PROOFS
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CHAPTER 1
Supply and demand determine the price of stock and other commodities.
FUNCTIONS, GRAPHS, AND LIMITS
1 2 3 4 5 6 Functions The Graph of a Function Linear Functions Functional Models Limits One-Sided Limits and Continuity Chapter Summary
Important Terms, Symbols, and Formulas Checkup for Chapter 1 Review Problems
Explore! Update Think About It
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2 CHAPTER 1 Functions, Graphs, and Limits
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SECTION
1.1 Functions
In many practical situations, the value of one quantity may depend on the value of a second. For example, the consumer demand for beef may depend on the current market price; the amount of air pollution in a metropolitan area may depend on the number of cars on the road; or the value of a rare coin may depend on its age. Such relationships can often be represented mathematically as functions. Loosely speaking, a function consists of two sets and a rule that associates elements in one set with elements in the other. For instance, suppose you want to determine the effect of price on the number of units of a particular commodity that will be sold at that price. To study this relationship, you need to know the set of admissible prices, the set of possible sales levels, and a rule for associating each price with a particular sales level. Here is the denition of function we shall use. A function is a rule that assigns to each object in a set A exactly one object in a set B. The set A is called the domain of the function, and the set of assigned objects in B is called the range. For most functions in this book, the domain and range will be collections of real numbers and the function itself will be denoted by a letter such as f. The value that the function f assigns to the number x in the domain is then denoted by f(x) (read as f of x), which is often given by a formula, such as f(x) x2 4.
Function
A
B
Input x
f machine
Output f(x)
(a) A function as a mapping
(b) A function as a machine
FIGURE 1.1 Interpretations of the function f(x). It may help to think of such a function as a mapping from numbers in A to numbers in B (Figure 1.1a), or as a machine that takes a given number from A and converts it into a number in B through a process indicated by the functional rule (Figure 1.1b). For instance, the function f (x) x2 4 can be thought of as an f machine that accepts an input x, then squares it and adds 4 to produce an output y x2 4. No matter how you choose to think of a functional relationship, it is important to remember that it assigns one and only one number in the range (output) to each number in the domain (input). Here is an example.
EXPLORE!
Store f(x) x2 4 into your graphing utility. Evaluate at x 3, 1, 0, 1, and 3. Make a table of values. Repeat using g(x) x2 1. Explain how the values of f(x) and g(x) differ for each x value.
EXAMPLE 1.1.1
Find f(3) if f(x)
Solution
x2
4. f(3) 32 4 13
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1-3
SECTION 1.1 Functions
3
Observe the convenience and simplicity of the functional notation. In Example 1.1.1, the compact formula f(x) x2 4 completely denes the function, and you can indicate that 13 is the number the function assigns to 3 by simply writing f(3) 13. It is often convenient to represent a functional relationship by an equation y f(x), and in this context, x and y are called variables. In particular, since the numerical value of y is determined by that of x, we refer to y as the dependent variable and to x as the independent variable. Note that there is nothing sacred about the symbols x and y. For example, the function y x2 4 can just as easily be represented by s t2 4 or by w u2 4. Functional notation can also be used to describe tabular data. For instance, Table 1.1 lists the average tuition and fees for private 4-year colleges at 5-year intervals from 1973 to 2003. TABLE 1.1 Average Tuition and Fees for 4-Year Private Colleges
Academic Year Ending in Tuition and Fees
Period n
1973 1978 1983 1988 1993 1998 2003
1 2 3 4 5 6 7
$1,898 $2,700 $4,639 $7,048 $10,448 $13,785 $18,273
SOURCE: Annual Survey of Colleges, The College Board, New York.
We can describe this data as a function f dened by the rule f(n) average tuition and fees at the beginning of the nth 5-year period
Thus, f(1) 1,898, f(2) 2,700, . . . , f(7) 18,273. Note that the domain of f is the set of integers A {1, 2, . . . , 7}. The use of functional notation is illustrated further in Examples 1.1.2 and 1.1.3. In Example 1.1.2, notice that letters other than f and x are used to denote the function and its independent variable.
Just-In-Time Review
Recall that x a/b 5 x a whenever a and b are positive integers. Example 1.1.2 uses the case when a 1 and b 2; x1/2 is another way of expressing x.
b
EXAMPLE 1.1.2
If g(t)
Solution
(t
2)1/2, nd (if possible) g(27), g(5), g(2), and g(1).
t 2. (If you need to brush up on fractional powRewrite the function as g(t) ers, consult the discussion of exponential notation in Appendix A. Then g(27) g(5) g(2) 27 2 5 2 2 2 25 3 0 5 1.7321 0
and
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4 CHAPTER 1 Functions, Graphs, and Limits
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EXPLORE!
x 2 in the Store g(x) function editor of your graph(x 2). ing utility as Y1 Now on your HOME SCREEN create Y1(27), Y1(5), and Y1(2), or, alternatively, Y1({27, 5, 2}), where the braces are used to enclose a list of values. What happens when you construct Y1(1)?
However, g(1) is undened since g(1) 1 2 1
and negative numbers do not have real square roots. Functions are often dened using more than one formula, where each individual formula describes the function on a subset of the domain. A function dened in this way is sometimes called a piecewise-dened function. Here is an example of such a function.
EXAMPLE 1.1.3
Find f
EXPLORE!
Create a simple piecewisedened function using the boolean algebra features of your graphing utility. Write Y1 2(X , 1) ( 1)(X $ 1) in the function editor. Examine the graph of this function, using the ZOOM Decimal Window. What values does Y1 assume at X 2, 0, 1, and 3?
1 , f(1), and f(2) if 2 1 f(x) x 1 3x 2 1 if x if x 1 1
Solution
Since x
1 satises x 2 f
1, use the top part of the formula to nd 1 2 1 1/2 1 3/2 2 3
1
However, x 1 and x 2 satisfy x bottom part of the formula: f(1) 3(1)2 1 4
1, so f(1) and f(2) are both found by using the and f(2) 3(2)2 1 13
EXPLORE!
Store f(x) 1/(x 3) in your graphing utility as Y1, and display its graph using a ZOOM Decimal Window. TRACE values of the function from X 2.5 to 3.5. What do you notice at X 3? Next store g(x) (x 2) into Y1, and graph using a ZOOM Decimal Window. TRACE values from X 0 to 3, in 0.1 increments. When do the Y values start to appear, and what does this tell you about the domain of g(x)?
Unless otherwise specied, if a formula (or several formulas, as in Example 1.1.3) is used to dene a function f, then we assume the domain of f to be the set of all numbers for which f(x) is dened (as a real number). We refer to this as the natural domain of f. Determining the natural domain of a function often amounts to excluding all numbers x that result in dividing by 0 or taking the square root of a negative number. This procedure is illustrated in Example 1.1.4.
Domain Convention
EXAMPLE 1.1.4
Find the domain and range of each of these functions 1 t a. f(x) b. g(t) x 3
Solution a.
2
Since division by any number other than 0 is possible, the domain of f is the set of all numbers x 3. The range of f is the set of all numbers y except 0, since 1 1 ; in particular, x 3 . for any y 0, there is an x such that y x 3 y
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SECTION 1.1 Functions
5
b.
Since negative numbers do not have real square roots, g(t) can be evaluated only when t 2 0, so the domain of g is the set of all numbers t such that t 2. The range of g is the set of all nonnegative numbers, for if y 0 is any such t 2; namely, t y2 2. number, there is a t such that y
Functions Used in Economics
There are several basic functions that occur frequently in applications involving economics. A demand function p D(x) is a function that relates the unit price p for a particular commodity to the number of units x demanded by consumers at that price. The total revenue is given by the product R(x) (number of items sold)(price per item) xp xD(x)
If C(x) is the total cost of producing the x units, then the prot derived from their sale at the unit price p is given by the function P(x) R(x) C(x) xD(x) C(x)
These terms are used in Example 1.1.5.
EXAMPLE 1.1.5
Market research indicates that consumers will buy x thousand units of a particular kind of coffee maker when the unit price is p 0.27x 2.23x2 51
dollars. The cost of producing the x thousand units is C(x) 3.5x 85
thousand dollars. a. What are the demand, revenue, and prot functions, D(x), R(x), and P(x), for this production process? b. For what values of x is production of the coffee makers protable?
Solution a.
The demand function is D(x) R(x)
0.27x xD(x)
51, so the revenue is 0.27x2 51x
thousand dollars, and the prot is P(x) R(x) C(x) 0.27x 2 51x 2.5x 2 47.5x (2.23x 2 85 3.5x 85)
b.
thousand dollars. Production is protable when P(x) P(x)
0. We nd that
2.5x 2 47.5x 85 2.5(x 2 19x 34) 2.5(x 2)(x 17)
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6 CHAPTER 1 Functions, Graphs, and Limits
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Just-In-Time Review
The product of two numbers is positive if they have the same sign and is negative if they have different signs. That is, ab . 0 if a . 0 and b . 0 and also if a , 0 and b , 0. On the other hand, ab , 0 if a , 0 and b . 0 or if a . 0 and b , 0.
Since the coefcient 2.5 is negative, it follows that P(x) 0 only if the terms (x 2) and (x 17) have different signs; that is, when x 2 0 and x 17 0. Thus, production is protable for 2 x 17. Example 1.1.6 illustrates how functional notation is used in a practical situation. Notice that to make the algebraic formula easier to interpret, letters suggesting the relevant practical quantities are used for the function and its independent variable. (In this example, the letter C stands for cost and q stands for quantity manufactured.)
EXAMPLE
1.1.6
EXPLORE!
Refer to Example 1.1.6, and store the cost function C(q) into Y1 as X3 30X2 500X 200 Construct a TABLE of values for C(q) using your calculator, setting TblStart at X 5 with an increment Tbl 1 unit. On the table of values observe the cost of manufacturing the 10th unit.
Suppose the total cost in dollars of manufacturing q units of a certain commodity is given by the function C(q) q3 30q2 500q 200. a. Compute the cost of manufacturing 10 units of the commodity. b. Compute the cost of manufacturing the 10th unit of the commodity.
Solution a.
The cost of manufacturing 10 units is the value of the total cost function when q 10. That is, Cost of 10 units C(10) (10)3 30(10)2 $3,200 500(10) 200
b.
The cost of manufacturing the 10th unit is the difference between the cost of manufacturing 10 units and the cost of manufacturing 9 units. That is, Cost of 10th unit C(10) C(9) 3,200 2,999 $201
Composition of Functions
There are many situations in which a quantity is given as a function of one variable that, in turn, can be written as a function of a second variable. By combining the functions in an appropriate way, you can express the original quantity as a function of the second variable. This process is called composition of functions or functional composition. For instance, suppose environmentalists estimate that when p thousand people live in a certain city, the average daily level of carbon monoxide in the air will be c(p) parts per million, and that separate demographic studies indicate the population in t years will be p(t) thousand. What level of pollution should be expected in t years? You would answer this question by substituting p(t) into the pollution formula c(p) to express c as a composite function of t. We shall return to the pollution problem in Example 1.1.11 with specific formulas for c( p) and p(t), but first you need to see a few examples of how composite functions are formed and evaluated. Here is a denition of functional composition. Given functions f(u) and g(x), the composition f(g(x)) is the function of x formed by substituting u g(x) for u in the formula for f(u).
Composition of Functions
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SECTION 1.1 Functions
7
Note that the composite function f(g(x)) makes sense only if the domain of f contains the range of g. In Figure 1.2, the denition of composite function is illustrated as an assembly line in which raw input x is rst converted into a transitional product g(x) that acts as input the f machine uses to produce f(g(x)).
Input x x x g Machine
Output . . . g(x) g(x)
Input
g(x) g(x)
Output f Machine f (g(x))
FIGURE 1.2 The composition f(g(x)) as an assembly line.
EXAMPLE 1.1.7
Find the composite function f(g(x)), where f(u)
Solution
u2
3u
1 and g(x)
x
1.
Replace u by x
1 in the formula for f(u) to get f(g(x)) (x (x2 x2 1)2 3(x 1) 2x 1) (3x 5x 5 1 3) 1
EXPLORE!
Store the functions f(x) x2 and g(x) x 3 into Y1 and Y2, respectively, of the function editor. Deselect (turn off) Y1 and Y2. Set Y3 Y1(Y2) and Y4 Y2(Y1). Show graphically (using ZOOM Standard) and analytically (by table values) that f(g(x)) represented by Y3 and g(f(x)) represented by Y4 are not the same functions. What are the explicit equations for both of these composites?
NOTE By reversing the roles of f and g in the denition of composite function, you can dene the composition g( f(x)), and, in general, f(g(x)) and g( f(x)) will not be the same. For instance, with the functions in Example 1.1.7, you rst write g(w) w
2
1 3x
and 1 to get (x2 x2 5x
f(x)
x2
3x
1
and then replace w by x
g( f(x)) which is equal to f(g(x)) this). x2
3x 3x
1) 2
1 3/2 (you should verify
5 only when x
Example 1.1.7 could have been worded more compactly as follows: Find the composite function f(x 1) where f(u) u2 3u 1. The use of this compact notation is illustrated further in Example 1.1.8.
EXPLORE!
Refer to Example 1.1.8. Store f(x) 3x2 1/x 5 into Y1. Write Y2 Y1(X 1). Construct a table of values for Y1 and Y2 for 0, 1, . . . , 6. What do you notice about the values for Y1 and Y2?
EXAMPLE
Find f(x
Solution
1.1.8
3x 2 1 x 5.
1) if f(x)
At rst glance, this problem may look confusing because the letter x appears both as the independent variable in the formula dening f and as part of the expression
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x 1. Because of this, you may nd it helpful to begin by writing the formula for f in more neutral terms, say as f( ) To nd f(x 3( )2 1 5 1 inside each box, getting 1 5
1), you simply insert the expression x f(x 1) 3(x 1)2 1 x
Occasionally, you will have to take apart a given composite function g(h(x)) and identify the outer function g(u) and inner function h(x) from which it was formed. The procedure is demonstrated in Example 1.1.9.
EXAMPLE 1.1.9
If f(x)
Solution
5 x 2
4(x
2)3, nd functions g(u) and h(x) such that f(x)
g(h(x)).
The form of the given function is f(x) 5 4( )3 2. Thus, f(x) and h(x) x g(h(x)), where 2
where each box contains the expression x g(u) 5 u 4u3
Actually, in Example 1.1.9, there are innitely many pairs of functions g(u) 5 and h(x) that combine to give g(h(x)) f(x). [For example, g(u) 4(u 1)3 u 1 and h(x) x 3.] The particular pair selected in the solution to this example is the most natural one and reects most clearly the structure of the original function f(x).
EXAMPLE 1.1.10
A difference quotient is an expression of the general form f(x h) h f(x)
where f is a given function of x and h is a number. Difference quotients will be used in Chapter 2 to dene the derivative, one of the fundamental concepts of calculus. Find the difference quotient for f(x) x2 3x.
Solution
outer function
inner function
You nd that f(x h) h f(x) [(x h)2 3(x h h)] [x 2 3x]
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1-9
SECTION 1.1 Functions
9
[x2 2xh 2x
2xh
h2
3x h
3h]
[x2
3x]
expand the numerator combine terms in the numerator divide by h
h2 3h h h 3
Example 1.1.11 illustrates how a composite function may arise in an applied problem.
EXAMPLE
1.1.11
An environmental study of a certain community suggests that the average daily level of carbon monoxide in the air will be c( p) 0.5p 1 parts per million when the population is p thousand. It is estimated that t years from now the population of the community will be p(t) 10 0.1t2 thousand. a. Express the level of carbon monoxide in the air as a function of time. b. When will the carbon monoxide level reach 6.8 parts per million?
Solution a.
Since the level of carbon monoxide is related to the variable p by the equation c( p) 0.5p 1 0.1t2 0.1t2) 0.05t2
and the variable p is related to the variable t by the equation p(t) 0.1t2) 10
it follows that the composite function c( p(t)) c(10 0.5(10 1 6
expresses the level of carbon monoxide in the air as a function of the variable t. b. Set c( p(t)) equal to 6.8 and solve for t to get 6 0.05t2 0.05t2 t2 t 6.8 0.8 0.8 0.05 16
16 4
discard t 4
That is, 4 years from now the level of carbon monoxide will be 6.8 parts per million.
PROBLEMS
1. f(x) 3. g(x) 5. h(t) 3x2 x 5x
1.1
2. h(t) 4. f(x) (2t x x2 (u 1)3; h( 1), h(0), h(1) ; f(2), f(0), f( 1) 1 1)3/2; g(0), g( 1), g(8)
In Problems 1 through 11, compute the indicated values of the given function. 2; f (0), f ( 2), f (1) 1 ; g( 1), g(1), g(2) x t 2 2t 4; h(2), h(0), h( 4)
6. g(u)
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7. f(t) 9. f(x)
(2t x 3 t t
1) |x
3/2
; f(1), f(5), f(13)
8. g(x) 10. h(x)
4
2|; f(1), f(2), f(3) if t 5 if 5 t if t 5
|x|; g( 2), g(0), g(2) 2x 4 if x 1 ; h(3), h(1), h(0), h( 3) x2 1 if x 1
11.
f(t)
1
5; f( 6), f( 5), f(16)
In Problems 12 through 15, determine whether the given function has the set of all real numbers as its domain. 12. f(x) x 1 x2 1 t2 1 13. g(x) 15. f(t) x 1 1 x2 t
14. h(t)
In Problems 16 through 21, determine the domain of the given function. 16. f(x) 18. f(t) x3 t t2 s2 t 4 3x2 1 2 2x 5 17. g(x) 19. 21. f(x) f(t) x2 x 2x 5 2 6
20. h(s)
t 2 9 t2
In Problems 22 through 29, nd the composite function f(g(x)). 22. f(u) 24. f(u) 26. 28. f(u) f(u) u2 4, g(x) x 1 (2u 10)2, g(x) x 5 1 , g(x) x 2 x 2 u 1 u 2, g(x) x 1 23. f(u) 25. f(u) 27. 29. f(u) f(u) f(x x2 1 x 3u2 2u 6, g(x) x 2 (u 1)3 2u2, g(x) x 1 1 , g(x) x 1 u2 u h) h 1, g(x) f(x) x2 1
In Problems 30 through 33, nd the difference quotient of f; namely, 30. f(x) 32. f(x) 2x 4x 3 x2 31. f(x) 33. f(x)
.
In Problems 34 through 37, obtain the composite functions f(g(x)) and g(f(x)), and nd all (if any) values of x such that f(g(x)) g(f(x)). 34. f(x) 36. f(x) x2 1, g(x) 1 4 , g(x) x 2 1 x x x 35. 37. f(x) f(x) x, g(x) 1 2x 3 , g(x) x 1 3x x 3 x 2
In Problems 38 through 45, nd the indicated composite function. 38. f(x 40. f(x 1) where f(x) 3) where f(x) x2 5 (2x 6)2 39. f(x 41. f(x 2) where f(x) 1) where f(x) 2x2 3x (x 1)5 1 3x2
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SECTION 1.1 Functions
11
42.
f
1 where f(x) x 2x
3x
2 x 2x 20
43. 45.
f(x 2 f (x
3x
1) where f(x) x x 1
x
44. f (x2
9) where f (x)
1) where f (x) g(h(x)). (x 1 x2
3
In Problems 46 through 51, nd functions h(x) and g(u) such that f(x) 46. f(x) 48. 50. f(x) f(x) (x
5
3x 3x x
2
12)
3
47. f(x) 49. f(x) f(x)
1)2 1 2 x
2(x
1)
3
5 4 1 (x 4)3
51.
4 2 x
52. MANUFACTURING COST Suppose the total cost in dollars of manufacturing q units of a certain commodity is given by the function C(q) q3 30q2 400q 500. a. Compute the cost of manufacturing 20 units. b. Compute the cost of manufacturing the 20th unit. 53. WORKER EFFICIENCY An efciency study of the morning shift at a certain factory indicates that an average worker who arrives on the job at 8:00 A.M. will have assembled f (x) x3 6x2 15x television sets x hours later. a. How many sets will such a worker have assembled by 10:00 A.M.? [Hint: At 10:00 A.M., x 2.] b. How many sets will such a worker assemble between 9:00 and 10:00 A.M.? 54. TEMPERATURE CHANGE Suppose that t hours past midnight, the temperature in Miami 1 2 was C(t) t 4t 10 degrees Celsius. 6 a. What was the temperature at 2:00 A.M.? b. By how much did the temperature increase or decrease between 6:00 and 9:00 P.M.? 55. POPULATION GROWTH It is estimated that t years from now, the population of a certain 6 suburban community will be P(t) 20 t 1 thousand. a. What will be the population of the community 9 years from now? b. By how much will the population increase during the 9th year? c. What happens to P(t) as t gets larger and larger? Interpret your result.
56. EXPERIMENTAL PSYCHOLOGY To study the rate at which animals learn, a psychology student performed an experiment in which a rat was sent repeatedly through a laboratory maze. Suppose that the time required for the rat to traverse the maze on the nth trial was approximately 12 T(n) 3 n minutes. a. What is the domain of the function T ? b. For what values of n does T(n) have meaning in the context of the psychology experiment? c. How long did it take the rat to traverse the maze on the 3rd trial? d. On which trial did the rat rst traverse the maze in 4 minutes or less? e. According to the function T, what will happen to the time required for the rat to traverse the maze as the number of trials increases? Will the rat ever be able to traverse the maze in less than 3 minutes? 57. BLOOD FLOW Biologists have found that the speed of blood in an artery is a function of the distance of the blood from the arterys central axis. According to Poiseuilles law,* the speed (in centimeters per second) of blood that is r centimeters from the central axis of an artery is given by the function S(r) C(R2 r2), where C is a constant and R is the radius of the artery. Suppose that for a certain artery, C 1.76 105 and R 1.2 10 2 centimeters. a. Compute the speed of the blood at the central axis of this artery.
*Edward Batschelet, Introduction to Mathematics for Life Scientists, 3rd ed., New York: Springer-Verlag, 1979, pp. 101103.
