicates that k(x) is continuous there. Algebraic evaluations of thetwo one-sided limits confirm this.

x " #1k(#1)x " #1,

h(#1)x " #1.

! Checkpoint Solutions

-10 10

-8

8

y = f (x)

(a) (b)

-8 8

-8

8

y = g(x)

(c)

-10 10

-8

8

y = h(x)

(d)

-5 3

-3

5

y = k(x)

Figure 9.17

SummaryThe following information is useful in discussing continuity of functions.

A. A polynomial function is continuous everywhere.

B. A rational function is a function of the form where f(x) and g(x) are polynomials.

1. If at any value of x, the function is continuous everywhere.2. If the function is discontinuous at

(a) If and then there is a vertical asymptote at

(b) If and then the graph has a missing point at (c, L).

C. A piecewise defined function may have a discontinuity at any x-value where the func-tion 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 func-tion

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.

1. (a) This is a polynomial function, so it is continuous at all values of x (discontinuous atnone).

(b) This is a rational function. It is discontinuous at and because thesevalues make its denominator 0.

2. x " a.

x " #2x " 1

f(x) " p(x)!q(x).

limxSc

f(x)g(x)

" L,g(c) " 0

x " c.f (c) ! 0,g(c) " 0x " c.g(c) " 0,

g(x) ! 0

f(x)g(x)

,


3. (a)

(b) The line is a horizontal asymptote.y " 1!2

limxS$)

x2 # 42x2 # 7

" limxS$)

1 #4x2

2 #7x2

"1 # 02 # 0

"12

Exercises9.2

Problems 1 and 2 refer to the following figure. For eachgiven x-value, use the figure to determine whether thefunction is continuous or discontinuous at that x-value.If the function is discontinuous, state which of the threeconditions that define continuity is not satisfied.

1. (a) (b) (c) (d)2. (a) (b) (c) (d)

In Problems 3–8, determine whether each function iscontinuous or discontinuous at the given x-value. Exam-ine the three conditions in the definition of continuity.

3.

4.

5.

6.

7.

8.

In Problems 9–16, determine whether the given functionis continuous. If it is not, identify where it is discontin-uous and which condition fails to hold. You can verify

f(x) " bx2 $ 1 if x ( 12x2 # 1 if x & 1

, x " 1

f(x) " bx # 3 if x ( 24x # 7 if x & 2

, x " 2

f(x) "x2 # 4x # 2

, x " 2

y "x2 # 9x $ 3

, x " #3

y "x2 # 9x $ 3

, x " 3

f(x) "x2 # 4x # 2

, x " #2

x " 5x " #2x " #4x " 2x " 0x " 3x " 1x " #5

x

y

642!4!6

2

4

!2

!4

y = f (x)

your conclusions by graphing each function with agraphing utility, if one is available.

9. 10.

11. 12.

13. 14.

15.

16.

In Problems 17–20, use the trace and table features of agraphing utility to investigate whether each of the fol-lowing functions has any discontinuities.

17. 18.

19.

20.

Each of Problems 21–24 contains a function and itsgraph. For each problem, answer parts (a) and (b).(a) Use the graph to determine, as well as you can,

(i) vertical asymptotes, (ii)(iii) (iv) horizontal asymptotes.

(b) Check your conclusions in (a) by using the functionsto determine items (i)–(iv) analytically.

21. f(x) "8

x $ 2

limxS #%

f(x),lim

xS $%f(x),

f(x) " bx2 $ 4 if x ! 18 if x " 1

f(x) " bx # 4 if x ( 3x2 # 8 if x & 3

y "x2 # 5x $ 4

x # 4y "

x2 # 5x # 6x $ 1

f(x) " bx3 $ 1 if x ( 12 if x & 1

f(x) " b3 if x ( 1x2 $ 2 if x & 1

y "2x # 1x2 $ 3

y "x

x2 $ 1

y "4x2 $ 4x $ 1

x $ 1!2g(x) "

4x2 $ 3x $ 2x $ 2

y " 5x2 # 2xf(x) " 4x2 # 1

y = f (x)

-10 5 10

5

10

x

y


22.

23.

24.

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 hori-

zontal asymptotes?You can verify your conclusions by graphing the func-tions with a graphing utility, if one is available.

25. 26.

27. 28.

29. 30.

31. 32.

In Problems 33 and 34, use a graphing utility to com-plete (a) and (b).(a) Graph each function using a window with 0 ! x

! 300 and #2 ! y ! 2. What does the graph indi-cate about

(b) Use the table feature with x-values larger than 10,000to investigate Does the table support yourconclusions in part (a)?