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b. Compute the speed of the blood midway between the arterys wall and central axis. 58. DISTRIBUTION COST Suppose that the number of worker-hours required to distribute new telephone books to x% of the households in a certain rural community is given by the function 600x W(x) . 300 x a. What is the domain of the function W? b. For what values of x does W(x) have a practical interpretation in this context? c. How many worker-hours were required to distribute new telephone books to the rst 50% of the households? d. How many worker-hours were required to distribute new telephone books to the entire community? e. What percentage of the households in the community had received new telephone books by the time 150 worker-hours had been expended? 59. ISLAND ECOLOGY Observations show that on an island of area A square miles, the average number of animal species is approximately equal to 3 s(A) 2.9 A. a. On average, how many animal species would you expect to nd on an island of area 8 square miles? b. If s1 is the average number of species on an island of area A and s2 is the average number of species on an island of area 2A, what is the relationship between s1 and s2? c. How big must an island be to have an average of 100 animal species? 60. POPULATION DENSITY Observations suggest that for herbivorous mammals, the number of animals N per square kilometer can be estimated by the 91.2 , where m is the average mass of formula N m0.73 the animal in kilograms. a. Assuming that the average elk on a particular reserve has mass 300 kg, approximately how many elk would you expect to nd per square kilometer in the reserve? b. Using this formula, it is estimated that there is less than one animal of a certain species per square kilometer. How large can the average animal of this species be? c. One species of large mammal has twice the average mass as a second species. If a particular reserve contains 100 animals of the larger
species, how many animals of the smaller species would you expect to nd there? 61. IMMUNIZATION Suppose that during a nationwide program to immunize the population against a certain form of inuenza, public health ofcials found that the cost of inoculating x% of 150x the population was approximately C(x) 200 x million dollars. a. What is the domain of the function C? b. For what values of x does C(x) have a practical interpretation in this context? c. What was the cost of inoculating the rst 50% of the population? d. What was the cost of inoculating the second 50% of the population? e. What percentage of the population had been inoculated by the time 37.5 million dollars had been spent? 62. POSITION OF A MOVING OBJECT A ball has been dropped from the top of a building. Its height (in feet) after t seconds is given by the function H(t) 16t2 256. a. What is the height of the ball after 2 seconds? b. How far will the ball travel during the third second? c. How tall is the building? d. When will the ball hit the ground? 63. AIR POLLUTION An environmental study of a certain suburban community suggests that the average daily level of carbon monoxide in the air will be c( p) 0.4p 1 parts per million when the population is p thousand. It is estimated that t years from now the population of the community will be p(t) 8 0.2t2 thousand. a. Express the level of carbon monoxide in the air as a function of time. b. What will the carbon monoxide level be 2 years from now? c. When will the carbon monoxide level reach 6.2 parts per million? 64. MANUFACTURING COST At a certain factory, the total cost of manufacturing q units during the daily production run is C(q) q2 q 900 dollars. On a typical workday, q(t) 25t units are manufactured during the rst t hours of a production run. a. Express the total manufacturing cost as a function of t.
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SECTION 1.1 Functions
13
b. How much will have been spent on production by the end of the third hour? c. When will the total manufacturing cost reach $11,000? In Problems 65 through 68, the demand function D(x) and the total cost function C(x) for a particular commodity are given in terms of the level of production x. In each case, nd: (a) The revenue R(x) and prot P(x). (b) All values of x for which production of the commodity is protable. 65. D(x) C(x) 66. D(x) C(x) 67. D(x) C(x) 68. D(x) C(x) 0.02x 29 1.43x2 18.3x 0.37x 47 1.38x2 15.15x 0.5x 39 1.5x2 9.2x 0.09x 51 1.32x2 11.7x 67 101.4 15.6 115.5
74. COST OF EDUCATION The accompanying Table 1.2 gives the average annual total xed costs (tuition, fees, room and board) for undergraduates by institution type in constant (ination-adjusted) 2002 dollars for the academic years 19871988 to 20022003. Dene the cost of education index (CEI) for a particular academic year to be the ratio of the total xed cost for that year to the total xed cost for the base academic year ending in 1990. For example, for 4-year public institutions in the academic year ending in 2000, the cost of education index is CEI(2000) 8,311 6,476 1.28
69. CONSUMER DEMAND An importer of Brazilian coffee estimates that local consumers will buy 4,374 approximately Q( p) kilograms of the p2 coffee per week when the price is p dollars per kilogram. It is estimated that t weeks from now the price of this coffee will be p(t) 0.04t2 0.2t 12 dollars per kilogram. a. Express the weekly consumer demand for the coffee as a function of t. b. How many kilograms of the coffee will consumers be buying from the importer 10 weeks from now? c. When will the demand for the coffee be 30.375 kilograms? 70. What is the domain of f(x) 71. What is the domain of f(x) 7x 2 4 ? x 3 2x 4 4x 2 3 ? 2x 2 x 3
a. Compute the CEI for your particular type of institution for each of the 16 academic years shown in the table. What is the average annual increase in CEI over the 16-year period for your type of institution? b. Compute the CEI for all four institution types for the academic year ending in 2003 and interpret your results. c. Write a paragraph on the cost of education index. Can it continue to rise as it has? What do you think will happen eventually? 75. VALUE OF EDUCATION The accompanying Table 1.3 gives the average income in constant (ination-adjusted) 2002 dollars by educational attainment for persons at least 18 years old for the decade 19912000. Dene the value of education index (VEI) for a particular level of education in a given year to be the ratio of average income earned in that year to the average income earned by the lowest level of education (no high school diploma) for the same year. For example, for a person with a bachelors degree in 1995, the value of education index is VEI(1995) 43,450 16,465 2.64
72. For f(x) 2 x 1 and g(x) x3 1.2, find g( f(4.8)). Use two decimal places. 73. For f(x) 2 x 1 and g(x) x3 1.2, find f(g(2.3)). Use two decimal places.
a. Compute the VEI for each year in the decade 19912000 for the level of education you hope to attain. b. Compare the VEI for the year 2000 for the four different educational levels requiring at least a high school diploma. Interpret your results.
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TABLE 1.2 Average Annual Total Fixed Costs of Education (Tuition, Fees, Room and Board) by Institutional Type in Constant (Ination-Adjusted) 2002 Dollars
Sector/Year 2-yr public 2-yr private 4-yr public 4-yr private 87-88 88-89 1,112 10,640 6,382 13,888 1,190 11,159 6,417 14,852 89-90 1,203 10,929 6,476 14,838 90-91 91-92 1,283 11,012 6,547 15,330 1,476 11,039 6,925 15,747 92-93 93-94 1,395 11,480 7,150 16,364 1,478 12,130 7,382 16,765 94-95 1,517 12,137 7,535 17,216 95-96 1,631 12,267 7,680 17,560 96-97 1,673 12,328 7,784 17,999 97-98 1,701 12,853 8,033 18,577 98-99 1,699 13,052 8,214 18,998 99-00 1,707 13,088 8,311 19,368 00-01 1,752 13,213 8,266 19,636 01-02 1,767 13,375 8,630 20,783 02-03 1,914 14,202 9,135 21,678
All data are unweighted averages, intended to reect the average prices set by institutions.
SOURCE: Annual Survey of Colleges. The College Board, New York, NY.
TABLE 1.3 Average Annual Income by Educational Attainment in Constant 2002 Dollars
Level of Education/Year 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000
No high school diploma High school diploma Some college Bachelors degree Advanced degree
16,582 24,007 27,017 41,178 60,525
16,344 23,908 26,626 41,634 62,080
15,889 24,072 26,696 43,529 69,145
16,545 24,458 26,847 44,963 67,770
16,465 25,180 28,037 43,450 66,581
17,135 25,289 28,744 43,505 69,993
17,985 25,537 29,263 45,150 70,527
17,647 25,937 30,304 48,131 69,777
17,346 26,439 30,561 49,149 72,841
18,727 27,097 31,212 51,653 72,175
SOURCE: U.S. Census Bureau website (www.census.gov/hhes/income/histinc/p28).
SECTION
1.2 The Graph of a Function
Graphs have visual impact. They also reveal information that may not be evident from verbal or algebraic descriptions. Two graphs depicting practical relationships are shown in Figure 1.3. The graph in Figure 1.3a describes the variation in total industrial production in a certain country over a 4-year period of time. Notice that the highest point on the graph occurs near the end of the third year, indicating that production was greatest at that time. The graph in Figure 1.3b represents population growth when environmental factors impose an upper bound on the possible size of the population. It indicates that the rate of population growth increases at rst and then decreases as the size of the population gets closer and closer to the upper bound.
Production Highest point
Population
Upper bound
0
1
2
3
4
Time (years)
0 Moment of most rapid growth (b)
Time
Moment of maximum production (a)
FIGURE 1.3 (a) A production function. (b) Bounded population growth.
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SECTION 1.2 The Graph of a Function 15
To represent a function y f (x) geometrically as a graph, it is common practice to use a rectangular coordinate system on which units for the independent variable x are marked on the horizontal axis and those for the dependent variable y are marked on the vertical axis.
The Graph of a Function The graph of a function f consists of all points (x, y) where x is in the domain of f and y f (x); that is, all points of the form (x, f (x)).
In Chapter 3, you will see efcient techniques involving calculus that can be used to draw accurate graphs of functions. For many functions, however, you can make a fairly good sketch by plotting a few points, as illustrated in Example 1.2.1.
y
EXAMPLE 1.2.1
Graph the function f (x)
Solution
x2.
Begin by constructing the table x
x
3 x2 9
2 4
1 1
1 2 1 4
0 0
1 2 1 4
1 1
2 4
3 9
y
FIGURE 1.4 The graph of
y x2.
Then plot the points (x, y) and connect them with the smooth curve shown in Figure 1.4. NOTE Many different curves pass through the points in Example 1.2.1. Several of these curves are shown in Figure 1.5. There is no way to guarantee that the curve we pass through the plotted points is the actual graph of f. However, in general, the more points that are plotted, the more likely the graph is to be reasonably accurate.
EXPLORE!
Store f(x) x2 into Y1 of the equation editor, using a bold graphing style. Represent g(x) x2 2 by Y2 Y1 2 and h(x) x2 3 by Y3 Y1 3. Use ZOOM decimal graphing to show how the graphs of g(x) and h(x) relate to that of f(x). Now deselect Y2 and Y3 and write Y4 Y1(X 2) and Y5 Y1(X 3). Explain how the graphs of Y1, Y4, and Y5 relate.
y
y
y
x
x
x
FIGURE 1.5 Other graphs through the points in Example 1.2.1. Example 1.2.2 illustrates how to sketch the graph of a function dened by more than one formula.
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16 CHAPTER 1 Functions, Graphs, and Limits
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EXPLORE!
Certain functions that are dened piecewise can be entered into a graphing calculator using indicator functions in sections. For example, the absolute value function, f(x) x x x if x $ 0 if x , 0
EXAMPLE 1.2.2
Graph the function 2x if 0 2 if 1 x 3 if x x x 4 1 4
f(x)
Solution
can be represented by Y1 X(X $ 0) ( X)(X , 0). Now represent the function in Example 1.2.2, using indicator functions, and graph it with an appropriate viewing window. [Hint: You will need to represent the interval, 0 , X , 1, by the boolean expression, (0 , X)(X , 1).]
When making a table of values for this function, remember to use the formula that is appropriate for each particular value of x. Using the formula f(x) 2x when 2 0 x 1, the formula f(x) when 1 x 4, and the formula f(x) 3 when x x 4, you can compile this table: 1 2 1
x f(x)
0 0
1 2
2 1
3 2 3
4 3
5 3
6 3
Now plot the corresponding points (x, f (x)) and draw the graph as in Figure 1.6. Notice that the pieces for 0 x 1 and 1 x 4 are connected to one another at (1, 2) but that the piece for x 4 is separated from the rest of the graph. [The open 1 dot at 4, indicates that the graph approaches this point but that the point is not 2 actually on the graph.]
y
3 2 1 x 1 2 3 4 5 6
0
FIGURE 1.6 The graph of f(x)
2x 2 x 3
0 1 x
x x 4
1 4
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SECTION 1.2 The Graph of a Function 17
Intercepts
EXPLORE!
Using your graphing utility, locate the x intercepts of f(x) x2 x 2. These intercepts can be located by rst using the ZOOM button and then conrmed by using the root nding feature of the graphing utility. Do the same for g(x) x2 x 4. What radical form do these roots have?
The points (if any) where a graph crosses the x axis are called x intercepts, and similarly, a y intercept is a point where the graph crosses the y axis. Intercepts are key features of a graph and can be determined using algebra or technology in conjunction with these criteria.
How to Find the x and y Intercepts
To nd any x intercept of a graph, set y 0 and solve for x. To nd any y intercept, set x 0 and solve for y. For a function f, the only y intercept is y0 f (0), but nding x intercepts may be difcult.
EXAMPLE 1.2.3
Graph the function f (x)
Solution
x2
x
2. Include all x and y intercepts. 0. Fac-
y
The y intercept is f (0) toring, we nd that
(1, 2) (2, 0) x
2. To nd the x intercepts, solve the equation f (x) x2 x 2 (x 1)(x 2) x 1, x 0 0 2
(0, 2) (1, 0) (2, 4)
factor uv 0 if and only if u v 0 0 or
(3, 4)
Thus, the x intercepts are ( 1, 0) and (2, 0). Next, make a table of values and plot the corresponding points (x, f(x)). x f(x) 3 10 2 4 1 0 0 2 1 2 2 0 3 4 4 10
(3, 10)
(4, 10)
FIGURE 1.7 The graph of
f (x) x2 x 2.
The graph of f is shown in Figure 1.7. NOTE The factoring in Example 1.2.3 is fairly straightforward, but in other problems, you may need to review the factoring procedure provided in Appendix A2.
Graphing Parabolas
The graphs in Figures 1.4 and 1.7 are called parabolas. In general, the graph of y Ax2 Bx C is a parabola as long as A 0. All parabolas have a U shape, and the parabola y Ax2 Bx C opens up if A 0 and down if A 0. The peak B or valley of the parabola is called its vertex, and it always occurs where x 2A (Figure 1.8; see also Problem 58). These features of the parabola are easily obtained by the methods of calculus developed in Chapter 3. Note that to get a reasonable sketch of the parabola y Ax2 Bx C, you need only determine three key features: B 1. The location of the vertex where x 2A 2. 3. Whether the parabola opens up (A Any intercepts 0) or down (A 0)
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For instance, in Example 1.2.3, the parabola y A
x2
x
1 is negative) and has its vertex (high point) at x
2 opens downward (since 1 1 B . 2A 2( 1) 2
y Vertex x = B 2A x x = B 2A
y
x
(a) If A
Vertex 0, the parabola opens up.
(b) If A
0, the parabola opens down.
FIGURE 1.8 The graph of the parabola y
Ax2
Bx
C.
In Chapter 3, we will develop a procedure in which the graph of a function of practical interest is rst obtained by calculus and then interpreted to obtain useful information about the function, such as its largest and smallest values. In Example 1.2.4 we preview this procedure by using what we know about the graph of a parabola to determine the maximum revenue obtained in a production process.
EXAMPLE 1.2.4
A manufacturer determines that when x hundred units of a particular commodity are produced, they can all be sold for a unit price given by the demand function p 60 x dollars. At what level of production is revenue maximized? What is the maximum revenue?
Solution
The revenue derived from producing x hundred units and selling them all at 60 x dollars is R(x) x(60 x) hundred dollars. Note that R(x) 0 only for 0 x 60. The graph of the revenue function R(x) x(60 x) x2 1 60x 0) and has its high point
is a parabola that opens downward (since A (vertex) at x B 2A 60 2( 1)
30 30 hundred units are
as shown in Figure 1.9. Thus, revenue is maximized when x produced, and the corresponding maximum revenue is R(30) 30(60 30) 900
hundred dollars. The manufacturer should produce 3,000 units and at that level of production should expect maximum revenue of $90,000.
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SECTION 1.2 The Graph of a Function 19
R(100 dollars) 900 500 R x (60 x)
0
x(100 units) 30 60
FIGURE 1.9 A revenue function. Note that we can also nd the largest value of R(x) ing the square: R(x) x2 (x2 (x Thus, R(30) 0 R(c) 900 (c 60x 60x 30)2 30)2 (x2 900)
900
x2
60x by complet1, the coefcient of x
60x) 900
900
factor out
complete the square inside parentheses by adding ( 60/2)2 900
900
900 and if c is any number other than 30, then 900 900 30.
since (c 30)2 0
so the maximum revenue is 900 when x
Intersections of Graphs
Just-In-Time Review
The quadratic formula is used in Example 1.2.5. Recall that this result says that the equation Ax2 Bx C 0 has real solutions if and only if B2 4AC $ 0, in which case, the solutions are r1 and r2 B B2 2A 4AC B B2 2A 4AC
Sometimes it is necessary to determine when two functions are equal. For instance, an economist may wish to compute the market price at which the consumer demand for a commodity will be equal to supply. Or a political analyst may wish to predict how long it will take for the popularity of a certain challenger to reach that of the incumbent. We shall examine some of these applications in Section 1.4. In geometric terms, the values of x for which two functions f (x) and g(x) are equal are the x coordinates of the points where their graphs intersect. In Figure 1.10, the graph of y f (x) intersects that of y g(x) at two points, labeled P and Q. To nd the points of intersection algebraically, set f (x) equal to g(x) and solve for x. This procedure is illustrated in Example 1.2.5.
y y = f(x)
P
0
Q
x
y = g(x)
A review of the quadratic formula may be found in Appendix A2.
FIGURE 1.10 The graphs of y
f (x) and y
g(x) intersect at P and Q.
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20 CHAPTER 1 Functions, Graphs, and Limits
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EXPLORE!
Refer to Example 1.2.5. Use your graphing utility to nd all points of intersection of the graphs of f(x) 3x 2 and g(x) x2. Also nd the roots of g(x) f(x) x2 3x 2. What can you conclude?
EXAMPLE 1.2.5 Find all points of intersection of the graphs of f (x)
Solution
3x
2 and g(x)
x2. 3x 2 0
You must solve the equation x2 3x 2. Rewrite the equation as x2 and apply the quadratic formula to obtain x The solutions are x 3 2 17 3.56 and x 3 2 17 0.56 ( 3) ( 3)2 2(1) 4(1)( 2) 3 2 17
(The computations were done on a calculator, with results rounded off to two decimal places.) Computing the corresponding y coordinates from the equation y x2, you nd that the points of intersection are approximately (3.56, 12.67) and ( 0.56, 0.31). [As a result of round-off errors, you will get slightly different values for the y coordinates if you substitute into the equation y 3x 2.] The graphs and the intersection points are shown in Figure 1.11.
y
(3.56, 12.67)
0 (0.56, 0.31)
x
FIGURE 1.11 The intersection of the graphs of f (x)
3x
2 and g(x)
x2.
Power Functions, Polynomials, and Rational Functions
A power function is a function of the form f(x) xn, where n is a real number. For 2 3 1/2 example, f(x) x , f(x) x , and f(x) x are all power functions. So are f(x) 1 3 x since they can be rewritten as f(x) x 2 and f(x) x1/3, respectively. and f(x) x2 A polynomial is a function of the form p(x) an x n an 1xn
1
...
a1x
a0
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SECTION 1.2 The Graph of a Function 21
EXPLORE!
Use your calculator to graph the third-degree polynomial f(x) x3 x2 6x 3. Conjecture the values of the x intercepts and conrm them using the root nding feature of your calculator.
where n is a nonnegative integer and a0, a1, . . . , an are constants. If an 0, the integer n is called the degree of the polynomial. For example, f(x) 3x5 6x2 7 is a polynomial of degree 5. It can be shown that the graph of a polynomial of degree n is an unbroken curve that crosses the x axis no more than n times. To illustrate some of the possibilities, the graphs of three polynomials of degree 3 are shown in Figure 1.12.
y y= x3
y y = x 3 2x 2
y
x
x
y = x 3 4x x
FIGURE 1.12 Three polynomials of degree 3. p(x) of two polynomials p(x) and q(x) is called a rational function. q(x) Such functions appear throughout this text in examples and exercises. Graphs of three rational functions are shown in Figure 1.13. You will learn how to sketch such graphs in Section 3.3 of Chapter 3. A quotient
y
y y= x x1
y
1 y= 2 x x
y= x
x x2 + 1 x
FIGURE 1.13 Graphs of three rational functions.
The Vertical Line Test
It is important to realize that not every curve is the graph of a function (Figure 1.14). For instance, suppose the circle x2 y2 5 were the graph of some function y f (x). Then, since the points (1, 2) and (1, 2) both lie on the circle, we would have f (1) 2 and f(1) 2, contrary to the requirement that a function assigns one and only one value to each number in its domain. Here is a geometric rule for determining whether a curve is the graph of a function.
The Vertical Line Test
A curve is the graph of a function if and only if no vertical line intersects the curve more than once.
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22 CHAPTER 1 Functions, Graphs, and Limits
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y
y
x (a) The graph of a function (b) Not the graph of a function
x
FIGURE 1.14 The vertical line test.
PROBLEMS
1.2
In Problems 1 and 2, classify each function as a polynomial, a power function, or a rational function. If the function is not one of these types, classify it as different. 1. a. f(x) b. f(x) c. f(x) d. f(x) x1.4 2x3 3x2 8 (3x 5)(4 x)2 3x 2 x 1 4x 7 2. a. f(x) b. f(x) c. f(x) d. f(x) 2 3x2 5x4 x 3x (x 3)(x 7) 5x 3 2x 2 3 2x 9 3 x2 3
In Problems 3 through 18, sketch the graph of the given function. Include all x and y intercepts. 3. 5. 7. 9. 11. 13. 15. 17. f (x) f (x) f (x) f (x) f (x) f(x) f(x) f(x) x x3 2x x(2x x2 x x x x2 1 4. 6. 8. 10. 12. 14. 16. 1 1 18. f (x) f (x) f (x) f (x) f (x) f (x) f(x) f(x) x2 x4 2 (x x2
1 5) 2x 15
1 if x 0 1 if x 0 x 3 if x 2x if x
3x 1)(x 2) 2x 8 x3 1 2x 1 if x 2 3 if x 2 9 x if x x 2 x 2 if x
2 2
In Problems 19 through 24, nd the points of intersection (if any) of the given pair of curves and draw the graphs. 19. y 3x 5 and y 21. y x2 and y 3x 23. 3y 2x 5 and y x 2 3x 3 9 20. y 3x 8 and y 3x 2 22. y x2 x and y x 1 24. 2x 3y 8 and 3x 5y
13
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SECTION 1.2 The Graph of a Function 23
In each of Problems 25 through 28, the graph of a function f(x) is given. In each case nd: (a) The y intercept. (b) All x intercepts. (c) The largest value of f(x) and the value(s) of x for which it occurs. (d) The smallest value of f(x) and the value(s) of x for which it occurs. 25.
y 4 2 0 4 2 2 4 2 4 x 4 2 2 4 4 2 0 2 4 x
26.
y
27.
y 4 2 0 4 2 2 4 2 4 x
28.
y 4 2 0 4 2 2 4 2 4 x
29. MANUFACTURING COST A manufacturer can produce cassette tape recorders at a cost of $40 apiece. It is estimated that if the tape recorders are sold for p dollars apiece, consumers will buy 120 p of them a month. Express the manufacturers monthly prot as a function of price, graph this function, and use the graph to estimate the optimal selling price. 30. RETAIL SALES A bookstore can obtain an atlas from the publisher at a cost of $10 per copy and estimates that if it sells the atlas for x dollars per copy, approximately 20(22 x) copies will be sold each month. Express the bookstores monthly prot from the sale of the atlas as a function of price, graph this function, and use the graph to estimate the optimal selling price.
31. CONSUMER EXPENDITURE Suppose x 200p 12,000 units of a particular commodity are sold each month when the market price is p dollars per unit. The total monthly consumer expenditure E is the total amount of money spent by consumers during each month. a. Express total monthly consumer expenditure E as a function of the unit price p, and sketch the graph of E( p). b. Discuss the economic signicance of the p intercepts of the expenditure function E( p). c. Use the graph in part (a) to determine the market price that generates the greatest total monthly consumer expenditure.
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24 CHAPTER 1 Functions, Graphs, and Limits
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32. MOTION OF A PROJECTILE If an object is thrown vertically upward from the ground with an initial speed of 160 feet per second, its height (in feet) t seconds later is given by the function H(t) 16t2 160t. a. Graph the function H(t). b. Use the graph in part (a) to determine when the object will hit the ground. c. Use the graph in part (a) to estimate how high the object will rise. 33. PROFIT Suppose that when the price of a certain commodity is p dollars per unit, then x hundred units will be purchased by consumers, where p 0.05x 38. The cost of producing x hundred units is C(x) 0.02x2 3x 574.77 hundred dollars. a. Express the prot P obtained from the sale of x hundred units as a function of x. Sketch the graph of the prot function. b. Use the prot curve found in part (a) to determine the level of production x that results in maximum prot. What unit price p corresponds to maximum prot? 34. BLOOD FLOW Recall from Problem 57, Section 1.1, that the speed of blood located r centimeters from the central axis of an artery is given by the function S(r) C(R2 r2), where C is a constant and R is the radius of the artery.* What is the domain of this function? Sketch the graph of S(r). 35. ROAD SAFETY When an automobile is being driven at v miles per hour, the average driver requires D feet of visibility to stop safely, where D 0.065v2 0.148v. Sketch the graph of D(v). 36. POSTAGE RATES Effective June 30, 2002, the cost of mailing a rst-class letter, at, or parcel was 37 cents for the rst ounce and 23 cents for each additional ounce or fraction of an ounce. Let P(w) be the postage required for mailing a parcel weighing w ounces, for 0 w 10. a. Describe P(w) as a piecewise-dened function. b. Sketch the graph of P. 37. REAL ESTATE A real estate company manages 150 apartments in Irvine, California. All the apartments can be rented at a price of $1,200 per month,
*Edward Batschelet, Introduction to Mathematics for Life Scientists, 3rd ed., New York: Springer-Verlag, 1979, pp. 101103.
but for each $100 increase in the monthly rent, there will be ve additional vacancies. a. Express the total monthly revenue R obtained from renting apartments as a function of the monthly rental price p for each unit. b. Sketch the graph of the revenue function found in part (a). c. What monthly rental price p should the company charge in order to maximize total revenue? What is the maximum revenue? 38. REAL ESTATE Suppose it costs $500 each month for the real estate company in Problem 37 to maintain and advertise each unit that is unrented. a. Express the total monthly revenue R obtained from renting apartments as a function of the monthly rental price p for each unit. b. Sketch the graph of the revenue function found in part (a). c. What monthly rental price p should the company charge in order to maximize total revenue? What is the maximum revenue? 39. AIR POLLUTION Lead emissions are a major source of air pollution. Using data gathered by the U.S. Environmental Protection Agency in the 1990s, it can be shown that the formula N(t) 35t 2 299t 3,347 estimates the total amount of lead emission N (in thousands of tons) occurring in the United States t years after the base year 1990. a. Sketch the graph of the pollution function N(t). b. Approximately how much lead emission did the formula predict for the year 1995? (The actual amount was about 3,924 thousand tons.) c. Based on this formula, when during the decade 19902000 would you expect the maximum lead emission to have occurred? d. Can this formula be used to predict the current level of lead emission? Explain. 40. ARCHITECTURE An arch over a road has a parabolic shape. It is 6 meters wide at the base and is just tall enough to allow a truck 5 meters high and 4 meters wide to pass. a. Assuming that the arch has an equation of the form y ax2 b, use the given information to nd a and b. Explain why this assumption is reasonable. b. Sketch the graph of the arch equation you found in part (a).
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SECTION 1.2 The Graph of a Function 25
In Problems 41 through 44, use the vertical line test to determine whether or not the given curve is the graph of a function. 41.
y
42.
y
b. Write an algebraic expression representing the cost y as a function of the number of days x. c. Graph the expression in part (b). 51. A company that manufactures lawn mowers has determined that a new employee can assemble N mowers per day after t days of training, where N(t) 45t2 5t2 t 8
x
x
43.
y
44.
y
x
x
45. What viewing rectangle should be used to get an adequate graph for the quadratic function f (x) 9x2 3,600x 358,200? 46. What viewing rectangle should be used to get an adequate graph for the quadratic function f (x) 4x2 2,400x 355,000? 47. a. Graph the functions y x2 and y x2 3. How are the graphs related? b. Without further computation, graph the function y x2 5. c. Suppose g(x) f (x) c, where c is a constant. How are the graphs of f and g related? Explain. 48. a. Graph the functions y x2 and y x2. How are the graphs related? b. Suppose g(x) f (x). How are the graphs of f and g related? Explain. 49. a. Graph the functions y x2 and y (x 2)2. How are the graphs related? b. Without further computation, graph the function y (x 1)2. c. Suppose g(x) f (x c), where c is a constant. How are the graphs of f and g related? Explain. 50. It costs $90 to rent a piece of equipment plus $21 for every day of use. a. Make a table showing the number of days the equipment is rented and the cost of renting for 2 days, 5 days, 7 days, and 10 days.
a. Make a table showing the numbers of mowers assembled for training periods of lengths t 2 days, 3 days, 5 days, 10 days, 50 days. b. Based on the table in part (a), what do you think happens to N(t) for very long training periods? c. Use your calculator to graph N(t). 52. Use your graphing utility to graph y x4, y x4 x, y x4 2x, and y x4 3x on the same coordinate axes, using [ 2, 2]1 by [ 2, 5]1. What effect does the added term involving x have on the shape of the graph? Repeat using y x4, y x4 x3, y x4 2x3, and y x4 3x3. Adjust the viewing rectangle appropriately. 9x 2 3x 4 . Determine the 53. Graph f(x) 4x 2 x 1 values of x for which the function is dened. 54. Graph f(x) 8x 2 9x 3 . Determine the values x2 x 1 of x for which the function is dened. 3x3 7x 4 and nd the x
55. Graph g(x) intercepts.
56. Show that the distance d between the two points (x1, y1) and (x2, y2) is given by the formula d (x 2 x 1)2 (y2 y1)2
[Hint: Apply the Pythagorean theorem to the right triangle whose hypotenuse is the line segment joining the two given points.] Then use the distance formula to nd the distance between these points: a. (5, 1) and (2, 3) b. (2, 6) and (2, 1)
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26 CHAPTER 1 Functions, Graphs, and Limits
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y
(x2 , y2) d
(x1, y1) x
57. Use the distance formula in Problem 56 to show that the circle with center (a, b) and radius R has the equation (x a)2 (y b)2 R2 58. Show that the vertex of the parabola y Ax2 Bx C (A 0) occurs at the B . point where x 2A B 2 C B2 [Hint: First verify that Ax2 Bx C A x 2A A 4A2 2 Then note that the largest or smallest value of f (x) Ax Bx C must B .] occur where x 2A
y y Vertex (high point) Vertex (low point) x B 2A B 2A
PROBLEM 56
x
PROBLEM 58
SECTION EXPLORE!
1.3 Linear Functions
In many practical situations, the rate at which one quantity changes with respect to another is constant. Here is a simple example from economics.
Input the cost function, Y1 50x {200, 300, 400}, into the equation editor, using braces to list various overhead costs. Set the WINDOW dimensions to [0, 5]1 by [ 100, 700]100 to view the effect of varying the overhead values.
EXAMPLE 1.3.1
A manufacturers total cost consists of a xed overhead of $200 plus production costs of $50 per unit. Express the total cost as a function of the number of units produced and draw the graph.
Solution
Let x denote the number of units produced and C(x) the corresponding total cost. Then, Total cost where (cost per unit)(number of units) Cost per unit Number of units Overhead 50 x 200 overhead
Hence, C(x) 50x 200
The graph of this cost function is sketched in Figure 1.15.
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SECTION 1.3 Linear Functions
27
C(x) 700 600 500 400 (2, 300) 300 200 100 x 1 2 3 4 5 (0, 200) (3, 350) C 50x 200
FIGURE 1.15 The cost function, C(x)
50x
200.
The total cost in Example 1.3.1 increases at a constant rate of $50 per unit. As a result, its graph in Figure 1.15 is a straight line that increases in height by 50 units for each 1-unit increase in x. In general, a function whose value changes at a constant rate with respect to its independent variable is said to be a linear function. This is because the graph of such a function is a straight line. In algebraic terms, a linear function is a function of the form f (x) a 1x a0
where a0 and a1 are constants. For example, the functions f(x) and f(x)
3 2x, f(x) 5x, 2 12 are all linear. Linear functions are traditionally written in the form y mx b
where m and b are constants. This standard notation will be used in the discussion that follows.
Linear Functions
A linear function is a function that changes at a constant rate with respect to its independent variable.
The graph of a linear function is a straight line. The equation of a linear function can be written in the form y where m and b are constants. mx b
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The Slope of a Line
A surveyor might say that a hill with a rise of 2 feet for every foot of run has a slope of m rise run 2 1 2
The steepness of a line can be measured by slope in much the same way. In particular, suppose (x1, y1) and (x2, y2) lie on a line as indicated in Figure 1.16. Between these points, x changes by the amount x2 x1 and y by the amount y2 y1. The slope is the ratio Slope change in y change in x y2 x2 y1 x1
It is sometimes convenient to use the symbol y instead of y2 y1 to denote the change in y. The symbol y is read delta y. Similarly, the symbol x is used to denote the change x2 x1.
The Slope of a Line
The slope of the nonvertical line passing through the points (x1, y1) and (x2, y2) is given by the formula Slope y x y2 x2 y1 x1
y (x2, y2) y2 y 1 = y (Rise) x2 x1 = x (Run) x
(x1, y1)
FIGURE 1.16 Slope
y2 x2
y1 x1
y . x
The use of this formula is illustrated in Example 1.3.2.
EXAMPLE
Solution
1.3.2
1).
Find the slope of the line joining the points ( 2, 5) and (3, y x
Slope The line is shown in Figure 1.17.
1 3
5 ( 2)
6 5
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SECTION 1.3 Linear Functions
29
y (2, 5)
y = 1 5 = 6
x (3, 1) x = 3 (2) = 5
FIGURE 1.17 The line joining ( 2, 5) and (3,
1).
The sign and magnitude of the slope of a line indicate the lines direction and steepness, respectively. The slope is positive if the height of the line increases as x increases and is negative if the height decreases as x increases. The absolute value of the slope is large if the slant of the line is severe and small if the slant of the line is gradual. The situation is illustrated in Figure 1.18.
EXPLORE!
Store the varying slope values {2, 1, 0.5, 0.5, 1, 2} into List 1, using the STAT menu and the EDIT option. Display a family of straight lines through the origin, similar to Figure 1.18, by placing Y1 L1 * X into your calculators equation editor. Graph using a ZOOM Decimal Window and TRACE the values for the different lines at X 1.
y m=2 m=1 m= 1 2
x 1 2
m= m = 1 m = 2
FIGURE 1.18 The direction and steepness of a line.
Horizontal and Vertical Lines
Horizontal and vertical lines (Figures 1.19a and 1.19b) have particularly simple equations. The y coordinates of all of the points on a horizontal line are the same. Hence, a horizontal line is the graph of a linear function of the form y b, where b is a constant. The slope of a horizontal line is zero, since changes in x produce no changes in y.
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The x coordinates of all the points on a vertical line are equal. Hence, vertical lines are characterized by equations of the form x c, where c is a constant. The slope of a vertical line is undened. This is because only the y coordinates of points change in y on a vertical line can change, and so the denominator of the quotient is change in x zero.
EXPLORE!
y
y
x=c (0, b) y=b x (c, 0) x
Determine what three slope values must be placed into List 1 so that Y1 L1*X 1 creates the screen pictured here.
(a)
(b)
FIGURE 1.19 Horizontal and vertical lines.
The Slope-Intercept Form of the Equation of a Line
y
The constants m and b in the equation y mx b of a nonvertical line have geometric interpretations. The coefcient m is the slope of the line. To see this, suppose that (x1, y1) and (x2, y2) are two points on the line y mx b. Then, y1 mx1 b and y2 mx2 b, and so Slope y2 y1 (mx2 b) (mx1 b) x2 x1 x2 x1 mx2 mx1 m(x2 x1) m x2 x1 x2 x1
m (0, b) 1 x
FIGURE 1.20 The slope
and y intercept of the line y mx b.
The constant b in the equation y mx b is the value of y corresponding to x 0. Hence, b is the height at which the line y mx b crosses the y axis, and the corresponding point (0, b) is the y intercept of the line. The situation is illustrated in Figure 1.20. Because the constants m and b in the equation y mx b correspond to the slope and y intercept, respectively, this form of the equation of a line is known as the slopeintercept form.
The Slope-Intercept Form of the Equation of a Line
equation y mx b
The
is the equation of the line whose slope is m and whose y intercept is (0, b).
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SECTION 1.3 Linear Functions
31
The slope-intercept form of the equation of a line is particularly useful when geometric information about a line (such as its slope or y intercept) is to be determined from the lines algebraic representation. Here is a typical example.
y
EXAMPLE 1.3.3
Find the slope and y intercept of the line 3y
Solution
(0, 2) (3, 0)
2x
6 and draw the graph. mx b. To do this,
First put the equation 3y solve for y to get
x
2x
6 in slope-intercept form y 2 x 3
3y It follows that the slope is
2x
6
or
y
2
FIGURE 1.21 The line
3y 2x 6.
2 and the y intercept is (0, 2). 3 To graph a linear function, plot two of its points and draw a straight line through them. In this case, you already know one point, the y intercept (0, 2). A convenient choice for the x coordinate of the second point is x 3, since the corresponding y 2 (3) 2 0. Draw a line through the points (0, 2) and (3, 0) coordinate is y 3 to obtain the graph shown in Figure 1.21.
The Point-Slope Form of the Equation of a Line
Geometric information about a line can be obtained readily from the slope-intercept formula y mx b. There is another form of the equation of a line, however, that is usually more efcient for problems in which the geometric properties of a line are known and the goal is to nd the equation of the line.
EXPLORE!
The Point-Slope Form of the Equation of a Line
y y0 m(x x0)
The equation
is an equation of the line that passes through the point (x0, y0) and that has slope equal to m.
Determine the y intercept values needed in List L1 so that the function Y1 0.5X L1 creates the screen shown here.
The point-slope form of the equation of a line is simply the formula for slope in disguise. To see this, suppose the point (x, y) lies on the line that passes through a given point (x0, y0) and that has slope m. Using the points (x, y) and (x0, y0) to compute the slope, you get y x which you can put in point-slope form y y0 m(x x0) y0 x0 m
by simply multiplying both sides by x x0. The use of the point-slope form of the equation of a line is illustrated in Examples 1.3.4 and 1.3.5.
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y
EXAMPLE 1.3.4
Find the equation of the line that passes through the point (5, 1) and whose slope is 1 equal to . 2
x (0,
3) 2
(5, 1)
Solution
Use the formula y
y0
m(x
x0) with (x0, y0) y 1 1 (x 2 5)
(5, 1) and m
1 to get 2
FIGURE 1.22 The line
y 1 x 2 3 . 2
which you can rewrite as y The graph is shown in Figure 1.22. For practice, solve the problem in Example 1.3.4 using the slope-intercept formula. Notice that the solution based on the point-slope formula is more efcient. In Chapter 2, the point-slope formula will be used extensively for nding the equation of the tangent line to the graph of a function at a given point. Example 1.3.5 illustrates how the point-slope formula can be used to nd the equation of a line through two given points. 1 x 2 3 2
y
EXAMPLE 1.3.5
Find the equation of the line that passes through the points (3,
Solution
2) and (1, 6).
10
First compute the slope
(1, 6)
m
6
( 2) 1 3
8 2
4
Then use the point-slope formula with (1, 6) as the given point (x0, y0) to get
x (3, 2)
y
6
4(x
1)
or
y
4x
10
Convince yourself that the resulting equation would have been the same if you had chosen (3, 2) to be the given point (x0, y0). The graph is shown in Figure 1.23. NOTE The general form for the equation of a line is Ax By C 0, where A, B, C are constants, with A and B not both equal to 0. If B 0, the line is vertical, and when B 0, the equation Ax By C 0 can be rewritten as y A x B C B
FIGURE 1.23 The line
y 4x 10.
Comparing this equation with the slope-intercept form y mx b, we see that the slope of the line is given by m A/B and the y intercept by b C/B. The line is horizontal (slope 0) when A 0.
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SECTION 1.3 Linear Functions
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Practical Applications
If the rate of change of one quantity with respect to a second quantity is constant, the function relating the quantities must be linear. The constant rate of change is the slope of the corresponding line. Examples 1.3.6 and 1.3.7 illustrate techniques you can use to nd the appropriate linear functions in such situations.
EXAMPLE 1.3.6
Since the beginning of the year, the price of whole wheat bread at a local discount supermarket has been rising at a constant rate of 2 cents per month. By November rst, the price had reached $1.56 per loaf. Express the price of the bread as a function of time and determine the price at the beginning of the year.
Solution
Let x denote the number of months that have elapsed since the rst of the year and y the price of a loaf of bread (in cents). Since y changes at a constant rate with respect to x, the function relating y to x must be linear, and its graph is a straight line. Since the price y increases by 2 each time x increases by 1, the slope of the line must be 2. The fact that the price was 156 cents ($1.56) on November rst, 10 months after the rst of the year, implies that the line passes through the point (10, 156). To write an equation dening y as a function of x, use the point-slope formula with to get y y0 m(x x0) m 2, x0 10, y0 156 156 2(x 10) or y 2x
y
136
The corresponding line is shown in Figure 1.24. Notice that the y intercept is (0, 136), which implies that the price of bread at the beginning of the year was $1.36 per loaf.
y (10, 156) y (0, 136) 2x 136
10 (Jan. 1) (Nov. 1)
x
FIGURE 1.24 The rising price of bread: y
2x
136.
Sometimes it is hard to tell how two quantities, x and y, in a data set are related by simply examining the data. In such cases, it may be useful to graph the data to see if the points (x, y) follow a clear pattern (say, lie along a line). Here is an example of this procedure.
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TABLE 1.4 Percentage of Civilian Unemployment 19912000
Number Percentage of Years of Year from 1991 Unemployed
EXAMPLE 1.3.7
Table 1.4 lists the percentage of the labor force that was unemployed during the decade 19912000. Plot a graph with time (years after 1991) on the x axis and percentage of unemployment on the y axis. Do the points follow a clear pattern? Based on these data, what would you expect the percentage of unemployment to be in the year 2005?
Solution
1991 1992 1993 1994 1995 1996 1997 1998 1999 2000
0 1 2 3 4 5 6 7 8 9
6.8 7.5 6.9 6.1 5.6 5.4 4.9 4.5 4.2 4.0
SOURCE: U.S. Bureau of Labor Statistics, Bulletin 2307; and Employment and Earnings, monthly.
The graph is shown in Figure 1.25. Note that except for the initial point (0, 6.8), the pattern is roughly linear. There is not enough evidence to infer that unemployment is linearly related to time, but the pattern does suggest that we may be able to get useful information by nding a line that best ts the data in some meaningful way. One such procedure, called least-squares approximation, requires the approximating line to be positioned so that the sum of squares of vertical distances from the data points to the line is minimized. The least-squares procedure, which will be developed in Section 7.4 of Chapter 7, can be carried out on your calculator. When this procedure is applied to the unemployment data in this example, it produces the best-tting line y 0.389x 7.338, as displayed in Figure 1.25. We can then use this formula to attempt a prediction of the unemployment rate in the year 2005 (when x 14): y(14) 0.389(14) 7.338 1.892
Thus, based on least-squares extrapolation from the given data, our best guess is that there will be roughly a 1.9% unemployment rate in 2005.
EXPLORE!
Place the data in Table 1.4 into L1 and L2 of the STAT data editor, where L1 is the number of years from 1991 and L2 is the percentage of unemployed. Following the Calculator Introduction for statistical graphing using the STAT and STAT PLOT keys, verify the scatterplot and best-t line displayed in Figure 1.25. 10 Percentage of unemployed 9 8
y
(1, 7.5) (2, 6.9) 7 (3, 6.1) (0, 6.8) 6 (4, 5.6) 5 4 3 2 1 0 1 2 3 4
Least-squares line y 0.389x 7.338
(5, 5.4) (6, 4.9) (7, 4.5) (9, 4) (8, 4.2) (14, 1.9)
5
6
7
8
9
10
11
12
13
14 2005
15
x
Years after 1991
FIGURE 1.25 Percentage of unemployed in the United States for 19912000.
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SECTION 1.3 Linear Functions
35
NOTE Care must be taken when making predictions by extrapolating from known data, especially when the data set is as small as the one in Example 1.3.7. In particular, the economy began to weaken after the year 2000, but the leastsquares line in Figure 1.25 predicts a steadily decreasing unemployment rate. Is this reasonable? In Problem 59, you are asked to explore this question by rst using the Internet to obtain unemployment data for years subsequent to 2000 and then comparing this new data with the values predicted by the least-squares line.
Parallel and Perpendicular Lines
In applications, it is sometimes necessary or useful to know whether two given lines are parallel or perpendicular. A vertical line is parallel only to other vertical lines and is perpendicular to any horizontal line. Cases involving nonvertical lines can be handled by these criteria.
Parallel and Perpendicular Lines
nonvertical lines L1 and L2. Then L1 and L2 are parallel if and only if m1
Let m1 and m2 be the slopes of the m2. 1 m1 .
L1 and L2 are perpendicular if and only if m2
These criteria are illustrated in Figure 1.26. Geometric proofs are outlined in Exercises 60 and 61. We close this section with an example illustrating one way the criteria can be used.
y L1 L2 L2
y L1
x (a) Parallel lines have m1 m2 (b) Perpendicular lines have m2
x 1/m1
FIGURE 1.26
EXAMPLE 1.3.8
Let L be the line 4x 3y 3. a. Find the equation of a line L1 parallel to L through P( 1, 4). b. Find the equation of a line L2 perpendicular to L through Q(2,
3).
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EXPLORE!
Write Y1 AX 2 and Y2 ( 1/A)X 5 in the equation editor of your graphing calculator. On the home screen, store different values into A, and then graph both lines using a ZOOM Square Window. What do you notice for different values of A (A 0)? Can you solve for the point of intersection in terms of the value A?
Solution
By rewriting the equation 4x see that L has slope m L
a.
3y 4 . 3
3 in the slope-intercept form y
4 x 3
1, we
Any line parallel to L must also have slope m contains P( 1, 4), so y 4 y 4 (x 3 4 x 3 1) 8 3 1 mL
4 . The required line L1 3
b.
A line perpendicular to L must have slope m line L2 contains Q(2, 3), we have y 3 y 3 (x 4 3 x 4 2) 9 2
3 . Since the required 4
The given line L and the required lines L1 and L2 are shown in Figure 1.27.
y L1 L y 4 x 3 P( 1, 4) 1 y L2 3 x 4 x y Q(2, 3) 4 x 3 8 3 9 2
FIGURE 1.27 Lines parallel and perpendicular to a given line L.
PROBLEMS
1. (2, 3) and (0, 4) 3. (2, 0) and (0, 2) 5. (2, 6) and (2, 4)
1.3
2. ( 1, 2) and (2, 5) 4. (5, 1) and ( 2, 1) 2 1 1 1 , and , 6. 3 5 7 8
In Problems 1 through 6, nd the slope (if possible) of the line that passes through the given pair of points.