33. 34. f(x) "5x3 # 7x1 # 3x3

f(x) "x2 # 4

3 $ 2x2

limxS $%

f(x).

limxS $%

f(x)?

limxS#)

5x3 # 84x2 $ 5x

limxS$)

3x2 $ 5x6x $ 1

limxS$)

4x2 $ 5xx2 # 4x

limxS#)

5x3 # 4x3x3 # 2

limxS#)

3x2 $ 2x2 # 4

limxS$)

x3 # 1x3 $ 4

limxS#)

4x2 # 2x

limxS$)

3x $ 1

f(x) "4x2

x2 # 4x $ 4

f(x) "2(x $ 1)3(x $ 5)(x # 3)2(x $ 2)2

f(x) "x # 3x # 2

In Problems 35 and 36, complete (a)–(c). Use analyticmethods to find (a) any points of discontinuity and (b)limits as and (c) Then explain why, forthese functions, a graphing utility is better as a supporttool for the analytic methods than as the primary tool forinvestigation.

35. 36.

For Problems 37 and 38, let

be a rational function.

37. If show that and hence that

is a horizontal asymptote.

38. (a) If show that and hence that

is a horizontal asymptote.(b) If find What does this say about

horizontal asymptotes?

APPLICATIONS

39. Sales volume Suppose that the weekly sales volume(in thousands of units) for a product is given by

where p is the price in dollars per unit. Is this functioncontinuous(a) for all values of p? (b) at (c) for all (d) What is the domain for this application?

40. Worker productivity Suppose that the average numberof minutes M that it takes a new employee to assembleone unit of a product is given by

where t is the number of days on the job. Is this func-tion continuous(a) for all values of t? (b) at (c) for all (d) What is the domain for this application?

41. Demand Suppose that the demand for a product isdefined by the equation

where p is the price and q is the quantity demanded.(a) Is this function discontinuous at any value of q?

What value?

p "200,000(q $ 1)2

t % 0?t " 14?

M "40 $ 30t2t $ 1

p % 0?p " 24?

y "32

(p $ 8)2!5

limxS)

f(x).m ' n,y " 0

limxS)

f(x) " 0,m & n,

y "an

bn

limxS)

f(x) "an

bn,m " n,

f(x) !anxn $ an#1xn#1 $ . . . $ a1x $ a0

bmxm $ bm#1xm#1 $ . . . $ b1x $ b0

f(x) "3000x

4350 # 2xf(x) "

1000x # 1000x $ 1000

x S #%.x S $%

-10 -5 5 10

-10

-5

5

10

x

y

y = f (x)

-20 -10 10 20

-10

5

10

15

20

25

x

y

y = f (x)

-20-10 10 20 30 40

5101520253035

x

y

y = f (x)


(b) Because q represents quantity, we know that Is this function continuous for

42. Advertising and sales The sales volume y (in thou-sands of dollars) is related to advertising expendituresx (in thousands of dollars) according to

(a) Is this function discontinuous at any points?(b) Advertising expenditures x must be nonnegative. Is

this function continuous for these values of x?

43. Annuities If an annuity makes an infinite series ofequal payments at the end of the interest periods, it iscalled a perpetuity. If a lump sum investment of isneeded to result in n periodic payments of R when theinterest rate per period is i, then

(a) Evaluate to find a formula for the lump sum

payment for a perpetuity.(b) Find the lump sum investment needed to make pay-

ments of $100 per month in perpetuity if interest is12%, compounded monthly.

44. Response to adrenalin Experimental evidence sug-gests that the response y of the body to the concentra-tion x of injected adrenalin is given by

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 and must a and b beboth positive, be both negative, or have oppositesigns?

45. Cost-benefit Suppose that the cost C of removingp percent of the impurities from the waste water in amanufacturing process is given by

Is this function continuous for all those p-values forwhich the problem makes sense?

46. Pollution Suppose that the cost C of removing p per-cent of the particulate pollution from the exhaust gasesat an industrial site is given by

C(p) "8100p

100 # p

C(p) "9800p

101 # p

y % 0,x % 0

y "x

a $ bx

limnS)

An

An " RB1 # (1 $ i)#n

iR An

y "200x

x $ 10

q % 0?q % 0. Describe any discontinuities for C(p). Explain what

each discontinuity means.

47. Pollution The percent p of particulate pollution thatcan be removed from the smokestacks of an industrialplant by spending C dollars is given by

Find the percent of the pollution that could be removedif spending C were allowed to increase without bound.Can 100% of the pollution be removed? Explain.