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SECTION 1.3 Linear Functions
37
In Problems 7 through 10, nd the slope and intercepts of the line shown. Then nd an equation for the line. 7.
y
8.
y
x
x
9.
y
10.
y
x
x
In Problems 11 through 18, nd the slope and intercepts of the line whose equation is given and sketch the graph of the line. 11. x 3 13. y 3x 15. 3x 2y 6 x y 1 17. 2 5 In Problems 19 through 34, write an equation for the line 19. Through (2, 0) with slope 1 21. Through (5, 23. 25. 27. 29. 31. 1 2 Through (2, 5) and parallel to the x axis Through (1, 0) and (0, 1) 1 2 1 , 1 and , Through 5 3 4 Through (1, 5) and (3, 5) Through (4, 1) and parallel to the line 2x 2) with slope 12. y 5 14. y 3x 6 16. 5y 3x 4 x 3 y 1 1 18. 5 2 with the given properties. 20. Through ( 1, 2) with slope 22. Through (0, 0) with slope 5 24. Through (2, 5) and parallel to the y axis 26. Through (2, 5) and (1, 2) 28. Through ( 2, 3) and (0, 5) 30. Through (1, 5) and (1, 4) 32. Through ( 2, 3) and parallel to the line x 3y 1 , 1 and perpendicular to the line 34. Through 2 2x 5y 3 2 3
y
3
5
33. Through (3, 5) and perpendicular to the line x y 4
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35. MANUFACTURING COST A manufacturers total cost consists of a xed overhead of $5,000 plus production costs of $60 per unit. Express the total cost as a function of the number of units produced and draw the graph. 36. CAR RENTAL A certain car rental agency charges $35 per day plus 55 cents per mile. a. Express the cost of renting a car from this agency for 1 day as a function of the number of miles driven and draw the graph. b. How much does it cost to rent a car for a 1-day trip of 50 miles? c. How many miles were driven if the daily rental cost was $72? 37. COURSE REGISTRATION Students at a state college may preregister for their fall classes by mail during the summer. Those who do not preregister must register in person in September. The registrar can process 35 students per hour during the September registration period. Suppose that after 4 hours in September, a total of 360 students (including those who preregistered) have been registered. a. Express the number of students registered as a function of time and draw the graph. b. How many students were registered after 3 hours? c. How many students preregistered during the summer? 38. MEMBERSHIP FEES Membership in a swimming club costs $250 for the 12-week summer season. If a member joins after the start of the season, the fee is prorated; that is, it is reduced linearly. a. Express the membership fee as a function of the number of weeks that have elapsed by the time the membership is purchased and draw the graph. b. Compute the cost of a membership that is purchased 5 weeks after the start of the season. 39. LINEAR DEPRECIATION A doctor owns $1,500 worth of medical books which, for tax purposes, are assumed to depreciate linearly to zero over a 10-year period. That is, the value of the books decreases at a constant rate so that it is equal to zero at the end of 10 years. Express the value of the books as a function of time and draw the graph. 40. LINEAR DEPRECIATION A manufacturer buys $20,000 worth of machinery that depreciates linearly so that its trade-in value after 10 years will be $1,000. a. Express the value of the machinery as a function of its age and draw the graph. b. Compute the value of the machinery after 4 years.
41.
42.
43.
44.
c. When does the machinery become worthless? The manufacturer might not wait this long to dispose of the machinery. Discuss the issues the manufacturer may consider in deciding when to sell. WATER CONSUMPTION Since the beginning of the month, a local reservoir has been losing water at a constant rate. On the 12th of the month the reservoir held 200 million gallons of water, and on the 21st it held only 164 million gallons. a. Express the amount of water in the reservoir as a function of time and draw the graph. b. How much water was in the reservoir on the 8th of the month? PRINTING COST A publisher estimates that the cost of producing between 1,000 and 10,000 copies of a certain textbook is $50 per copy; between 10,001 and 20,000, the cost is $40 per copy; and between 20,001 and 50,000, the cost is $35 per copy. a. What function F(N) gives the total cost of producing N copies of the text for 1,000 N 50,000? b. Sketch the graph of the function F(N) you found in part (a). STOCK PRICES A certain stock had an initial public offering (IPO) price of $10 per share, and is traded 24 hours a day. Sketch the graph of the share price over a 2-year period for each of the following cases: a. The price increases steadily to $50 a share over the rst 18 months and then decreases steadily to $25 per share over the next 6 months. b. The price takes just 2 months to rise at a constant rate to $15 a share, then slowly drops to $8 over the next 9 months before steadily rising to $20. c. The price steadily rises to $60 a share during the rst year, at which time, an accounting scandal is uncovered. The price gaps down to $25 a share, then steadily decreases to $5 over the next 3 months before rising at a constant rate to close at $12 at the end of the 2-year period. AN ANCIENT FABLE In Aesops fable about the race between the tortoise and the hare, the tortoise trudges along at a slow, constant rate from start to nish. The hare starts out running steadily at a much more rapid pace, but halfway to the nish line, stops to take a nap. Finally, the hare wakes up, sees the tortoise near the nish line, desperately resumes his old rapid pace, and is nosed out by a hair. On the same coordinate plane, graph the respective
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distances of the tortoise and the hare from the starting line regarded as functions of time. 45. GROWTH OF A CHILD The average height H in centimeters of a child of age A years can be estimated by the linear function H 6.5A 50. Use this formula to answer these questions. a. What is the average height of a 7-year-old child? b. How old must a child be to have an average height of 150 cm? c. What is the average height of a newborn baby? Does this answer seem reasonable? d. What is the average height of a 20-year-old? Does this answer seem reasonable? 46. CAR POOLING To encourage motorists to form car pools, the transit authority in a certain metropolitan area has been offering a special reduced rate at toll bridges for vehicles containing four or more persons. When the program began 30 days ago, 157 vehicles qualied for the reduced rate during the morning rush hour. Since then, the number of vehicles qualifying has been increasing at a constant rate, and today 247 vehicles qualied. a. Express the number of vehicles qualifying each morning for the reduced rate as a function of time and draw the graph. b. If the trend continues, how many vehicles will qualify during the morning rush hour 14 days from now? 47. TEMPERATURE CONVERSION a. Temperature measured in degrees Fahrenheit is a linear function of temperature measured in degrees Celsius. Use the fact that 0 Celsius is equal to 32 Fahrenheit and 100 Celsius is equal to 212 Fahrenheit to write an equation for this linear function. b. Use the function you obtained in part (a) to convert 15 Celsius to Fahrenheit. c. Convert 68 Fahrenheit to Celsius. d. What temperature is the same in both the Celsius and Fahrenheit scales? 48. ENTOMOLOGY It has been observed that the number of chirps made by a cricket each minute depends on the temperature. Crickets wont chirp if the temperature is 38F or less, and observations yield the following data:
Number of chirps (C) Temperature T (F) 0 38 5 39 10 40 20 42 60 50
a. Express T as a linear function of C. b. How many chirps would you expect to hear if the temperature is 75F? If you hear 37 chirps in a 30-second period of time, what is the approximate temperature? 49. APPRECIATION OF ASSETS The value of a certain rare book doubles every 10 years. In 1900, the book was worth $100. a. How much was it worth in 1930? In 1990? What about the year 2000? b. Is the value of the book a linear function of its age? Answer this question by interpreting an appropriate graph. 50. AIR POLLUTION In certain parts of the world, the number of deaths N per week have been observed to be linearly related to the average concentration x of sulfur dioxide in the air. Suppose there are 97 deaths when x 100 mg/m3 and 110 deaths when x 500 mg/m3. a. What is the functional relationship between N and x? b. Use the function in part (a) to nd the number of deaths per week when x 300 mg/m3. What concentration of sulfur dioxide corresponds to 100 deaths per week? c. Research data on how air pollution affects the death rate in a population.* Summarize your results in a one-paragraph essay. 51. COLLEGE ADMISSIONS The average scores of incoming students at an eastern liberal arts college in the SAT mathematics examination have been declining at a constant rate in recent years. In 1995, the average SAT score was 575, while in 2000 it was 545. a. Express the average SAT score as a function of time. b. If the trend continues, what will the average SAT score of incoming students be in 2005? c. If the trend continues, when will the average SAT score be 527?
*You may nd the following articles helpful: D. W. Dockery, J. Schwartz, and J. D. Spengler, Air Pollution and Daily Mortality: Associations with Particulates and Acid Aerosols, Environ. Res., Vol. 59, 1992, pp. 362373; Y. S. Kim, Air Pollution, Climate, Socioeconomics Status and Total Mortality in the United States, Sci. Total Environ., Vol. 42, 1985, pp. 245256.
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52. NUTRITION Each ounce of Food I contains 3 g of carbohydrate and 2 g of protein, and each ounce of Food II contains 5 g of carbohydrate and 3 g of protein. Suppose x ounces of Food I are mixed with y ounces of Food II. The foods are combined to produce a blend that contains exactly 73 g of carbohydrate and 46 g of protein. a. Explain why there are 3x 5y g of carbohydrate in the blend and why we must have 3x 5y 73. Find a similar equation for protein. Sketch the graphs of both equations. b. Where do the two graphs in part (a) intersect? Interpret the signicance of this point of intersection. 53. ACCOUNTING For tax purposes, the book value of certain assets is determined by depreciating the original value of the asset linearly over a xed period of time. Suppose an asset originally worth V dollars is linearly depreciated over a period of N years, at the end of which it has a scrap value of S dollars. a. Express the book value B of the asset t years into the N-year depreciation period as a linear function of t. [Hint: Note that B V when t 0 and B S when t N.] b. Suppose a $50,000 piece of ofce equipment is depreciated linearly over a 5-year period, with a scrap value of $18,000. What is the book value of the equipment after three years? 54. ALCOHOL ABUSE CONTROL Ethyl alcohol is metabolized by the human body at a constant rate (independent of concentration). Suppose the rate is 10 milliliters per hour. a. How much time is required to eliminate the effects of a liter of beer containing 3% ethyl alcohol? b. Express the time T required to metabolize the effects of drinking ethyl alcohol as a function of the amount A of ethyl alcohol consumed. c. Discuss how the function in part (b) can be used to determine a reasonable cutoff value for the amount of ethyl alcohol A that each individual may be served at a party. 55. Graph y 144 25 13 630 and y on the x x 7 2 45 229 same set of coordinate axes using [ 10, 10]1 by [ 10, 10]1. Are the two lines parallel?
56. Graph y
63 139 346 54 x and y x on the 270 19 695 14 same set of coordinate axes using [ 10, 10]1 by [ 10, 10]1 for a starting range. Adjust the range settings until both lines are displayed. Are the two lines parallel?
57. EQUIPMENT RENTAL A rental company rents a piece of equipment for a $60.00 at fee plus an hourly fee of $5.00 per hour. a. Make a chart showing the number of hours the equipment is rented and the cost for renting the equipment for 2 hours, 5 hours, 10 hours, and t hours of time. b. Write an algebraic expression representing the cost y as a function of the number of hours t. Assume t can be measured to any decimal portion of an hour. (In other words, assume t is any nonnegative real number.) c. Graph the expression from part (b). d. Use the graph to approximate, to two decimal places, the number of hours the equipment was rented if the bill is $216.25 (before taxes). 58. ASTRONOMY The following table gives the length of year L (in earth years) of each planet in the solar system along with the mean (average) distance D of the planet from the sun, in astronomical units (1 astronomical unit is the mean distance of the earth from the sun).
Planet
Mean Distance from Sun, D
Length of Year, L
Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto
0.388 0.722 1.000 1.523 5.203 9.545 19.189 30.079 39.463
0.241 0.615 1.000 1.881 11.862 29.457 84.013 164.783 248.420
SOURCE: Kendrick Frazier, The Solar System, Alexandria, VA: Time/Life Books, 1985, p. 37.
a. Plot the point (D, L) for each planet on a coordinate plane. Do these quantities appear to be linearly related?
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L2 . Interpret D3 what you nd by expressing L as a function of D. c. What you have discovered in part (b) is one of Keplers laws, named for the German astronomer Johannes Kepler (15711630). Read an article about Kepler and describe his place in the history of science. 59. UNEMPLOYMENT RATE In the solution to Example 1.3.7, we observed that the line that best ts the data in that example in the sense of leastsquares approximation has the equation y 0.389x 7.338. Interpret the slope of this line in terms of the rate of unemployment. The data in the example stop at the year 2000. Use the Internet to nd unemployment data for the years since 2000. Does the least-squares approximating line do a good job of predicting this new data? Explain. b. For each planet, compute the ratio 60. PARALLEL LINES Show that two nonvertical lines are parallel if and only if they have the same slope. 61. PERPENDICULAR LINES Show that if a nonvertical line L1 with slope m1 is perpendicular to a
line L2 with slope m2, then m2 1/m1. [Hint: Find expressions for the slopes of the lines L1 and L2 in the accompanying gure. Then apply the Pythagorean theorem along with the distance formula from Problem 56, Section 1.2, to the right triangle OAB to obtain the required relationship between m1 and m2.]
y A
L1 (a, b)
0 (a, c) L2
x B
PROBLEM 61
SECTION
1.4 Functional Models
Practical problems in business, economics, and the physical and life sciences are often too complicated to be precisely described by simple formulas, and one of our basic goals is to develop mathematical methods for dealing with such problems. Toward this end, we shall use a procedure called mathematical modeling, which may be described in terms of four stages: Stage 1 (Formulation): Given a real-world situation (for example, the U.S. trade decit, the AIDS epidemic, global weather patterns), we make enough simplifying assumptions to allow a mathematical formulation. This may require gathering and analyzing data and using knowledge from a variety of different areas to identify key variables and establish equations relating those variables. This formulation is called a mathematical model. Stage 2 (Analysis of the Model): We use mathematical methods to analyze or solve the mathematical model. Calculus will be the primary tool of analysis in this text, but in practice, a variety of tools, such as statistics, numerical analysis, and computer science, may be brought to bear on a particular model. Stage 3 (Interpretation): After the mathematical model has been analyzed, any conclusions that may be drawn from the analysis are applied to the original realworld problem, both to gauge the accuracy of the model and to make predictions. For instance, analysis of a model of a particular business may predict that prot will be maximized by producing 200 units of a certain commodity.
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Stage 4 (Testing and Adjustment): In this nal stage, the model is tested by gathering new data to check the accuracy of any predictions inferred from the analysis. If the predictions are not conrmed by the new evidence, the assumptions of the model are adjusted and the modeling process is repeated. Referring to the business example described in stage 3, it may be found that prot begins to wane at a production level signicantly less than 200 units, which would indicate that the model requires modication.
Real-world problem Adjustments Testing Predictions Interpretation Formulation Mathematical model Analysis
FIGURE 1.28 A diagram illustrating the mathematical modeling procedure. The four stages of mathematical modeling are displayed in Figure 1.28. In a good model, the real-world problem is idealized just enough to allow mathematical analysis but not so much that the essence of the underlying situation has been compromised. For instance, if we assume that weather is strictly periodic, with rain occurring every 10 days, the resulting model would be relatively easy to analyze but would clearly be a poor representation of reality. In preceding sections, you have seen models representing quantities such as manufacturing cost, price and demand, air pollution levels, and population size, and you will encounter many more in subsequent chapters. Some of these models, especially those analyzed in the Think About It essays at the end of each chapter, are more detailed and illustrate how decisions are made about assumptions and predictions. Constructing and analyzing mathematical models is one of the most important skills you will learn in calculus, and the process begins with learning how to set up and solve word problems. Examples 1.4.1 through 1.4.8 illustrate a variety of techniques for addressing such problems.
Elimination of Variables
In Examples 1.4.1 and 1.4.2 the quantity you are seeking is expressed most naturally in terms of two variables. You will have to eliminate one of these variables before you can write the quantity as a function of a single variable.
EXAMPLE 1.4.1
The highway department is planning to build a picnic area for motorists along a major highway. It is to be rectangular with an area of 5,000 square yards and is to be fenced
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x y y
off on the three sides not adjacent to the highway. Express the number of yards of fencing required as a function of the length of the unfenced side.
Solution
Picnic area
Highway
It is natural to start by introducing two variables, say x and y, to denote the lengths of the sides of the picnic area (Figure 1.29). Expressing the number of yards F of required fencing in terms of these two variables, we get F x 2y
FIGURE 1.29 Rectangular
picnic area.
Since the goal is to express the number of yards of fencing as a function of x alone, you must nd a way to express y in terms of x. To do this, use the fact that the area is to be 5,000 square yards and write xy area Solve this equation for y y 5,000 x 5,000
and substitute the resulting expression for y into the formula for F to get F(x) x 2 5,000 x x 10,000 x
A graph of the relevant portion of this rational function is sketched in Figure 1.30. Notice that there is some length x for which the amount of required fencing is minimal. In Chapter 3, you will compute this optimal value of x using calculus.
EXPLORE!
10,000 into the x equation editor of your graphing calculator. Use the table feature (TBLSET) to explore where a minimal value of f(x) might occur. For example, alternatively try setting the initial value of x (TblStart) at zero, increasing at increments ( Tbl) of 100, 50, and 10. Finally, use what you have discovered to establish a proper viewing window to view f(x) and to answer the question as to the dimensions of the picnic area with minimal fencing. Store f(x) x
F(x) 300 290 280 270 260 250 240 230 220 210 200 Minimum length x 20 40 60 80 100 120 140 160 180 200 x 20 40 60 80 100 120 140 160 180 F(x) 520 290 227 205 200 203 211 223 236
FIGURE 1.30 The length of fencing: F(x)
x
10,000 . x
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EXAMPLE
r
1.4.2
h
A cylindrical can is to have capacity (volume) of 24 cubic inches. The cost of the material used for the top and bottom of the can is 3 cents per square inch, and the cost of the material used for the curved side is 2 cents per square inch. Express the cost of constructing the can as a function of its radius.
Solution
2r
Let r denote the radius of the circular top and bottom, h the height of the can, and C the cost (in cents) of constructing the can. Then,
h
C
cost of top
cost of bottom
cost of side
where, for each component of cost,
2r
Cost
(cost per square inch)(number of square inches) (cost per square inch)(area) r 2, and the cost per square inch of 3 r2
FIGURE 1.31 Cylindrical
can for Example 1.4.2.
The area of the circular top (or bottom) is the top (or bottom) is 3 cents. Hence, Cost of top 3 r2 and
Cost of bottom
EXPLORE!
Use the table feature (TBLSET) of your graphing calculator to discover an appropriate viewing window for graphing C(r) 6 r2 96 r
To nd the area of the curved side, imagine the top and bottom of the can removed and the side cut and spread out to form a rectangle, as shown in Figure 1.31. The height of the rectangle is the height h of the can. The length of the rectangle is the circumference 2 r of the circular top (or bottom) of the can. Hence, the area of the rectangle (or curved side) is 2 rh square inches. Since the cost of the side is 2 cents per square inch, it follows that Cost of side Putting it all together, C 3 r2 3 r2 4 rh 6 r2 4 rh 2(2 rh) 4 rh
Now employ the TRACE, ZOOM or minimum-nding methods of your calculator to determine the radius for which the cost is minimum. What are the dimensions of the can? Using the TRACE or ZOOM features of your calculator, nd the radius for which the cost is $3.00. Is there a radius for which the cost is $2.00?
Since the goal is to express the cost as a function of the radius alone, you must nd a way to express the height h in terms of r. To do this, use the fact that the volume V r2h is to be 24 . That is, set r2h equal to 24 and solve for h to get r 2h 24 or h 24 r2
Now substitute this expression for h into the formula for C: C(r) or C(r) 6 r2 6 r2 4 r 96 r 24 r2
A graph of the relevant portion of this cost function is sketched in Figure 1.32. Notice that there is some radius r for which the cost is minimal. In Chapter 3, you will learn how to nd this optimal radius using calculus.
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C(r) r 600 500 400 300 200 100 r 0.5 1 1.5 2 2.5 3 Minimum cost 0.5 1.0 1.5 2.0 2.5 3.0 C(r) 608 320 243 226 238 270
FIGURE 1.32 The cost function: C(r)
6 r2
96 . r
EXAMPLE
1.4.3
During a drought, residents of Marin County, California, were faced with a severe water shortage. To discourage excessive use of water, the county water district initiated drastic rate increases. The monthly rate for a family of four was $1.22 per 100 cubic feet of water for the rst 1,200 cubic feet, $10 per 100 cubic feet for the next 1,200 cubic feet, and $50 per 100 cubic feet thereafter. Express the monthly water bill for a family of four as a function of the amount of water used.
Solution
Let x denote the number of hundred-cubic-feet units of water used by the family during the month and C(x) the corresponding cost in dollars. If 0 x 12, the cost is simply the cost per unit times the number of units used: C(x) 1.22x
If 12 x 24, each of the rst 12 units costs $1.22, and so the total cost of these 12 units is 1.22(12) 14.64 dollars. Each of the remaining x 12 units costs $10, and hence the total cost of these units is 10(x 12) dollars. The cost of all x units is the sum C(x) 14.64 10(x 12) 10x 105.36
If x 24, the cost of the rst 12 units is 1.22(12) 14.64 dollars, the cost of the next 12 units is 10(12) 120 dollars, and that of the remaining x 24 units is 50(x 24) dollars. The cost of all x units is the sum C(x) 14.64 120 50(x 24) 50x 1,065.36
Combining these three formulas, we can express the total cost as the piecewisedened function C(x) 1.22x 10x 105.36 50x 1,065.36 if 0 x 12 if 12 x 24 if x 24
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C(x) 450 400 350 300 250 200 150 100 50 x 6 12 18 24 30 x 0 12 24 30 C(x) 0 14.64 134.64 434.64
FIGURE 1.33 The cost of water in Marin County. The graph of this function is shown in Figure 1.33. Notice that the graph consists of three line segments, each one steeper than the preceding one. What aspect of the practical situation is reected by the increasing steepness of the lines?
Proportionality
In constructing mathematical models, it is often important to consider proportionality relationships. Three of the most important kinds of proportionality are dened as follows:
Proportionality
The quantity Q is said to be: directly proportional to x if Q kx for some constant k inversely proportional to x if Q k/x for some constant k jointly proportional to x and y if Q kxy for some constant k
Here is an example from biology.
R( p)
EXAMPLE
1.4.4
When environmental factors impose an upper bound on its size, population grows at a rate that is jointly proportional to its current size and the difference between its current size and the upper bound. Express the rate of population growth as a function of the size of the population.
p
Solution
0
b
Let p denote the size of the population, R( p) the corresponding rate of population growth, and b the upper bound placed on the population by the environment. Then and so Difference between population and bound R( p) kp(b p) b p
FIGURE 1.34 The rate of
bounded population growth: R( p) kp(b p).
where k is the constant of proportionality. A graph of this factored polynomial is sketched in Figure 1.34. In Chapter 3, you will use calculus to compute the population size for which the rate of population growth is greatest.
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Modeling in Business and Economics
Business and economic models often involve issues such as pricing, cost control, and optimization of prot. In Chapter 3, we shall examine a variety of such models. Here is an example in which prot is expressed as a function of the selling price of a particular product.
EXAMPLE
1.4.5
A manufacturer can produce blank premium-quality videotapes at a cost of $2 per cassette. The cassettes have been selling for $5 apiece, and at that price, consumers have been buying 4,000 cassettes a month. The manufacturer is planning to raise the price of the cassettes and estimates that for each $1 increase in the price, 400 fewer cassettes will be sold each month. a. Express the manufacturers monthly prot as a function of the price at which the cassettes are sold. b. Sketch the graph of the prot function. What price corresponds to maximum prot? What is the maximum prot?
Solution a.
Begin by stating the desired relationship in words: Prot (number of cassettes sold)(prot per cassette)
EXPLORE!
Store the function Y1 400X2 6,800X 12,000 into the equation editor of your graphing calculator. Use the TBLSET feature to set the initial value of x at 5 in TblStart with unit (1) increment for Tbl. Then, construct an appropriate viewing window for the graph of this prot function. Now employ the TRACE, ZOOM or minimum nding method on your calculator to conrm the optimal cost and prot as depicted in Figure 1.35.