48. Cost-benefit The percent p of impurities that can beremoved from the waste water of a manufacturingprocess at a cost of C dollars is given by

Find the percent of the impurities that could be removedif cost were no object (that is, if cost were allowed toincrease without bound). Can of the impuritiesbe removed? Explain.

49. Federal income tax The tax owed by a married cou-ple filing jointly and their tax rates can be found in thefollowing tax rate schedule.


From this schedule, the tax rate R(x) is a function oftaxable income x, as follows.

Identify any discontinuities in R(x).

R(x) " f0.10 if 0 ( x ( 14,3000.15 if 14,300 ' x ( 58,1000.25 if 58,100 ' x ( 117,2500.28 if 117,250 ' x ( 178,6500.33 if 178,650 ' x ( 319,1000.35 if x & 319,100


The tax is:


$014,30058,100

117,250178,650319,100

$14,30058,100

117,250178,650319,100


$014,30058,100

117,250178,650319,100

10%$1,430.00 + 15%

8,000.00 + 25%22,787.50 + 28%39,979.50 + 33%86,328.00 + 35%

100%

p "100C

8100 $ C

p "100C

7300 $ C



50. Calories and temperature Suppose that the number ofcalories of heat required to raise 1 gram of water (orice) from to is given by

(a) What can be said about the continuity of the func-tion f(x)?

(b) What happens to water at that accounts for thebehavior of the function at

51. Electrical usage costs The monthly charge in dollarsfor x kilowatt-hours (kWh) of electricity used by a res-idential consumer of Excelsior Electric MembershipCorporation from November through June is given bythe function

(a) What is the monthly charge if 1100 kWh of elec-tricity is consumed in a month?

(b) Find and if the limits exist.

(c) Is C continuous at and at

52. Postage costs First-class postage is 37 cents for thefirst ounce or part of an ounce that a letter weighs andis an additional 23 cents for each additional ounce orpart of an ounce above 1 ounce. Use the table or graphof the postage function, f(x), to determine the following.(a)

(b) f(2.5)(c) Is f(x) continuous at 2.5?(d)

(e) f(4)(f) Is f(x) continuous at 4?

PostageWeight x f (x)

$0.370.600.831.061.294 ' x ( 5

3 ' x ( 42 ' x ( 31 ' x ( 20 ' x ( 1

limxS4

f(x)

limxS2.5

f (x)

x " 500?x " 100

limxS500

C(x),limxS100

C(x)

C(x) " c10 $ 0.094x if 0 ( x ( 10019.40 $ 0.075(x # 100) if 100 ' x ( 50049.40 $ 0.05(x # 500) if x & 500

0°C?0°C

f(x) " b 12x $ 20 if #40 ( x ' 0x $ 100 if 0 ( x

x°C#40°C

53. Modeling Public debt of the United States Theinterest paid on the public debt of the United States ofAmerica as a percent of federal expenditures forselected years is shown in the following table.

Interest Paid as a Interest Paid as aPercent of Percent of

Year Federal Expenditures Year Federal Expenditures

1930 0 1975 9.81940 10.5 1980 12.71950 13.4 1985 18.91955 9.4 1990 21.11960 10.0 1995 22.01965 9.6 2000 20.31970 9.9 2003 14.7

Source: Bureau of Public Debt, Department of the Treasury

If t is the number of years past 1900, use the table tocomplete the following.(a) Use the data in the table to find a fourth-degree

function d(t) that models the percent of federalexpenditures devoted to payment of interest on thepublic debt.

(b) Use d(t) to predict the percent of federal expendi-tures devoted to payment of interest in 2009.

(c) Calculate

(d) Can d(t) be used to predict the percent of federalexpenditures devoted to payment of interest on thepublic debt for large values of t? Explain.

(e) For what years can you guarantee that d(t) cannotbe used to predict the percent of federal expendi-tures devoted to payment of interest on the publicdebt? Explain.

54. Students per computer By using data from QualityEducation Data Inc., Denver, CO, the number of stu-dents per computer in U.S. public schools (1983–2002)can be modeled with the function

where x is the number of years past the school year end-ing in 1981.(a) Is this function continuous for school years from

1981 onward?(b) Find the long-range projection of this model by

finding Explain what this tells us aboutthe validity of the model.

limxS)

f(x).

f(x) "375.5 # 14.9x

x $ 0.02

limtS$)

d(t).

0.37

0.60

0.83

1.06

1.29

1 2 3 4 5

Postage Functionf (x)

x

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