Since the goal is to express prot as a function of price, the independent variable is price and the dependent variable is prot. Let p denote the price at which each cassette will be sold and let P( p) be the corresponding monthly prot. Next, express the number of cassettes sold in terms of the variable p. You know that 4,000 cassettes are sold each month when the price is $5 and that 400 fewer will be sold each month for each $1 increase in price. Since the number of $1 increases is the difference p 5 between the and new old selling prices, you must have Number of cassettes sold 4,000 4,000 6,000 400(number of $1 increases) 400( p 5) 400p
The prot per cassette is simply the difference between the selling price p and the cost $2. Thus, Prot per cassette and the total prot is P( p) (number of cassettes sold)(prot per cassette) (6,000 400p)( p 2) 400p2 6,800p 12,000 p 2
b.
The graph of P( p) is the downward opening parabola shown in Figure 1.35. Maximum prot will occur at the value of p that corresponds to the highest point on the prot graph. This is the vertex of the parabola, which we know occurs where p B 2A (6,800) 2( 400) 8.5
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P( p) p 18,000 16,000 14,000 12,000 10,000 8,000 6,000 4,000 2,000 p 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Maximum profit 2 4 6 8 10 12 14 P(p) 0 8,800 14,400 16,800 16,000 12,000 4,800
FIGURE 1.35 The prot function P( p)
(6,000
400p)( p
2).
Thus, prot is maximized when the manufacturer charges $8.50 for each cassette, and the maximum monthly prot is Pmax P(8.5) $16,900 400(8.5)2 6,800(8.5) 12,000
Market Equilibrium
Recall from Section 1.1 that the demand function D(x) for a commodity relates the number of units x that are produced to the unit price p D(x) at which all x units are demanded (sold) in the marketplace. Similarly, the supply function S(x) gives the corresponding price p S(x) at which producers are willing to supply x units to the marketplace. Usually, as the price of a commodity increases, more units of the commodity will be supplied and fewer will be demanded. Likewise, as the production level x increases, the supply price p S(x) also increases but the demand price p D(x) decreases. This means that a typical supply curve is rising, while a typical demand curve is falling, as indicated in Figure 1.36.
p (price)
p = S(x) Equilibrium point Shortage
pe
Surplus
p = D(x) xe x (units)
FIGURE 1.36 Market equilibrium occurs when supply equals demand.
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The law of supply and demand says that in a competitive market environment, supply tends to equal demand, and when this occurs, the market is said to be in equilibrium. Thus, market equilibrium occurs precisely at the production level xe, where S(xe) D(xe). The corresponding unit price pe is called the equilibrium price; that is, pe D(xe) S(xe)
When the market is not in equilibrium, it has a shortage when demand exceeds supply [D(x) S(x)] and a surplus when supply exceeds demand [S(x) D(x)]. This terminology is illustrated in Figure 1.36 and in Example 1.4.6.
EXPLORE!
Following Example 1.4.6, store S(x) x2 14 into Y1 and D(x) 174 6x into Y2. Use a viewing window [5, 35]5 by [0, 200]50 to observe the shortage and surplus sectors. Check if your calculator can shade these sectors by a command such as SHADE (Y2, Y1). What sector is this?
EXAMPLE
1.4.6
Market research indicates that manufacturers will supply x units of a particular commodity to the marketplace when the price is p S(x) dollars per unit and that the same number of units will be demanded (bought) by consumers when the price is p D(x) dollars per unit, where the supply and demand functions are given by S(x)
a.
x2
14
and
D(x)
174
6x
At what level of production x and unit price p is market equilibrium achieved? b. Sketch the supply and demand curves, p S(x) and p D(x), on the same graph and interpret.
Solution a.
Market equilibrium occurs when S(x) x 14 x2 6x 160 (x 10)(x 16) x 10 or x
2
D(x) 174 6x 0 0 16
subtract 174 factor
6x from both sides
Since only positive values of the production level x are meaningful, we reject x 16 and conclude that equilibrium occurs when xe 10. The corresponding equilibrium price can be obtained by substituting x 10 into either the supply function or the demand function. Thus, pe
b.
D(10)
174
6(10)
114
The supply curve is a parabola and the demand curve is a line, as shown in Figure 1.37. Notice that no units are supplied to the market until the price reaches $14 per unit and that 29 units are demanded when the price is 0. For 0 x 10, there is a market shortage since the supply curve is below the demand curve. The supply curve crosses the demand curve at the equilibrium point (10, 114), and for 10 x 29, there is a market surplus.
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p (dollars) 174 p Demand 174 6x p Supply x 2 14
pe
114 Shortage
Surplus
14 xe 10 29
x (units)
FIGURE 1.37 Supply, demand, and equilibrium point for Example 1.4.6.
Break-Even Analysis
Intersections of graphs arise in business in the context of break-even analysis. In a typical situation, a manufacturer wishes to determine how many units of a certain commodity have to be sold for total revenue to equal total cost. Suppose x denotes the number of units manufactured and sold, and let C(x) and R(x) be the corresponding total cost and total revenue, respectively. A pair of cost and revenue curves is sketched in Figure 1.38.
y
Revenue: y = R(x)
Profit Cost: y = C(x) P Break-even point
Loss x
0
FIGURE 1.38 Cost and revenue curves, with a break-even point at P. Because of xed overhead costs, the total cost curve is initially higher than the total revenue curve. Hence, at low levels of production, the manufacturer suffers a loss. At higher levels of production, however, the total revenue curve is the higher one and the manufacturer realizes a prot. The point at which the two curves cross is called the break-even point, because when total revenue equals total cost, the manufacturer breaks even, experiencing neither a prot nor a loss. Here is an example.
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EXAMPLE
1.4.7
EXPLORE!
Following Example 1.4.7, put C(x) 7,500 60x into Y1 and R(x) 110x into Y2. Use the viewing window [0, 250]50 by [ 1,000, 20,000]5,000 with TRACE and ZOOM or the intersection-nding features of your graphing calculator to conrm the break-even point.
A manufacturer can sell a certain product for $110 per unit. Total cost consists of a xed overhead of $7,500 plus production costs of $ 60 per unit. a. How many units must the manufacturer sell to break even? b. What is the manufacturers prot or loss if 100 units are sold? c. How many units must be sold for the manufacturer to realize a prot of $1,250?
Solution
If x is the number of units manufactured and sold, the total revenue is given by R(x) 110x and the total cost by C(x) 7,500 60x. a. To nd the break-even point, set R(x) equal to C(x) and solve: 110x 50x x 7,500 7,500 150 60x
so that
It follows that the manufacturer will have to sell 150 units to break even (see Figure 1.39).
y
16,500 y
(150, 16,500) C(x) y R(x)
7,500
x 50 100 150
FIGURE 1.39 Revenue R(x)
b.
110x and cost C(x)
7,500
60x.
The prot P(x) is revenue minus cost. Hence, P(x) R(x) C(x) 110x (7,500 60x) 50x 7,500
The prot from the sale of 100 units is P(100) 50(100) 2,500 7,500
c.
The minus sign indicates a negative prot (that is, a loss), which was expected since 100 units is less than the break-even level of 150 units. It follows that the manufacturer will lose $2,500 if 100 units are sold. To determine the number of units that must be sold to generate a prot of $1,250, set the formula for prot P(x) equal to 1,250 and solve for x. You get
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50x
P(x) 7,500 50x x
1,250 1,250 8,750 8,750 50
175
from which you can conclude that 175 units must be sold to generate the desired prot. Example 1.4.8 illustrates how break-even analysis can be used as a tool for decision making.
EXPLORE!
Refer to Example 1.4.8. Place C1(x) 25 0.6x into Y1 and C2 (x) 30 0.5x into Y2 of the equation editor of your graphing calculator. Use the viewing window [ 25, 250]25 by [ 10, 125]50 to determine the range of mileage for which each agency gives the better deal. Would a person be better off using C1(x), C2(x), or C3(x) 23 0.55x if more than 100 miles are to be driven?
EXAMPLE 1.4.8
A certain car rental agency charges $25 plus 60 cents per mile. A second agency charges $30 plus 50 cents per mile. Which agency offers the better deal?
Solution
The answer depends on the number of miles the car is driven. For short trips, the rst agency charges less than the second, but for long trips, the second agency charges less. Break-even analysis can be used to nd the number of miles for which the two agencies charge the same. Suppose a car is to be driven x miles. Then the rst agency will charge C1(x) 25 0.60x dollars and the second will charge C2(x) 30 0.50x dollars. If you set these expressions equal to one another and solve, you get so that 25 0.60x 30 0.1x 5 or 0.50x x 50
This implies that the two agencies charge the same amount if the car is driven 50 miles. For shorter distances, the rst agency offers the better deal, and for longer distances, the second agency is better. The situation is illustrated in Figure 1.40.
y (dollars) Break-even point
y
C1 (x) y C2 (x)
30 25
Choose first agency
50
Choose second agency
x (miles)
FIGURE 1.40 Car rental costs at competing agencies.
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PROBLEMS
1.4
11. Express the surface area of the can as a function of its radius. PACKAGING A closed cylindrical can has surface area 120 square inches. Express the volume of the can as a function of its radius. PACKAGING A closed cylindrical can has a radius r and height h. a. If the surface area S of the can is a constant, express the volume V of the can in terms of S and r. b. If the volume V of the can is a constant, express the surface area S in terms of V and r. PACKAGING A cylindrical can is to hold 4 cubic inches of frozen orange juice. The cost per square inch of constructing the metal top and bottom is twice the cost per square inch of constructing the cardboard side. Express the cost of constructing the can as a function of its radius if the cost of the side is 0.02 cent per square inch. PACKAGING A cylindrical can with no top has been made from 27 square inches of metal. Express the volume of the can as a function of its radius. POPULATION GROWTH In the absence of environmental constraints, population grows at a rate proportional to its size. Express the rate of population growth as a function of the size of the population. RADIOACTIVE DECAY A sample of radium decays at a rate proportional to the amount of radium remaining. Express the rate of decay of the sample as a function of the amount remaining.
1. FENCING A farmer wishes to fence off a rectangular eld with 1,000 feet of fencing. If the long side of the eld is along a stream (and does not require fencing), express the area of the eld as a function of its width. 2. LANDSCAPING A landscaper wishes to make a rectangular ower garden that is twice as long as it is wide. Express the area of the garden as a function of its width. 3. The sum of two numbers is 18. Express the product of the numbers as a function of the smaller number. 4. The product of two numbers is 318. Express the sum of the numbers as a function of the smaller number. 5. SALES REVENUE Each unit of a certain commodity costs p 35x 15 cents when x units of the commodity are produced. If all x units are sold at this price, express the revenue derived from the sales as a function of x. 6. FENCING A city recreation department plans to build a rectangular playground 3,600 square meters in area. The playground is to be surrounded by a fence. Express the length of the fencing as a function of the length of one of the sides of the playground, draw the graph, and estimate the dimensions of the playground requiring the least amount of fencing. 7. AREA Express the area of a rectangular eld whose perimeter is 320 meters as a function of the length of one of its sides. Draw the graph and estimate the dimensions of the eld of maximum area. 8. PACKAGING A closed box with a square base is to have a volume of 1,500 cubic inches. Express its surface area as a function of the length of its base. 9. PACKAGING A closed box with a square base has a surface area of 4,000 square centimeters. Express its volume as a function of the length of its base. In Problems 10 through 14, you need to know that a cylinder of radius r and height h has volume V r2h and lateral (side) surface area S 2 rh. A circular disk of radius r has area A r2. 10. PACKAGING A soda can holds 12 uid ounces (approximately 6.89 cubic inches).
12.
13.
14.
15.
16.
17. TEMPERATURE CHANGE The rate at which the temperature of an object changes is proportional to the difference between its own temperature and the temperature of the surrounding medium. Express this rate as a function of the temperature of the object. 18. THE SPREAD OF AN EPIDEMIC The rate at which an epidemic spreads through a community is jointly proportional to the number of people who have caught the disease and the number who have not. Express this rate as a function of the number of people who have caught the disease.
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19. POLITICAL CORRUPTION The rate at which people are implicated in a government scandal is jointly proportional to the number of people already implicated and the number of people involved who have not yet been implicated. Express this rate as a function of the number of people who have been implicated. 20. PRODUCTION COST At a certain factory, setup cost is directly proportional to the number of machines used and operating cost is inversely proportional to the number of machines used. Express the total cost as a function of the number of machines used. 21. TRANSPORTATION COST A truck is hired to transport goods from a factory to a warehouse. The drivers wages are gured by the hour and so are inversely proportional to the speed at which the truck is driven. The cost of gasoline is directly proportional to the speed. Express the total cost of operating the truck as a function of the speed at which it is driven. PEDIATRIC DRUG DOSAGE Several different formulas have been proposed for determining the appropriate dose of a drug for a child in terms of the adult dosage. Suppose that A milligrams (mg) is the adult dose of a certain drug and C is the appropriate dosage for a child of age N years. Then Cowlings rule says that N 1 C A 24 while Friends rule says that C 2NA 25
25. PEDIATRIC DRUG DOSAGE As an alternative to Friends rule and Cowlings rule, pediatricians sometimes use the formula SA C 1.7 to estimate an appropriate drug dosage for a child whose surface area is S square meters, when the adult dosage of the drug is A milligrams (mg). In turn, the surface area of the childs body is estimated by the formula S 0.0072W 0.425H 0.725 where W and H are, respectively, the childs weight in kilograms (kg) and height in centimeters (cm). a. The adult dosage for a certain drug is 250 mg. How much of the drug should be given to a child who is 91 cm tall and weighs 18 kg? b. A drug is prescribed for two children, one of whom is twice as tall and twice as heavy as the other. Show that the larger child should receive approximately 2.22 times as much of the drug as the smaller child. 26. AUCTION BUYERS PREMIUM Usually, when you purchase a lot in an auction, you pay not only your winning bid price but also a buyers premium. At one auction house, the buyers premium is 17.5% of the winning bid price for purchases up to $50,000. For larger purchases, the buyers premium is 17.5% of the rst $50,000 plus 10% of the purchase price above $50,000. a. Find the total price a buyer pays (bid price plus buyers premium) at this auction house for purchases of $1,000, $25,000, and $100,000. b. Express the total purchase price of a lot at this auction house as a function of the nal (winning) bid price. Sketch the graph of this function. 27. TRANSPORTATION COST A bus company has adopted the following pricing policy for groups that wish to charter its buses. Groups containing no more than 40 people will be charged a xed amount of $2,400 (40 times $60). In groups containing between 40 and 80 people, everyone will pay $60 minus 50 cents for each person in excess of 40. The companys lowest fare of $40 per person will be offered to groups that have 80 members or more. Express the bus companys revenue as a function of the size of the group. Draw the graph.
Problems 22 through 24 require these formulas. 22. If an adult dose of ibuprofen is 300 mg, what dose does Cowlings rule suggest for an 11-year-old child? What dose does Friends rule suggest for the same child? 23. Assume an adult dose of A 300 mg, so that Cowlings rule and Friends rule become functions of the childs age N. Sketch the graphs of these two functions. 24. For what childs age is the dosage suggested by Cowlings rule the same as that predicted by Friends rule? For what ages does Cowlings rule suggest a larger dosage than Friends rule? For what ages does Friends rule suggest the larger dosage?
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28. INCOME TAX The accompanying table represents the 2004 federal income tax rate schedule for single taxpayers. a. Express an individuals income tax as a function of the taxable income x for 0 x 146,750 and draw the graph. b. The graph in part (a) should consist of four line segments. Compute the slope of each segment. What happens to these slopes as the taxable income increases? Explain the behavior of the slopes in practical terms
If the Taxable Income Is Over But Not Over The Income Tax Is Of the Amount Over
from each corner and folding up the aps to form the sides. Express the volume of the resulting box as a function of the length x of a side of the removed squares. Draw the graph and estimate the value of x for which the volume of the resulting box is greatest.
x
0 $7,150 $29,050 $70,350
$7,150 $29,050 $70,350 $146,750
10% $715 $4,000 $14,325 15% 25% 28%
0 $7,150 $29,050 $70,350
18
PROBLEM 32 33. POSTER DESIGN A rectangular poster contains 25 square centimeters of print surrounded by margins of 2 centimeters on each side and 4 centimeters on the top and bottom. Express the total area of the poster (printing plus margins) as a function of the width of the printed portion. 34. RETAIL SALES A bookstore can obtain a certain gift book from the publisher at a cost of $3 per book. The bookstore has been offering the book at the price of $15 per copy, and at this price, has been selling 200 copies a month. The bookstore is planning to lower its price to stimulate sales and estimates that for each $1 reduction in the price, 20 more books will be sold each month. Express the bookstores monthly prot from the sale of this book as a function of the selling price, draw the graph, and estimate the optimal selling price. 35. RETAIL SALES A manufacturer has been selling lamps at the price of $30 per lamp, and at this price consumers have been buying 3,000 lamps a month. The manufacturer wishes to raise the price and estimates that for each $1 increase in the price, 1,000 fewer lamps will be sold each month. The manufacturer can produce the lamps at a cost of $18 per lamp. Express the manufacturers monthly prot as a function of the price that the lamps are sold, draw the graph, and estimate the optimal selling price.
SOURCE: Bankrate.com.
29. ADMISSION FEES A local natural history museum charges admission to groups according to the following policy. Groups of fewer than 50 people are charged a rate of $3.50 per person, while groups of 50 people or more are charged a reduced rate of $3 per person. a. Express the amount a group will be charged for admission as a function of its size and draw the graph. b. How much money will a group of 49 people save in admission costs if it can recruit 1 additional member? 30. CONSTRUCTION COST A closed box with a square base is to have a volume of 250 cubic meters. The material for the top and bottom of the box costs $2 per square meter, and the material for the sides costs $1 per square meter. Express the construction cost of the box as a function of the length of its base. 31. CONSTRUCTION COST An open box with a square base is to be built for $48. The sides of the box will cost $3 per square meter, and the base will cost $4 per square meter. Express the volume of the box as a function of the length of its base. 32. CONSTRUCTION COST An open box is to be made from a square piece of cardboard, 18 inches by 18 inches, by removing a small square
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36. DISTANCE A truck is 300 miles due east of a car and is traveling west at the constant speed of 30 miles per hour. Meanwhile, the car is going north at the constant speed of 60 miles per hour. Express the distance between the car and truck as a function of time. 37. PRODUCTION COST A company has received an order from the city recreation department to manufacture 8,000 Styrofoam kickboards for its summer swimming program. The company owns several machines, each of which can produce 30 kickboards an hour. The cost of setting up the machines to produce these particular kickboards is $20 per machine. Once the machines have been set up, the operation is fully automated and can be overseen by a single production supervisor earning $19.20 per hour. Express the cost of producing the 8,000 kickboards as a function of the number of machines used, draw the graph, and estimate the number of machines the company should use to minimize cost. 38. AGRICULTURAL YIELD A Florida citrus grower estimates that if 60 orange trees are planted, the average yield per tree will be 400 oranges. The average yield will decrease by 4 oranges per tree for each additional tree planted on the same acreage. Express the growers total yield as a function of the number of additional trees planted, draw the graph, and estimate the total number of trees the grower should plant to maximize yield. 39. HARVESTING Farmers can get $3 per bushel for their potatoes on July rst, and after that, the price drops by 2 cents per bushel per day. On July rst, a farmer has 140 bushels of potatoes in the eld and estimates that the crop is increasing at the rate of 1 bushel per day. Express the farmers revenue from the sale of the potatoes as a function of the time at which the crop is harvested, draw the graph, and estimate when the farmer should harvest the potatoes to maximize revenue. MARKET EQUILIBRIUM In Problems 4043, supply and demand functions, S(x) and D(x) are given for a particular commodity in terms of the level of production x. In each case: (a) Find the value of xe for which equilibrium occurs and the corresponding equilibrium price pe. (b) Sketch the graphs of the supply and demand curves, p S(x) and p D(x), on the same graph. (c) For what values of x is there a market shortage? A market surplus?
40. 41. 42. 43.
S(x) S(x) S(x) S(x)
4x 3x x2 2x
200 and D(x) 150 and D(x) x 3 and D(x) 7.43 and D(x)
3x 480 2x 275 21 3x2 0.21x2 0.84x
50
44. SUPPLY AND DEMAND When electric blenders are sold for p dollars apiece, manufacp2 turers will supply blenders to local retailers, 10 while the local demand will be 60 p blenders. At what market price will the manufacturers supply of electric blenders be equal to the consumers demand for the blenders? How many blenders will be sold at this price? 45. SUPPLY AND DEMAND Producers will supply x units of a certain commodity to the market when the price is p S(x) dollars per unit, and consumers will demand (buy) x units when the price is p D(x) dollars per unit, where 385 S(x) 2x 15 and D(x) x 1 a. Find the equilibrium production level xe and the equilibrium price pe. b. Draw the supply and demand curves on the same graph. c. Where does the supply curve cross the y axis? Describe the economic signicance of this point. 46. SPY STORY The hero of a popular spy story has escaped from the headquarters of an international diamond smuggling ring in the tiny Mediterranean country of Azusa. Our hero, driving a stolen milk truck at 72 kilometers per hour, has a 40minute head start on his pursuers, who are chasing him in a Ferrari going 168 kilometers per hour. The distance from the smugglers headquarters to the border, and freedom, is 83.8 kilometers. Will he make it? 47. AIR TRAVEL Two jets bound for Los Angeles leave New York 30 minutes apart. The rst travels 550 miles per hour, while the second goes 650 miles per hour. At what time will the second plane pass the rst? 48. BREAK-EVEN ANALYSIS A furniture manufacturer can sell dining room tables for $70 apiece. The manufacturers total cost consists of a xed overhead of $8,000 plus production costs of $30 per table.
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49.
50.
51.
52.
a. How many tables must the manufacturer sell to break even? b. How many tables must the manufacturer sell to make a prot of $6,000? c. What will be the manufacturers prot or loss if 150 tables are sold? d. On the same set of axes, graph the manufacturers total revenue and total cost functions. Explain how the overhead can be read from the graph. PUBLISHING DECISION An author must decide between two publishers who are vying for his new book. Publisher A offers royalties of 1% of net proceeds on the rst 30,000 copies and 3.5% on all copies in excess of that gure, and expects to net $2 on each copy sold. Publisher B will pay no royalties on the rst 4,000 copies sold but will pay 2% on the net proceeds of all copies sold in excess of 4,000 copies, and expects to net $3 on each copy sold. Suppose the author expects to sell N copies. State a simple criterion based on N for deciding how to choose between the publishers. CHECKING ACCOUNT The charge for maintaining a checking account at a certain bank is $12 per month plus 10 cents for each check that is written. A competing bank charges $10 per month plus 14 cents per check. Find a criterion for deciding which bank offers the better deal. PHYSIOLOGY The pupil of the human eye is roughly circular. If the intensity of light I entering the eye is proportional to the area of the pupil, express I as a function of the radius r of the pupil. RECYCLING To raise money, a service club has been collecting used bottles that it plans to deliver to a local glass company for recycling. Since the project began 80 days ago, the club has collected 24,000 pounds of glass for which the glass company currently offers 1 cent per pound. However, since bottles are accumulating faster than they can be recycled, the company plans to reduce by 1 cent each day the price it will pay for 100 pounds of used glass. Assuming that the club can continue to collect bottles at the same rate and that transportation costs make more than one trip to the glass company infeasible, express the clubs revenue from its recycling project as a function of the number of additional days the project runs. Draw the graph and estimate when the club should conclude the project and deliver the bottles to maximize its revenue.
53. BIOCHEMISTRY In biochemistry, the rate R of an enzymatic reaction is found to be given by the equation Rm[S] R Km [S] where Km is a constant (called the Michaelis constant), Rm is the maximum possible rate, and [S] is the substrate concentration.* Rewrite this 1 equation so that y is expressed as a function R 1 of x , and sketch the graph of this function. [S] (This graph is called the Lineweaver-Burk doublereciprocal plot.) 54. SUPPLY AND DEMAND Producers will supply q units of a certain commodity to the market when the price is p S(q) dollars per unit, and consumers will demand (buy) q units when the price is p D(q) dollars per unit, where S(q) aq b and D(q) cq d
where a, b, c, and d are all constants. a. What can you say about the signs of the constants a, b, c, and d if the supply and demand curves are as shown in the accompanying gure? b. Express the equilibrium production level qe and the equilibrium price pe in terms of the coefcients a, b, c, and d. c. Use your answer in part (b) to determine what happens to the equilibrium production level qe as a increases. What happens to qe as d increases?
p (dollars)
p = S(q) p = D(q) q (units)
PROBLEM 54
*Mary K. Campbell, Biochemistry, Philadelphia: Saunders College Publishing, 1991, pp. 221226.
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55. PUBLISHING PROFIT It costs a book publisher $74,200 to prepare a book for publication (typesetting, illustrating, editing, and so on); printing and binding costs are $5.50 per book. The book is sold to bookstores for $19.50 per copy. a. Make a table showing the cost of producing 2,000, 4,000, 6,000, and 8,000 books. Use four signicant digits. b. Make a table showing the revenue from selling 2,000, 4,000, 6,000, and 8,000 books. Use four signicant digits. c. Write an algebraic expression representing the cost y as a function of the number of books x that are produced.
d. Write an algebraic expression representing the revenue y as a function of the number of books x sold. e. Graph both functions on the same coordinate axes. f. Use TRACE and ZOOM to nd where cost equals revenue. g. Use the graph to determine how many books need to be made to produce revenue of at least $85,000. How much prot is made for this number of books?
SECTION
1.5 Limits
As you will see in subsequent chapters, calculus is an enormously powerful branch of mathematics with a wide range of applications, including curve sketching, optimization of functions, analysis of rates of change, and computation of area and probability. What gives calculus its power and distinguishes it from algebra is the concept of limit, and the purpose of this section is to provide an introduction to this important concept. Our approach will be intuitive rather than formal. The ideas outlined here form the basis for a more rigorous development of the laws and procedures of calculus and lie at the heart of much of modern mathematics.
Intuitive Introduction to the Limit
Roughly speaking, the limit process involves examining the behavior of a function f (x) as x approaches a number c that may or may not be in the domain of f. Limiting behavior occurs in a variety of practical situations. For instance, absolute zero, the temperature Tc at which all molecular activity ceases, can be approached but never actually attained in practice. Similarly, economists who speak of prot under ideal conditions or engineers proling the ideal specications of a new engine are really dealing with limiting behavior. To illustrate the limit process, consider a manager who determines that when x % of her companys plant capacity is being used, the total cost of operation is C hundred thousand dollars, where C(x) 8x2 x2 636x 320 68x 960
The company has a policy of rotating maintenance in an attempt to ensure that approximately 80% of capacity is always in use. What cost should the manager expect when the plant is operating at this ideal capacity? It may seem that we can answer this question by simply evaluating C(80), but 0 attempting this evaluation results in the meaningless fraction . However, it is still 0 possible to evaluate C(x) for values of x that approach 80 from the right (x 80, when capacity is temporarily overutilized) and from the left (x 80, when capacity is underutilized). A few such calculations are summarized in the following table.
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SECTION 1.5 Limits 59
x approaches 80 from the left x C(x) 79.8 6.99782 79.99 6.99989 79.999 6.99999 80
x approaches 80 from the right 80.0001 7.000001 80.001 7.00001 80.04 7.00043
The values of C(x) displayed on the lower line of this table suggest that C(x) approaches the number 7 as x gets closer and closer to 80. Thus, it is reasonable for the manager to expect a cost of $700,000 when 80% of plant capacity is utilized. The functional behavior in this example can be described by saying C(x) has the limiting value 7 as x approaches 80 or, equivalently, by writing
x80
lim C(x)
7
More generally, the limit of f (x) as x approaches the number c can be dened informally as follows.
Limit
If f (x) gets closer and closer to a number L as x gets closer and closer to c from both sides, then L is the limit of f (x) as x approaches c. The behavior is expressed by writing
xc
lim f(x)
L
Geometrically, the limit statement lim f(x)
xc
L means that the height of the graph
y f (x) approaches L as x approaches c, as shown in Figure 1.41. This interpretation is illustrated along with the tabular approach to computing limits in Example 1.5.1.
y f(x) L f(x) xcx x
0
FIGURE 1.41 If lim f (x)
xc
L, the height of the graph of f approaches L as x approaches c.
EXAMPLE
1.5.1
x x 1 1
Use a table to estimate the limit
x1
lim
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EXPLORE!
1 , using x 1 the modied decimal viewing window [0, 4.7]1 by [ 1.1, 2.1]1. Trace values near x 1. Also construct a table of values, using an initial value of 0.97 for x with an incremental change of 0.01. Describe what you observe. Now use an initial value of 0.997 for x with an incremental change of 0.001. Specically what happens as x approaches 1 from either side? What would be the most appropriate value for f(x) at x 1 to ll up the hole in the graph? Graph f(x) x
Solution
Let f(x) x x 1 1
and compute f (x) for a succession of values of x approaching 1 from the left and from the right:
x1x x f (x) 0.99 0.50126 0.999 0.50013 0.9999 0.50001 1 1.00001 0.499999 1.0001 0.49999 1.001 0.49988
The numbers on the bottom line of the table suggest that f (x) approaches 0.5 as x approaches 1; that is, lim x x 1 1 0.5
x1
The graph of f (x) is shown in Figure 1.42. The limit computation says that the height of the graph of y f (x) approaches L 0.5 as x approaches 1. This corresponds to the hole in the graph of f (x) at (1, 0.5). We will compute this same limit using an algebraic procedure in Example 1.5.6.
y
1 L (1, 0.5) 0.5 y x x 1 x 1 1
c
FIGURE 1.42 The function f (x)
x x
1 tends toward L 1
0.5 as x approaches c
1.
It is important to remember that limits describe the behavior of a function near a particular point, not necessarily at the point itself. This is illustrated in Figure 1.43. For all three functions graphed, the limit of f (x) as x approaches 3 is equal to 4. Yet the functions behave quite differently at x 3 itself. In Figure 1.43a, f (c) is equal to the limit 4; in Figure 1.43b, f (c) is different from 4; and in Figure 1.43c, f (c) is not dened at all.
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SECTION 1.5 Limits 61
y
y
y
4
4
4
0
x3x (a)
x
0
x3x (b)
x
0
x3x (c)
x
FIGURE 1.43 Three functions for which lim f(x)
x3
4.
Figure 1.44 shows the graph of two functions that do not have a limit as x approaches 2. The limit does not exist in Figure 1.44a because f (x) tends toward 5 as x approaches 2 from the right and tends toward a different value, 3, as x approaches 2 from the left. The function in Figure 1.44b has no nite limit as x approaches 2 because the values of f (x) increase without bound as x tends toward 2 and hence tend to no nite number L. Such so-called innite limits will be discussed later in this section.
EXPLORE!
2 using the (x 2)2 window [0, 4]1 by [ 5, 40]5. Trace the graph on both sides of x 2 to view the behavior of f (x) about x 2. Also display the table value of the function with the incremental change of x set to 0.01 and the initial value x 1.97. What happens to the values of f (x) as x approaches 2? Graph f (x) y y
5 3
0
x2x (a)
x
0
x2x (b)
x
FIGURE 1.44
Two functions for which lim f(x) does not exist.
x2
Properties of Limits
Limits obey certain algebraic rules that can be used in computations. These rules, which should seem plausible on the basis of our informal denition of limit, are proved formally in more theoretical courses. They are important because they simplify the calculation of limits of algebraic functions.
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EXPLORE!
Graph the function f(x) 3 5 x#2 x.2
Algebraic Properties of Limits
xc xc
If lim f (x) and lim g(x) exist, then
xc xc xc
lim [ f(x)
g(x)] g(x)]
xc xc
lim f (x) lim f (x)
xc
lim g(x) lim g(x) for any constant k
lim [ f (x)
xc xc
using the dot graphing style and writing Y1 3(X # 2) 5(X . 2)
xc
lim [kf(x)]
k lim f(x)
xc
lim [ f(x)g(x)] f(x) xc g(x) lim
xc
[lim f(x)][lim g(x)]
xc xc xc
in the equation editor of your graphing calculator. Use your TRACE key to determine the values of y when x is near 2. Does it make a difference from which side x 2 is approached? Also evaluate f(2).
lim f(x) if lim g(x)
xc
lim g(x)
xc
0
lim [ f (x)] p
[lim f (x)] p
if [lim f (x)] p exists
xc
That is, the limit of a sum, a difference, a multiple, a product, a quotient, or a power exists and is the sum, difference, multiple, product, quotient, or power of the individual limits, as long as all expressions involved are dened. Here are two elementary limits that we will use along with the limit rules to compute limits involving more complex expressions.
Limits of Two Linear Functions
xc
For any constant k,
xc
lim k
k
and
lim x
c x as x
That is, the limit of a constant is the constant itself, and the limit of f (x) approaches c is c. In geometric terms, the limit statement lim k
xc
k says that the height of the graph
xc
of the constant function f (x)
k approaches k as x approaches c. Similarly, lim x
c
says that the height of the linear function f (x) These statements are illustrated in Figure 1.45.
y
x approaches c as x approaches c.
y
y=k
(c, k)
c
(c, c)
0
x c (a) lim k = k
xc
0 c (b) lim x = c
xc
x
FIGURE 1.45 Limits of
two linear functions.
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SECTION 1.5 Limits 63
Computation of Limits
Examples 1.5.2 through 1.5.6 illustrate how the properties of limits can be used to calculate limits of algebraic functions. In Example 1.5.2, you will see how to nd the limit of a polynomial.
EXPLORE!
Graph f ( x) x2 x 2 x 1 using the viewing window [0, 2]0.5 by [0, 5]0.5. Trace to x 1 and notice there is no corresponding y value. Create a table with an initial value of 0.5 for x, increasing in increments of 0.1. Notice that an error is displayed for x 1, conrming that f(x) is undened at x 1. What would be the appropriate y value if this gap were lled? Change the initial value of x to 0.9, and the increment size to 0.01 to get a better approximation. Finally, zoom in on the graph about x 1 to conjecture a limiting value for the function at x 1.
EXAMPLE 1.5.2
Find lim (3x3
x 1
4x
8).
Solution
Apply the properties of limits to obtain lim (3x 3 4x 8) 3 lim x 3( 1)
x 1 3 3
x 1
4 lim x
x 1
x 1
lim 8
4( 1)
8
9
In Example 1.5.3, you will see how to nd the limit of a rational function whose denominator does not approach zero.
EXAMPLE 1.5.3
Find lim
Solution
3x3 8 . x1 x 2 2) 0, you can use the quotient rule for limits to get
x1
Since lim (x
x1
3x3 8 lim x1 x 2
lim (3x3
x1
8) 2)
3lim x3
x1 x1
x1 x1
lim 8
lim (x
lim x
lim 2
3 1
8 2
5
In general, you can use the properties of limits to obtain these formulas, which can be used to evaluate many limits that occur in practical problems.
Limits of Polynomials and Rational Functions
polynomials, then
xc
If p(x) and q(x) are
lim p(x)
p(c)
and lim p(x) q(x) p(c) q(c) if q(c) 0
xc
In Example 1.5.4, the denominator of the given rational function approaches zero, while the numerator does not. When this happens, you can conclude that the limit does not exist. The absolute value of such a quotient increases without bound and hence does not approach any nite number.
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y
EXAMPLE 1.5.4
Find lim x x2 x 1 . 2
1 2 x
Solution
The quotient rule for limits does not apply in this case since the limit of the denominator is
x2
lim (x
2) 1)
0 3, which is not equal to zero, you
FIGURE 1.46 The graph of
f (x) x x 1 . 2
Since the limit of the numerator is lim (x
x2
EXPLORE!
x 1 using an x 2 enlarged decimal window [ 9.4, 9.4]1 by [ 6.2, 6.2]1. Use the TRACE key to approach x 2 from the left side and the right side. Also create a table of values, using an initial value of 1.97 for x and increasing in increments of 0.01. Describe what you observe. Graph y
can conclude that the limit of the quotient does not exist. x 1 The graph of the function f(x) in Figure 1.46 gives you a better idea of x 2 what is actually happening in this example. Note that f (x) increases without bound as x approaches 2 from the right and decreases without bound as x approaches 2 from the left.
In Example 1.5.5, the numerator and the denominator of the given rational function both approach zero. When this happens, you should try to simplify the function algebraically to nd the desired limit.
EXAMPLE
Find lim
Solution
1.5.5
1 3x 2 .
x2 x
2
x1
y
1 0 2 x
As x approaches 1, both the numerator and the denominator approach zero, and you can draw no conclusion about the size of the quotient. To proceed, observe that the given function is not dened when x 1 but that for all other values of x, you can divide the numerator and denominator by x 1 to obtain x2 x
2
1 3x 2
(x (x
1)(x 1)(x
1) 2)
x x
1 2
x
1
(1, 2)
FIGURE 1.47 The graph of
f(x) x2 x
2
(Since x 1, you are not dividing by zero.) Now take the limit as x approaches (but is not equal to) 1 to get lim x2 x
2
1 3x 2
.
1 3x 2 x2
x1
lim (x lim (x 1
1) 2)
x1
x1
2 1
2
The graph of the function f(x)
is shown in Figure 1.47. Note that it x 3x 2 is like the graph in Figure 1.46 with a hole at the point (1, 2).
2
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1-65
SECTION 1.5 Limits 65
In general, when both the numerator and denominator of a quotient approach zero as x approaches c, your strategy will be to simplify the quotient algebraically (as in Example 1.5.5 by canceling x 1). In most cases, the simplied form of the quotient will be valid for all values of x except x c. Since you are interested in the behavior of the quotient near x c and not at x c, you may use the simplied form of the quotient to calculate the limit. In Example 1.5.6, we use this technique to obtain the limit we estimated using a table in Example 1.5.1.
EXAMPLE
Find lim x x
x1
1.5.6
1 1 .
Just-In-Time Review
Recall that (a b)(a b) a2 b2
Solution
In Example 1.5.6, we use this identity with a x and b 1.
Both the numerator and denominator approach 0 as x approaches 1. To simplify the quotient, we rationalize the numerator (that is, multiply numerator and denominator by x 1) to get x x 1 1 ( x 1)( x 1) (x 1)( x 1) (x x 1 1)( x 1) 1 x 1 x 1
and then take the limit to obtain
x1
lim
x x 1
1
x1
lim
1 x 1
1 2
Limits Involving Innity
Long-term behavior is often a matter of interest in business and economics or the physical and life sciences. For example, a biologist may wish to know the population of a bacterial colony or a population of fruit ies after an indenite period of time, or a business manager may wish to know how the average cost of producing a particular commodity is affected as the level of production increases indenitely. In mathematics, the innity symbol is used to represent either unbounded growth or the result of such growth. Here are denitions of limits involving innity we will use to study long-term behavior. If the values of the function f(x) approach the number L as x increases without bound, we write
x
Limits at Innity
lim f(x)
L
Similarly, we write
x
lim f(x)
M
when the functional values f(x) approach the number M as x decreases without bound. Geometrically, the limit statement lim f(x)
x
L means that as x increases
without bound, the graph of f (x) approaches the horizontal line y L, while lim f(x) M means that the graph of f (x) approaches the line y M as x decreases
x
without bound. The lines y L and y M that appear in this context are called horizontal asymptotes of the graph of f (x). There are many different ways for a graph
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to have horizontal asymptotes, one of which is shown in Figure 1.48. We will have more to say about asymptotes in Chapter 3 as part of a general discussion of graphing with calculus.
y y L L
y
f (x)
x
lim f (x)
L
x
x
lim f (x)
M
M
y
M
FIGURE 1.48
A graph illustrating limits at innity and horizontal asymptotes.
The algebraic properties of limits listed earlier in this section also apply to limits at innity. In addition, since any reciprocal power 1/xk for k 0 becomes smaller and smaller in absolute value as x either increases or decreases without bound, we have these useful rules:
Reciprocal Power Rules
defered for all x, then
x
If A and k are constants with k A xk
0 and x k is
lim
A xk
0
and
x
lim
0
The use of these rules is illustrated in Example 1.5.7.
EXAMPLE
Find lim
x
1.5.7
x2 x 2x2
1
Solution
To get a feeling for what happens with this limit, we evaluate the function f(x) at x 1 x2 x 2x 2
100, 1,000, 10,000, and 100,000 and display the results in the table: x x f (x) 100 0.49749 1,000 0.49975 10,000 0.49997 100,000 0.49999
The functional values on the bottom line in the table suggest that f (x) tends toward 0.5 as x grows larger and larger. To conrm this observation analytically, we divide each term in f (x) by the highest power that appears in the denominator 1 x 2x2; namely,
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1-67
SECTION 1.5 Limits 67
EXPLORE!
x2 Graph f (x) 1 x 2x 2 using the viewing window [ 20, 20]5 by [0, 1]1. Now TRACE the graph to the right for large values of x, past x 30, 40, and beyond. What do you notice about the corresponding y values and the behavior of the graph? What would you conjecture as the value of f(x) as x ?
by x2. This enables us to nd lim f(x) by applying reciprocal power rules as follows:
x
x
lim
1
x x
2
2x 2
x
lim
1/x 2
2
x 2/x 2 x/x 2 2x 2/x 2
x
lim 1 lim 1/x
x
x
lim 1/x 1 0
x
lim 2
several algebraic properties of limits reciprocal power rule
0
2
0.5
The graph of f (x) is shown in Figure 1.49. For practice, verify that lim f(x)
x
0.5 also.
y y
1/2
0
x
FIGURE 1.49 The graph of f (x)
1
x2 . x 2x 2
Here is a general description of the procedure for evaluating a limit of a rational function at innity.
Procedure for Evaluating a Limit at Innity of f(x)
k
p(x) q(x)
Step 1. Divide each term in f (x) by the highest power x that appears in the denominator polynomial q(x). Step 2. Compute lim f(x) or lim f(x) using algebraic properties of limits and
x x
the reciprocal power rules.
EXAMPLE 1.5.8
Find lim
x
2x2 3x2
3x 5x
1 . 2
Solution
The highest power in the denominator is x2. Divide the numerator and denominator by x2 to get
x
lim
2x2 3x2
3x 5x
1 2
x
lim
2 3
3/x 5/x
1/x2 2/x2
2 3
0 0
0 0
2 3
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EXAMPLE
1.5.9
If a crop is planted in soil where the nitrogen level is N, then the crop yield Y can be modeled by the Michaelis-Menten function Y(N) AN B N N 0
where A and B are positive constants. What happens to crop yield as the nitrogen level is increased indenitely?
Solution
We wish to compute lim Y(N) N AN/N lim N B/N N/N A A lim N B/N 1 0 1 A
N
N
lim
AN B
Thus, the crop yield tends toward the constant value A as the nitrogen level N increases indenitely. For this reason, A is called the maximum attainable yield.
Innite Limits We say that lim f(x) is an innite limit if f (x) increases or decreases
xc
without bound as xc. Technically, such a limit does not exist, but more information can be given about the behavior of the function by writing
xc
lim f(x)
if f(x) increases without bound as xc or
xc
lim f(x)
if f(x) decreases without bound as xc. This notation is illustrated in Example 1.5.10 for the case where x .
EXAMPLE
Find lim
x
1.5.10
x
3
x
2x 3
1
.
Solution
The highest power in the denominator is x. Divide numerator and denominator by x to get
x
lim
x3
2x x 3
1
x
lim
x2 1
2 1/x 3/x
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1-69
SECTION 1.5 Limits 69
Since lim x2 2 1 x and lim 1 3 x 1
x
x
it follows that lim x3 x 2x 3 1
x
PROBLEMS
1.
y
1.5
xa
In Problems 1 through 6, nd lim f(x) if it exists. 2.
y
3.
y
c b b b
a
x
a
x
a
x
4.
y
5.
y
6.
y
c b b b
a
x
a
x
a
x
In Problems 7 through 26, nd the indicated limit if it exists. 7. lim (3x 2
x2 x0 x3
5x 6x4 1)2(x
2) 7) 1)
8. 10. 12.
x 1
lim (x 3 lim (1
2x 2 5x3) 1)(1
x
3)
9. lim (x5 11. lim (x
x 1/2 x 1
lim (x2
2x)2
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70 CHAPTER 1 Functions, Graphs, and Limits
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13.
x1/3
lim
x x
1 2
x 5 x2 17. lim x1 x x2 19. lim 15. lim
x5
3 x 1 1 3x 10 x5 x 5 (x 1)(x 4) 21. lim x4 (x 1)(x 4) 2 x x 6 23. lim 2 x 2 x 3x 2 x 2 25. lim x4 x 4 For Problems 27 through 36, nd lim x or . 27. f (x) 29. f (x) 31. 33. 35. f(x) f(x) f(x) x3 4x2 (1 2x)(x x 2 2x 2x 2 5x 2x 1 2 3x 2x 2 3x 6x 2x 9 4 5) 3 1 7 2 f(x) and lim
x
2x 3 x 1 2x 3 16. lim x3 x 3 9 x2 18. lim x3 x 3 x2 x 6 20. lim x2 x 2 x(x2 1) 22. lim x0 x2 x2 4x 5 24. lim x1 x2 1 x 3 26. lim x9 x 9 14. lim
x1
f(x). If the limiting value is innite, indicate whether it is 28. f (x) 30. f (x) 32. 34. 36. f(x) f(x) f(x) 1 x 2x2 3x3 (1 x2)3 1 3x 3 2x 3 6x 2 x2 x 5 1 2x x 3 1 2x 3 x 1
x x
In Problems 37 and 38, the graph of a function f(x) is given. Use the graph to determine lim f(x) and lim f(x). 37.
y 1 0 x 1
38.
y 2 0 2 3 3 x
In Problems 39 through 42, complete the table by evaluating f(x) at the specied values of x. Then use the table to estimate the indicated limit or show it does not exist. 39.
x f(x)
f(x)
1.9
x2
1.99
x;
x2
lim f(x)
2 2.001 2.01 2.1
40.
x f(x)
f(x)
x
1 ; x
x0
lim f(x)
0 0.0009 0.009 0.09
1.999
0.09
0.009
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SECTION 1.5 Limits 71
41.
x f(x)
f(x)
0.9
x3 x
0.99
1 ; lim f(x) 1 x1
0.999 1 1.001 1.01 1.1
42.
x f(x)
f(x)
1.1
x3 x
1 ; lim f(x) 1 x 1
1.001 1 0.999 0.99 0.9
1.01
In Problems 43 through 50, nd the indicated limit or show that it does not exist using the following facts about limits involving the functions f(x) and g(x):
xc
lim f(x) lim g(x)
5 2 3g(x)]
and and
x
lim f(x) lim g(x) 4
3
xc
x
43. lim [2 f (x)
xc
44. lim f (x) g(x)
xc
f(x) 45. lim xc g(x) 47. lim
x
46. lim
2 f(x) xc 5g(x)
x
g(x) 2f(x)
g(x)
48. lim
2f(x) g(x) x f(x)
Growth rate (generations/hr)
49. A wire is stretched horizontally, as shown in the accompanying gure. An experiment is conducted in which different weights are attached at the center and the corresponding vertical displacements are measured. When too much weight is added, the wire snaps. Based on the data in the following table, what do you think is the maximum possible displacement for this kind of wire?
Weight W (lb) 15 16 17 18 17.5 17.9 17.99
51. PER CAPITA EARNINGS Studies indicate that t years from now, the population of a certain country will be p 0.2t 1,500 thousand people, and that the gross earnings of the country will be E million dollars, where E(t) 9t2 0.5t 179 a. Express the per capita earnings of the country P E/p as a function of time t. (Take care with the units.) b. What happens to the per capita earnings in the long run (as t )? 52. BACTERIAL GROWTH The accompanying graph shows how the growth rate R(T) of a bacterial colony changes with temperature T.*
R 1.5 1.0 0.5 0 T
0
Displacement 1.7 1.75 1.78 Snaps 1.79 1.795 Snaps y (in.)
10 20 30 40 Temperature (C)
50
PROBLEM 52
y W
PROBLEM 49 50. PRODUCTION A business manager determines that t months after production begins on a new product, the number of units produced will be P thousand, where 6t2 5t P(t) (t 1)2 What happens to production in the long run (as t )?
a. Over what range of values of T does the growth rate R(T) double? b. What can be said about the growth rate for 25 T 45? c. What happens when the temperature reaches roughly 45C? Does it make sense to compute lim R(T)?
T50
d. Write a paragraph describing how temperature affects the growth rate of a species.
*Source: Michael D. La Grega, Phillip L. Buckingham, and Jeffrey C. Evans, Hazardous Waste Management. New York: McGraw-Hill, 1994, pp. 565566. Reprinted by permission.
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53. ANIMAL BEHAVIOR In some animal species, the intake of food is affected by the amount of vigilance maintained by the animal while feeding. In essence, it is hard to eat hardy while watching for predators that may eat you. In one model,* if the animal is foraging on plants that offer a bite of size S, the intake rate of food, I(S), is given by a function of the form I(S) aS S c
56. Solve Problems 17 through 26 by using the TRACE feature of your calculator to make a table of x and f (x) values near the number x is approaching. 57. The accompanying graph represents a function f (x) that oscillates between 1 and 1 more and more frequently as x approaches 0 from either the right or the left. Does lim f (x) exist? If so, what is its value? x0 [Note: For students with experience in trigonometry, 1 the function f(x) sin behaves in this way.] x
where a and c are positive constants. a. What happens to the intake I(S) as bite size S increases indenitely? Interpret your result. b. Read an article on various ways that the food intake rate may be affected by scanning for predators. Then write a paragraph on how mathematical models may be used to study such behavior in zoology. The reference cited in this problem offers a good starting point. 54. EXPERIMENTAL PSYCHOLOGY To study the rate at which animals learn, a psychology student performed an experiment in which a rat was sent repeatedly through a laboratory maze. Suppose the time required for the rat to traverse the maze on the nth trial was approximately 5n 17 T(n) n minutes. What happens to the time of traverse as the number of trials n increases indenitely? Interpret your result. 55. AVERAGE COST A business manager determines that the total cost of producing x units of a particular commodity may be modeled by the function C(x) 7.5x 120,000 C(x) (dollars). The average cost is A(x) . Find x lim A(x) and interpret your result.
x
y 1 x 1
PROBLEM 57
58. If $1,000 is invested at 5% compounded n times per year, the balance after 1 year will be 1 1,000(1 0.05x)1/x, where x is the length of n the compounding period. For example, if n 4 the 1 compounding period is year long. For what is 4 called continuous compounding of interest, the balance after 1 year is given by the limit B
x0
lim 1,000(1
0.05x)1/x
Estimate the value of this limit by lling in the second line of the following table:
x 1,000(1 0.05x)1/x 1 0.1 0.01 0.001 0.0001
*A. W. Willius and C. Fitzgibbon, Costs of Vigilance in Foraging Ungulates, Animal Behavior, Vol. 47, Pt. 2 (Feb. 1994).
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SECTION 1.6 One-Sided Limits and Continuity 73
59. Evaluate the limit lim a nx n bmx m a n 1x n bm 1x m
1 1
x
a 1x b1x
a0 b0
for constants a0, a1, . . . , a n and b0, b1, . . . , bm in each of the following cases:
a. n m b. n m c. n m [Note: There are two possible answers, depending on the signs of an and bm.]
SECTION
1.6 One-Sided Limits and Continuity
The dictionary denes continuity as an unbroken or uninterrupted succession. Continuous behavior is certainly an important part of our lives. For instance, the growth of a tree is continuous, as are the motion of a rocket and the volume of water owing into a bathtub. In this section, we shall discuss what it means for a function to be continuous, and shall examine a few important properties of such functions.
One-Sided Limits
Informally, a continuous function is one whose graph can be drawn without the pen leaving the paper (Figure 1.50a). Not all functions have this property, but those that do play a special role in calculus. A function is not continuous where its graph has a hole or gap (Figure 1.50b), but what do we really mean by holes and gaps in a graph? To describe such features mathematically, we require the concept of a onesided limit of a function; that is, a limit in which the approach is either from the right or from the left, rather than from both sides as required for the two-sided limit introduced in Section 1.5.
y hole
y gap
x (a) A continuous graph
a
b
x
(b) A graph with holes or gaps is not continuous
FIGURE 1.50 For instance, Figure 1.51 shows the graph of inventory I as a function of time t for a company that immediately restocks to level L1 whenever the inventory falls to a certain minimum level L2 (this is called just-in-time inventory). Suppose the rst restocking time occurs at t t1. Then as t tends toward t1 from the left, the limiting value of I(t) is L2, while if the approach is from the right, the limiting value is L1.
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74 CHAPTER 1 Functions, Graphs, and Limits
1-74
I (units in inventory) L1
L2 t t1 t t2 t3 t
FIGURE 1.51 One-sided limits in a just-in-time inventory example. Here is the notation we will use to describe one-sided limiting behavior.
One-Sided Limits
(x
xc
c), we write lim f(x)
If f (x) approaches L as x tends toward c from the left L. Likewise, if f (x) approaches M as x tends toward
xc
c from the right (c
x), then lim f(x)
M.
If this notation is used in our inventory example, we would write
tt 1
lim I(t)
L2
and
tt 1
lim I(t)
L1
Here are two more examples involving one-sided limits.
y
EXAMPLE
1.6.1
1 x2 2x 1 if 0 if x x 2 2
For the function
5
f(x)
0 3 x
x2
evaluate the one-sided limits lim f(x) and lim f(x).
x2
1
2
Solution
The graph of f(x) is shown in Figure 1.52. Since f(x) FIGURE 1.52 The graph of
f (x) 1 x2 2x 1 if 0 if x x 2 2
x2
1 3
x2 for 0
x
2, we have
lim f(x) 2, so
x2
lim (1
x )
2
Similarly, f(x)
2x
1 if x
x2
lim f(x)
EXPLORE!
Refer to Example 1.6.2. Graph x 2 using the window f(x) x 4 [0, 9.4]1 by [ 4, 4]1 to verify the limit results as x approaches 4 from the left and the right. Now trace f(x) for large positive or negative values of x. What do you observe?
x2
lim (2x
1)
5
EXAMPLE
Find lim
Solution
1.6.2
2 as x approaches 4 from the left and from the right. 4 x 4 the quantity f(x) x x 2 4
x x
First, note that for 2
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SECTION 1.6 One-Sided Limits and Continuity 75
is negative, so as x approaches 4 from the left, f(x) decreases without bound. We denote this fact by writing
x4
lim
x x
2 4 4), f(x) increases without bound
Likewise, as x approaches 4 from the right (with x and we write
x4
lim
x x
2 4
The graph of f is shown in Figure 1.53.
y
x
4
x4
lim f (x)
y
1
1 4 x
x4
lim f (x)
FIGURE 1.53 The graph of f (x)
x x
2 . 4
Notice that the two-sided limit lim f(x) does not exist for the function in Examx4
ple 1.6.2 since the functional values f (x) do not approach a single value L as x tends toward 4 from each side. In general, we have the following useful criterion for the existence of a limit.
Existence of a Limit
The two-sided limit lim f(x) exists if and only if
xc xc
the two one-sided limits lim f(x) and lim f(x) both exist and are equal, and then
xc
EXPLORE!
Re-create the piecewise linear function f(x) dened in the Explore! Box on page 62. Verify graphically that lim f(x) 3 and lim f(x) 5.
x2 x2
xc
lim f(x)
xc
lim f(x)
xc
lim f(x)
EXAMPLE 1.6.3
Determine whether lim f(x) exists, where
x1
f(x)
x x
2
1 4x 1
if x if x
1 1
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76 CHAPTER 1 Functions, Graphs, and Limits Solution
1-76
Computing the one-sided limits at x
x1
1, we nd 1 2 since f (x)
x 1 when x 1
lim f(x)
x1
lim (x
1)
(1)
and
x1
lim f(x)
x1
lim ( x 2 (1)2 4(1)
4x 1
1) 2
since f (x)
x2
4x
1 when x
1
Since the two one-sided limits are equal, it follows that the two-sided limit of f (x) at x 1 exists, and we have
x1
lim f (x)
x1
lim f (x)
x1
lim f (x)
2
The graph of f (x) is shown in Figure 1.54.
y y 2 x2 4x 1
x y x 1 1
FIGURE 1.54 The graph of f (x)
x x2
1 4x 1
if x if x
1 . 1
Continuity
At the beginning of this section, we observed that a continuous function is one whose graph has no holes or gaps. A hole at x c can arise in several ways, three of which are shown in Figure 1.55.
y
y
y
c (a) f (c) is not defined
x (b) lim f (x)
xc
c f (c)
x (c) lim f (x)
xc
x
xc
c
x
lim f (x)
FIGURE 1.55 Three ways the graph of a function can have a hole at x The graph of f (x) will have a gap at x
xc
c.
c if the one-sided limits lim f(x) and
xc
lim f(x) are not equal. Three ways this can happen are shown in Figure 1.56.
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SECTION 1.6 One-Sided Limits and Continuity 77
y
y
y
c
x
x
c
x
x
c
x
(a) A finite gap: lim f (x) lim f (x)
xc xc
(b) An infinite gap: lim f (x) is finite
xc
(c) An infinite gap: lim f (x)
xc
but lim f (x)
xc
and lim f (x)
xc
FIGURE 1.56 Three ways for the graph of a function to have a gap at x
c.
So what properties will guarantee that f (x) does not have a hole or gap at x c? The answer is surprisingly simple. The function must be dened at x c, it must have a nite, two-sided limit at x c, and lim f(x) must equal f (c). To summarize:
xc
Continuity
A function f is continuous at c if all three of these conditions
are satised: a. f(c) is dened b. lim f(x) exists
xc xc
c. lim f(x)
f(c)
If f(x) is not continuous at c, it is said to have a discontinuity there.
Continuity of Polynomials and Rational Functions
Recall that if p(x) and q(x) are polynomials, then
xc
lim p(x)
p(c) 0
and
xc
lim
p(x) q(x)
p(c) if q(c) q(c)
These limit formulas can be interpreted as saying that a polynomial or a rational function is continuous wherever it is dened. This is illustrated in Examples 1.6.4 through 1.6.7.
EXAMPLE 1.6.4
Show that the polynomial p(x)
Solution
3x3
x
5 is continuous at x
1.
Verify that the three criteria for continuity are satised. Clearly p(1) is dened; in fact, p(1) 7. Moreover, lim p(x) exists and lim p(x) 7. Thus,
x1 x1 x1
lim p(x)
7 1.
p(1)
as required for p(x) to be continuous at x
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78 CHAPTER 1 Functions, Graphs, and Limits
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EXPLORE!
x 1 using the x 2 enlarged decimal window [ 9.4, 9.4]1 by [ 6.2, 6.2]1. Is the function continuous? Is it continuous at x 2? How about at x 3? Also examine this function using a table with an initial value of x at 1.8, increasing in increments of 0.2. Graph f (x)
EXAMPLE 1.6.5
Show that the rational function f(x)
Solution
x x
1 is continuous at x 2
3.
Note that f(3)
3 3
1 2
4. Since lim (x
x3
2)
0, you nd that 1) 2) 4 1
x3
lim f(x)
x x3 x lim
1 2
x3
lim (x lim (x 3.
4
f(3)
x3
as required for f(x) to be continuous at x
EXPLORE!
Store h(x) of Example 1.6.6 (c) into the equation editor as Y1 (X 1)(X , 1) (2 X)(X $ 1). Use a decimal window with a dot graphing style. Is this function continuous at x 1? Use the TRACE key to display the value of the function at x 1 and to nd the limiting values of y as x approaches 1 from the left side and from the right side.
EXAMPLE
1.6.6
x 2 1 if x x if x 1 1
Discuss the continuity of each of the following functions: 1 x2 1 a. f(x) b. g(x) c. h(x) x x 1
Solution
The functions in parts (a) and (b) are rational and are therefore continuous wherever they are dened (that is, wherever their denominators are not zero). 1 a. f(x) is dened everywhere except x 0, and thus it is continuous for all x x 0 (Figure 1.57a).
y
y
y
1 y= x x 0 1 (1, 2) 0
y=
x2 1 x+1 y=x+1 x 0 1 y=2x x
(a) Continuous for x 0
(b) Continuous for x 1
(c) Continuous for x 1
FIGURE 1.57 Functions for Example 1.6.6.
b.
Since x 1 is the only value of x for which g(x) is undened, g(x) is continuous except at x 1 (Figure 1.57b).
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SECTION 1.6 One-Sided Limits and Continuity 79
EXPLORE!
x3 8 , using a x 2 standard window. Does this graph appear continuous? Now use a modied decimal window [ 4.7, 4.7]1 by [0, 14.4]1, and describe what you observe. Which case in Example 1.6.6 does this resemble? Graph f (x)
c.
This function is dened in two pieces. First check for continuity at x 1, the value of x that separates the two pieces. You nd that lim h(x) does not exist, since h(x) approaches 2 from the left and 1 from the right. Thus, h(x) is not continuous at 1 (Figure 1.57c). However, since the polynomials x 1 and 2 x are each continuous for every value of x, it follows that h(x) is continuous at every number x other than 1.
x1
EXAMPLE 1.6.7
For what value of the constant A is the following function continuous for all real x? f (x)
Solution
Ax x2
5 3x
4
if x if x
1 1
Since Ax 5 and x2 3x 4 are both polynomials, it follows that f(x) will be continuous everywhere except possibly at x 1. Moreover, f (x) approaches A 5 as x approaches 1 from the left and approaches 2 as x approaches 1 from the right. Thus, for lim f(x) to exist, we must have A 5 2 or A 3, in which case
x1 x1
lim f(x)
2
f(1) 3.
This means that f is continuous for all x only when A
Continuity on an Interval
For many applications of calculus, it is useful to have denitions of continuity on open and closed intervals.
Continuity on an Interval
A function f(x) is said to be continuous on an open interval a x b if it is continuous at each point x c in that interval. Moreover, f is continuous on the closed interval a x b if it is continuous on the open interval a x b and
xa
lim f(x)
f(a)
and
xb
lim f(x)
f(b)
In other words, continuity on an interval means that the graph of f is one piece throughout the interval.
y
EXAMPLE 1.6.8
Discuss the continuity of the function f(x) x x 2 3 2 x 3.
2
3
x
on the open interval
Solution
2
x
3 and on the closed interval
FIGURE 1.58 The graph of
f (x) x x 2 . 3
The rational function f(x) is continuous for all x except x 3. Therefore, it is continuous on the open interval 2 x 3 but not on the closed interval 2 x 3, since it is discontinuous at the endpoint 3 (where its denominator is zero). The graph of f is shown in Figure 1.58.
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The Intermediate Value Property
An important feature of continuous functions is the intermediate value property, which says that if f(x) is continuous on the interval a x b and L is a number between f(a) and f(b), then f(c) L for some number c between a and b (see Figure 1.59). In other words, a continuous function attains all values between any two of its values. For instance, a girl who weighs 5 lb at birth and 100 lb at age 12 must have weighed exactly 50 lb at some time during her 12 years of life, since her weight is a continuous function of time.
y f(c) = L for some c between a and b
y = f (x)
(a, f(a)) L
(b, f(b)) x
a
c
b
FIGURE 1.59 The intermediate value property. The intermediate value property has a variety of applications. In Example 1.6.9, we show how it can be used to estimate a solution of a given equation.
y
EXAMPLE 1.6.9
Show that the equation x2
Solution
x
1
1 x 1
has a solution for 1
x
2.
1 1 2 2 3
x
1 3 2 . Then f(1) and f(2) . Since f(x) is conx 1 2 3 tinuous for 1 x 2 and the graph of f is below the x axis at x 1 and above the x axis at x 2, it follows from the intermediate value property that the graph must cross the x axis somewhere between x 1 and x 2 (see Figure 1.60). In other words, there is a number c such that 1 c 2 and f (c) 0, so Let f(x) x2 x 1 c2 c 1 1 c 1
FIGURE 1.60 The graph
of y x2 x 1 1 x 1 .
NOTE The root-location procedure described in Example 1.6.9 can be continued to estimate the root c to any desired degree of accuracy. For instance, the midpoint of the interval 1 x 2 is d 1.5 and f(1.5) 0.65, so the root c must lie in the interval 1.5 x 2 (since f(2) 0), and so on. Thats nice, you say, but I can use the solve utility on my calculator to nd a much more accurate estimate for c with much less effort. You are right, of course, but how do you think your calculator makes its estimation? Perhaps not by the method just described, but certainly by some similar algorithmic procedure. It is important to understand such procedures as you use the technology that utilizes them.
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SECTION 1.6 One-Sided Limits and Continuity 81
PROBLEMS
x2
1.6
In Problems 14, nd the one-sided limits lim f(x) and lim f(x) from the given graph of f and determine x2 x2 whether lim f(x) exists. 1.
y 4 2 0 2 4 x 2 4 4 2 4 2 0 2 4 x 2 4
2.
y
4
2
3.
y 4 2 0 2 4 x 2 4
4.
y 4 2 0 2 4 x 2 4
4
2
4
2
In Problems 514, nd the indicated one-sided limit. If the limiting value is innite, indicate whether it is 5. 7. 9. 11.
x4
or
.
lim (3x2 lim 3x
9) 9 x)
6. 8. 10. 12. 14. 3 3
x3
x0
lim (x lim x
1 2 x3 x 3 13. lim f(x) and lim f(x),
x3 x3
x 3 x2 x 2 x2 4 lim x2 x 2 x x lim x1 x 1 2x 1 3 lim x5 x 5 lim f(x) and lim f(x), lim
x 1 x 1
where f(x)
2x 2 x if x 3 x if x
where f(x)
1 if x x 1 x 2 2x if x
1 1
In Problems 1526, decide if the given function is continuous at the specied value of x. 15. f(x) 17. f(x) 5x2 6x 1 at x x 2 at x 1 x 1 2 16. f(x) 18. f(x) x3 2x2 x 2x 4 at x 3x 2 5 2 at x 0
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19. 21. 23.
f(x) f(x) f(x)
x x x x x 2 x2 2x
1 1 2 4 1
at x at x if x if x if x if x
1 4 2 2 3 3 at x 2
20. 22. 24.
f(x) f(x) f(x)
2x 3x x x x x x2 x x2
1 6 2 4 1 1 1 1 3
at x at x if x if x if x if x
2 2 0 0 at x 1 1 0
25.
f(x)
1 4
at x
3
26.
f(x)
at x
1
In Problems 27 through 40, list all the values of x for which the given function is not continuous. 27. f(x) 29. 31. 33. 35. 37. 39. f(x) f(x) f(x) f(x) f(x) f(x) 3x2 x x 3x x 1 2 3 1 6x 9 28. f(x) 30. 32. 34. 36. 3 1 2 x if x if x if x if x 1 1 0 0 38. 40. f(x) f(x) f(x) f(x) f(x) f(x) x5 3x 2x x2 x (x x x2 x2 9 2 x2
2
x3 1 6 1 1 x 5)(x 2x x if x if x 3x x 1 2 2 2 3 if x if x 1 1 1)
3x 2 (x 3)(x 6) x x2 2x 6x 3x x2 x
41. WEATHER Suppose air temperature on a certain day is 30F. Then the equivalent windchill temperature (in F) produced by a wind with speed v miles per hour (mph) is given by* 30 W(v) 1.25v 7 18.67 v 62.3 if 0 if 4 if v v 4 v 45 45
42. ELECTRIC FIELD INTENSITY If a hollow sphere of radius R is charged with one unit of static electricity, then the eld intensity E(x) at a point P located x units from the center of the sphere satises: 0 1 2x 2 1 x2 if 0 if x if x x R R R
a. What is the windchill temperature when v 20 mph? When v 50 mph? b. What wind speed produces a windchill temperature of 0F? c. Is the windchill function W(v) continuous at v 4? What about at v 45?
E(x)
Sketch the graph of E(x). Is E(x) continuous for x 0?
*Adapted from UMAP Module No. 658, Windchill, by W. Bosch and L. G. Cobb, 1984, pp. 244247.
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SECTION 1.6 One-Sided Limits and Continuity 83
43. POSTAGE The postage function p(x) can be described as follows: 37 60 83 290 if 0 if 1 if 2 if 11 x x x x 1 2 3 12
0 300 200 100
I (units in inventory)
p(x)
where x is the weight of a letter in ounces and p(x) is the corresponding postage in cents. Sketch the graph of p(x) for 0 x 6. For what values of x is p(x) discontinuous for 0 x 6? 44. WATER POLLUTION A ruptured pipe in a North Sea oil rig produces a circular oil slick that is y meters thick at a distance x meters from the rupture. Turbulence makes it difcult to directly measure the thickness of the slick at the source (where x 0), but for x 0, it is found that y 0.5(x 2 3x) x 3 x 2 4x
6
12
18
24
t (months)
PROBLEM 46 47. COST-BENEFIT ANALYSIS In certain situations, it is necessary to weigh the benet of pursuing a certain goal against the cost of achieving that goal. For instance, suppose that in order to remove x% of the pollution from an oil spill, it costs C thousands of dollars, where C(x) 12x 100 x
Assuming the oil slick is continuously distributed, how thick would you expect it to be at the source? 45. ENERGY CONSUMPTION The accompanying graph shows the amount of gasoline in the tank of Sues car over a 30-day period. When is the graph discontinuous? What do you think happens at these times?
Q (gals) 10
0
5
10
15
20
25
30
t (days)
a. How much does it cost to remove 25% of the pollution? 50%? b. Sketch the graph of the cost function. c. What happens as x 100 ? Is it possible to remove all the pollution? 48. EARNINGS On January 1, 2005, Sam started working for Acme Corporation with an annual salary of $48,000, paid each month on the last day of that month. On July 1, he received a commission of $2,000 for his work, and on September 1, his base salary was raised to $54,000 per year. Finally, on December 21, he received a Christmas bonus of 1% of his base salary. a. Sketch the graph of Sams cumulative earnings E as a function of time t (days) during the year 2005. b. For what values of t is the graph of E(t) discountinuous? 49. COST MANAGEMENT A business manager determines that when x% of her companys plant capacity is being used, the total cost of operation is C hundred thousand dollars, where C(x) 8x2 x2 636x 320 68x 960
PROBLEM 45 46. INVENTORY The accompanying graph shows the number of units in inventory at a certain business over a 2-year period. When is the graph discontinuous? What do you think is happening at those times?
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a. Find C(0) and C(100). b. Explain why the result of part (a) cannot be used along with the intermediate value property to show that the cost of operation is exactly $700,000 when a certain percentage of plant capacity is being used. 50. AIR POLLUTION It is estimated that t years from now the population of a certain suburban community will be p thousand people, where p(t) 20 7 t 2
53. Discuss the continuity of the function 1 f(x) x 1 on the open interval 0 x and on the closed interval 0 x 1. 54. Discuss the continuity of the function x 2 3x if x 2 f(x) 4 2x if x 2
x
1
55. 56. 57.
An environmental study indicates that the average level of carbon monoxide in the air will be c parts per million when the population is p thousand, where c(p) 0.4 p
2
p
21
What happens to the level of pollution c in the long run (as t )? In Problems 51 and 52 nd the values of the constant A such that the function f(x) will be continuous for all x. 51. 52. f (x) f (x) Ax 3 if x 2 3 x 2x 2 if x 2 1 3x if x 4 Ax 2 2x 3 if x 4
58.
59. 60.
on the open interval 0 x 2 and on the closed interval 0 x 2. 3 9x 2/3 29 has Show that the equation x 8 at least one solution for the interval 0 x 8. 3 Show that the equation x x 2 2x 1 must have at least one solution on the interval 0 x 1. 2x 2 5x 2 Investigate the behavior of f(x) x2 4 when x is near to (a) 2 and (b) 2. Does the limit exist at these values of x? Is the function continuous at these values of x? Explain why there must have been some time in your life when your weight in pounds was the same as your height in inches. Explain why there is a time every hour when the hour hand and minute hand of a clock coincide. At age 15, Nan is twice as tall as her 5-year-old brother Dan, but on Dans 21st birthday, they nd that he is 6 inches taller. Explain why there must have been a time when they were exactly the same height.
CHAPTER SUMMARY
Important Terms, Symbols, and Formulas
Function (2) Functional notation: f(x) (2) Domain and range of a function (2) Domain convention (4) Independent and dependent variables (3) Functions used in economics: Demand (5) Supply (48) Cost (5) Revenue (5) Prot (5) Composition of functions: g(h(x)) (6) Graph of a function: the points (x, f(x)) (15) x and y intercepts (17) Piecewise-dened functions (4) Power function (20) Polynomial (20) Rational function (21) Vertical line test (21) Linear function; constant rate of change (27) y y2 y1 Slope: m (28) x x2 x1 Slope-intercept formula: y mx b (30) Point-slope formula: y y0 m(x x0) (31)
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Chapter Summary 85
xc
xc
Joint proportionality: Q kxy (46) Market equilibrium; law of supply and demand Shortage and surplus (49) Break-even analysis (50) Limit of a function: lim f(x) (59)
xc
(49)
One-sided limits: lim f(x) and lim f(x) (74)
xc xc
Existence of a limit: lim f(x) exists if and only if
xc xc
lim f(x) and lim f(x) exist and are equal.
xc
(75)
Limits at innity: lim f(x) and lim f(x)
x x
(65)
Continuity of f(x) at x c: lim f(x) f(c) (77)
xc
Reciprocal power rules: A A lim k 0 and lim k x x x x
0
k
0 (66) p(x) : q(x)
Limits at innity of a rational function f(x)
Continuity on an interval (79) Discontinuity (77) Continuity of polynomials and rational functions (77) Intermediate value property (80)
Checkup for Chapter 1
1. Specify the domain of the function 2x 1 f(x) 4 x2 2. Find the composite function g(h(x)), where x 2 1 and h(x) g(u) 2u 1 2x 1 3. Find an equation for each of these lines: a. Through the point ( 1, 2) with slope b. With slope 2 and y intercept 3 1 2 c. lim d. x2 x 2x3 x2 x 2 3x 2x 5 7 1
x1
x
lim
6. Determine whether this function f(x) is continuous at x 1: 2x 1 if x 1 2 x 2x 3 f (x) if x 1 x 1 7. PRICE OF GASOLINE Since the beginning of the year, the price of unleaded gasoline has been increasing at a constant rate of 2 cents per gallon per month. By June rst, the price had reached $1.80 per gallon. a. Express the price of unleaded gasoline as a function of time and draw the graph. b. What was the price at the beginning of the year? c. What will be the price on October rst?
4. Sketch the graph of each of these functions. Be sure to show all intercepts and any high or low points. a. f(x) 3x 5 b. f(x) x2 3x 4 5. Find each of these limits. If the limit is innite, indicate whether it is or . x2 2x 3 a. lim x 1 x 1 x 2 2x 3 b. lim x1 x 1
CHAPTER SUMMARY
Criteria for lines to be parallel or perpendicular (35) Mathematical modeling (41) Direct proportionality: Q kx (46) k Inverse proportionality: Q (46) x
Divide all terms in f(x) by the highest power xk in the denominator q(x) and use the reciprocal power rules. (67) Horizontal asymptote (65) or lim f(x) Innite limit: lim f(x) (68)
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CHAPTER SUMMARY
8. SUPPLY AND DEMAND Suppose it is known that producers will supply x units of a certain commodity to the market when the price is p S(x) dollars per unit and that the same number of units will be demanded (bought) by consumers when the price is p D(x) dollars per unit, where S(x) x2 A and D(x) Bx 59 for constants A and B. It is also known that no units will be supplied until the unit price is at least $3 and that market equilibrium occurs when x 7 units. a. Use this information to nd A and B and the equilibrium unit price. b. Sketch the supply and demand curves on the same graph. c. What is the difference between the supply price and the demand price when 5 units are produced? When 10 units are produced?
9. BACTERIAL POPULATION The population (in thousands) of a colony of bacteria t minutes after the introduction of a toxin is given by the function t2 7 if 0 t 5 f(t) 8t 72 if t 5 a. When does the colony die out? b. Explain why the population must be 10,000 sometime between t 1 and t 7. 10. MUTATION In a study of mutation in fruit ies, researchers radiate ies with X-rays and determine that the mutation percentage M increases linearly with the X-ray dosage D, measured in kiloRoentgens (kR). When a dose of D 3 kR is used, the percentage of mutations is 7.7%, while a dose of 5 kR results in a 12.7% mutation percentage. Express M as a function of D. What percentage of the ies will mutate even if no radiation is used?
Review Problems
1. Specify the domain of each of these functions: a. f(x) x 2 2x 6 b. f(x) c. f(x) x x2 x2 x 9 3 2 4. a. Find f (x b. Find f (x 2 c. Find f (x 2) if f(x) 1) if f (x) 1) f (x) if f (x) x2 x x 4. 2 x 2 x. 1 .
5. Find functions h(x) and g(u) such that f(x) a. f (x) b. f (x) (x 2 (3x 3x 1)2 4)5 5 2(3x 2)3
g(h(x)).
2. As advances in technology result in the production of increasingly powerful and compact calculators, the price of calculators currently on the market drops. Suppose that x months from now, the price 30 of a certain model will be P(x) 40 x 1 dollars. a. What will be the price 5 months from now? b. By how much will the price drop during the fth month? c. When will the price be $43? d. What happens to the price in the long run (as x becomes very large)? 3. Find the composite function g(h(x)). a. g(u) u2 2u 1, h(x) 1 x b. g(u) c. g(u) 1 2u 1 1 , h(x) x 2x 2 4
6. ENVIRONMENTAL ANALYSIS An environmental study of a certain community suggests that the average daily level of smog in the air will be Q(p) 0.5p 19.4 units when the population is p thousand. It is estimated that t years from now, the population will be p(t) 8 0.2t 2 thousand. a. Express the level of smog in the air as a function of time. b. What will the smog level be 3 years from now? c. When will the smog level reach 5 units? 7. Find c so that the curve y 3x2 2x c passes through the point (2, 4). 8. Graph these functions. a. f (x) x2 2x 8 b. f (x) 3 4x 2x2
u, h(x)
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Chapter Summary 87
10. Find equations for these lines: a. Slope 5 and y intercept (0, 4) b. Slope 2 and contains (1, 3) 2 c. x intercept (3, 0) and y intercept 0, 3 d. Contains (5, 4) and is parallel to 2x y 3 e. Contains ( 1, 3) and is perpendicular to 5x 3y 7 11. EDUCATIONAL FUNDING A private college in the southwest has launched a fund-raising campaign. Suppose that college ofcials estimate that it 10x will take f(x) weeks to reach x% of 150 x their goal. a. Sketch the relevant portion of the graph of this function. b. How long will it take to reach 50% of the campaigns goal? c. How long will it take to reach 100% of the goal? 12. CONSUMER EXPENDITURE The demand for a certain commodity is given by D(x) 50x 800; that is, x units of the commodity will be demanded by consumers when the price is p D(x) dollars per unit. Total consumer expenditure E(x) is the amount of money consumers pay to buy x units of the commodity. a. Express consumer expenditure as a function of x, and sketch the graph of E(x). b. Use the graph in part (a) to determine the level of production x at which consumer expenditure is largest. What price p corresponds to maximum consumer expenditure? 13. MICROBIOLOGY A spherical cell of radius r 4 3 r and surface area S 4 r2. has volume V 3 Express V as a function of S. If S is doubled, what happens to V? 14. NEWSPAPER CIRCULATION The circulation of a newspaper is increasing at a constant rate.
15. Find the points of intersection (if any) of the given pair of curves and draw the graphs. a. y 3x 5 and y 2x 10 b. y x 7 and y 2 x c. y x2 1 and y 1 x2 d. y x2 and y 15 2x 16. OPTIMAL SELLING PRICE A manufacturer can produce bookcases at a cost of $80 apiece. Sales gures indicate that if the bookcases are sold for x dollars apiece, approximately 150 x will be sold each month. Express the manufacturers monthly prot as a function of the selling price x, draw the graph, and estimate the optimal selling price. 17. OPTIMAL SELLING PRICE A retailer can obtain cameras from the manufacturer at a cost of $150 apiece. The retailer has been selling the cameras at the price of $340 apiece, and at this price, consumers have been buying 40 cameras a month. The retailer is planning to lower the price to stimulate sales and estimates that for each $5 reduction in the price, 10 more cameras will be sold each month. Express the retailers monthly prot from the sale of the cameras as a function of the selling price. Draw the graph and estimate the optimal selling price. 18. STRUCTURAL DESIGN A cylindrical can with no top is to be constructed for 80 cents. The cost of the material used for the bottom is 3 cents per square inch, and the cost of the material used for the curved side is 2 cents per square inch. Express the volume of the can as a function of its radius. 19. MANUFACTURING EFFICIENCY A manufacturing rm has received an order to make 400,000 souvenir silver medals commemorating the 35th anniversary of the landing of Apollo 11 on the moon. The rm owns several machines, each of which can produce 200 medals per hour. The cost of setting up the machines to produce the medals is $80 per machine, and the total operating cost is $5.76 per hour. Express the cost of producing the 400,000 medals as a function of the number of machines used. Draw the graph and estimate the number of machines the rm should use to minimize cost.
CHAPTER SUMMARY
9. Find the slope and y intercept of the given line and draw the graph. a. y 3x 2 b. 5x 4y 20 c. 2y 3x 0 x y 4 d. 3 2
Three months ago the circulation was 3,200. Today it is 4,400. a. Express the circulation as a function of time and draw the graph. b. What will be the circulation 2 months from now?
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CHAPTER SUMMARY
20. INVENTORY ANALYSIS A businessman maintains inventory over a particular 30-day month as follows: days 19 30 units days 1015 days 1623 days 2430 17 units 12 units
from the power plant. Express the cost of installing the cable as a function of x.
P x 900
steadily decreasing from 12 units to 0 units Sketch the graph of the inventory as a function of time t (days). At what times is the graph discontinuous? 21. BREAK-EVEN ANALYSIS A manufacturer can sell a certain product for $80 per unit. Total cost consists of a xed overhead of $4,500 plus production costs of $50 per unit. a. How many units must the manufacturer sell to break even? b. What is the manufacturers prot or loss if 200 units are sold? c. How many units must the manufacturer sell to realize a prot of $900? 22. PRODUCTION MANAGEMENT During the summer, a group of students builds kayaks in a converted garage. The rental for the garage is $1,500 for the summer, and the materials needed to build a kayak cost $125. The kayaks can be sold for $275 apiece. a. How many kayaks must the students sell to break even? b. How many kayaks must the students sell to make a prot of at least $1,000? 23. LEARNING Some psychologists believe that when a person is asked to recall a set of facts, the rate at which the facts are recalled is proportional to the number of relevant facts in the subjects memory that have not yet been recalled. Express the recall rate as a function of the number of facts that have been recalled. 24. COST EFFICIENT DESIGN A cable is to be run from a power plant on one side of a river 900 meters wide to a factory on the other side, 3,000 meters downstream. The cable will be run in a straight line from the power plant to some point P on the opposite bank and then along the bank to the factory. The cost of running the cable across the water is $5 per meter, while the cost over land is $4 per meter. Let x be the distance from P to the point directly across the river
3,000
PROBLEM 24 25. CONSTRUCTION COST A window with a 20foot perimeter (frame) is to be comprised of a semicircular stained glass pane above a rectangular clear pane, as shown in the accompanying gure. Clear glass costs $3 per square foot and stained glass costs $10 per square foot. Express the cost of the window as a function of the radius of the stained glass pane.
x y
2x
PROBLEM 25 26. MANUFACTURING OVERHEAD A furniture manufacturer can sell end tables for $70 apiece. It costs the manufacturer $30 to produce each table, and it is estimated that revenue will equal cost when 200 tables are sold. What is the overhead associated with the production of the tables? [Note: Overhead is the cost when 0 units are produced.] 27. MANUFACTURING COST A manufacturer is capable of producing 5,000 units per day. There is a xed (overhead) cost of $1,500 per day and a variable cost of $2 per unit produced. Express the daily cost C as a function of the number of units produced, and sketch the graph of C(x). Is C(x) continuous? If not, where do its discontinuities occur? 28. At what time between 3 P.M. and 4 P.M. will the minute hand coincide with the hour hand? [Hint: 1 The hour hand moves as fast as the minute hand.] 12
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Chapter Summary 89
In Problems 30 through 43, either nd the given limit or show it does not exist. If the limit is innite, indicate whether it is or . x 2 3x x2 x 2 30. lim 31. lim x2 x x1 1 x2 1 32. lim 34. 36. 38.
x1
1 x2 2 x3 x
3
1 x 1 x2 1 x2 3x 2x 3) x2 1 x2 3 5
33. lim
x3 x2 2
x0
8 x 1 x3 x 5 3x 2 x 1
3
x
lim
35. lim 2 37. 39. lim lim
x0
lim lim
x
x2 x
4
x
x
2x x 1
7
x(x 40. lim x 7 42.
x0
41. 43.
x
lim
x2 1
1 1 x x2 3x 1 1 x
lim
x 1
x0
lim x
In Problems 44 through 47, list all values of x for which the given function is not continuous. 44. f(x) 5x 3 3x x 45. f(x) x2 x x3 x2 x 1 3 2x 6x 3 33 9 if x if x 3 3
46. g(x)
x 3 5x (x 2)(2x 3)
47. h(x)
48. In each of these cases, nd the value of the constant A that makes the given function f(x) continuous for all x. 2x 3 if x 1 a. f(x) Ax 1 if x 1 x2 1 if x 1 x 1 b. f(x) Ax 2 x 3 if x 1
49. The radius of the earth is roughly 4,000 miles, and an object located x miles from the center of the earth weighs w(x) lb, where w(x) Ax B x2 if x if x 4,000 4,000
and A and B are positive constants. Assuming that w(x) is continuous for all x, what must be true about A and B? Sketch the graph of w(x).
CHAPTER SUMMARY
29. PROPERTY TAX A homeowner is trying to decide between two competing property tax propositions. With Proposition A, the homeowner will pay $100 plus 8% of the assessed value of her home, while Proposition B requires a payment of $1,900
plus 2% of the assessed value. Assuming the homeowners only consideration is to minimize her tax payment, develop a criterion based on the assessed value V of her home for deciding between the propositions.
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CHAPTER SUMMARY
50. Graph f(x)
3x 2 6x 9 . Determine the values x2 x 2 of x where the function is undened. 21 84 654 54 x x and y on the 9 35 279 10 same set of coordinate axes using [ 10, 10]1 by [ 10, 10]1. Are the two lines parallel?
55. The accompanying graph represents a function g(x) that oscillates more and more frequently as x approaches 0 from either the right or the left but with decreasing magnitude. Does lim g(x) exist? If so,
x0
51. Graph y
what is its value? [Note: For students with experience in trigonometry, the function x sin (1 x) behaves in this way.] g(x)
x 3 and g(x) 5x2 4, nd (a) 52. For f(x) f(g( 1.28)) and (b) g( f ( 2)). Use three decimal place accuracy. 53. Graph y x2 x2 discontinuity. 1 1 if x if x 1 . Find the points of 1 x2
y
x
54. Graph the function f(x)
3x 10 2. Find 1 x the x and y intercepts. For what values of x is this function dened?
PROBLEM 55
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EXPLORE! UPDATE
Explore! Updates are included at the end of each chapter of this textbook. These updates provide additional instruction and exercises for exploring calculus using a graphing calculator or giving hints and solutions for selected Explore! boxes found in each chapter. An attempt has been made to use function keys that are standard on most handheld graphing utilities. The exact names of the function keys on your particular calculator may vary, depending on the brand. Please consult your calculator user manual for more specic details. Complete solutions for all Explore! boxes throughout the text can be accessed via the book specic website www.mhhe.com/hoffmann.
Solution for Explore! Exercise on Page 2
Write f (x) x2 4 in the function editor (Y key) of your graphing calculator. While on a cleared homescreen (2nd MODE CLEAR), locate the symbol for Y1 through the VARS key, arrowing right to Y-VARS and selecting 1:Function and 1:Y1. (Also see the Calculator Introduction at the front of the text.) Y1({ 3, 1, 0, 1, 3}) yields the desired functional values, all at once. Or you can do each value individually, such as Y1( 3).
NOTE An Explore! tip is that it is easier to view a table of values, especially for several functions. Set up the desired table parameters through TBLSET (2nd WINDOW). Now enter g(x) x2 1 into Y2 of the equation editor (Y ). Observing the values of Y1 and Y2, we notice that they differ by a xed constant of 5, since the two functions are simply vertical translations of f (x) x2.
Solution for Explore! Exercise on Page 4 (middle)
The graph of the piecewise-dened function Y1 2(X 1) ( 1)(X 1) is shown in the following gure, and the table to its right shows the functional values at X 2, 0, 1, and 3. Recall that the inequality symbols can be accessed through the TEST (2nd MATH) key.
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EXPLORE! UPDATE
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EXPLORE! UPDATE
Solution for Explore! Exercise on Page 4 (bottom)
The function f(x) 1/(x 3) appears to have a break at x 3 and not be dened there. So the domain consists of all real numbers x 3. The function g(x) (x 2) is not dened for x 2, as indicated by the following graph in the middle screen and the table on the right.
Solution for Explore! Exercise on Page 44
Use TBLSET with TblStart 1 and Tbl 1 to obtain the following table on the left which indicates a minimal cost in the low hundreds occurring within the interval [1, 4]. Then, graph using WINDOW dimensions of [1, 4]1 by [100, 400]50 to obtain the following middle screen, where we have utilized the minimum-nding option in CALC (see Calculator Introduction) to locate an apparent minimal value of about 226 cents at a radius of 2 inches. Thus, no radius will create a cost of $2.00, which is less than the minimal cost. Graphing at the front of the text Y2 300 and utilizing the intersection-nding feature shows that the cost is $3.00 when the radius is 1.09 in. or 3.33 in.
Solution for Explore! Exercise on Page 79
The graph of f (x) (x3 8)/(x 2) appears continuous based on the window, [ 6, 6]1 by [ 2, 10]1. However, examination of this graph using a modied decimal window, [ 4.7, 4.7]1 by [0, 14.4]1 shows an exaggerated hole at x 2.
The function f(x) is not continuous, specically at x 2, where it is not dened. The situation is similar to Figure 1.57(b) on page 78. What value of y would ll up the hole in the graph here? Answer: lim f(x) 12.
x2
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THINK ABOUT IT
ALLOMETRIC MODELS
When developing a mathematical model, the rst task is to identify quantities of interest, and the next is to nd equations that express relationships between these quantities. Such equations can be quite complicated, but there are many important relationships that can be expressed in the relatively simple form y Cx k, in which one quantity y is expressed as a constant multiple of a power function of another quantity x. In biology, the study of the relative growth rates of various parts of an organism is called allometry, from the Greek words allo (other or different) and metry (measure). In allometric models, equations of the form y Cx k are often used to describe the relationship between two biological measurements. For example, the size a of the antlers of an elk from tip to tip has been shown to be related to h, the shoulder height of the elk, by the allometric equation a 0.026h1.7
where a and h are both measured in centimeters (cm).* This relationship is shown in the accompanying gure.
130 110 Antler size 90 70 50 80
100 120 140 Shoulder height
Whenever possible, allometric models are developed using basic assumptions from biological (or other) principles. For example, it is reasonable to assume that the body volume and hence the weight of most animals is proportional to the cube of the linear dimension of the body, such as height for animals that stand up or length for four-legged animals. Thus, it is reasonable to expect the weight of a snake to be proportional to the cube of its length, and indeed, observations of the hognose snake of
*Frederick R. Adler, Modeling the Dynamics of Life, Pacic Grove, CA: Brooks-Cole Publishing, 1998, p. 61.
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THINK ABOUT IT
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Kansas indicate that the weight w (grams) and length L (meters) of a typical such snake are related by the equation* w 440L3
At times, observed data may be quite different from the results predicted by the model. In such a case, we seek a better model. For the western hognose snake, it turns out that the equations w 446L2.99 and w 429L2.90 provide better approximations to the weight of male and female western hognose snakes, respectively. However, there is no underlying biological reason why we should use exponents less than three. It just turns out these equations are slightly better ts. The basal metabolic rate M of an animal is the rate of heat produced by its body when it is resting and fasting. Basal metabolic rates have been studied since the 1830s. Allometric equations of the form M cwr, for constants c and r, have long been used to build models relating the basal metabolic rate to the body weight w of an animal. The development of such a model is based on the assumption that basal metabolic rate M is proportional to S, the surface area of the body, so that M aS where a is a constant. To set up an equation relating M and w, we need to relate the weight w of an animal to its surface area S. Assuming that all animals are shaped like spheres or cubes, and that the weight of an animal is proportional to its volume, we can show (see Exercises 1 and 2) that the surface area is proportional to w2/3, so that S bw2/3 where b is a constant. Putting the equations M aS and S bw2/3 together, we obtain the allometric equation M abw2/3 kw2/3
where k ab. However, this is not the end of the story. When more rened modeling assumptions are used, it is found that the basal metabolic rate M is better approximated if the exponent 3/4 is used in the allometric equation rather than the exponent 2/3. Observations further suggest that the constant 70 be used in this equation (see M. Kleiber, The Fire of Life, An Introduction to Animal Energetics, Wiley, 1961). This gives us the equation M 70w3/4
where M is measured in kilocalories per day and w is measured in kilograms. Additional information about allometric models may be found in our Web Resources Guide at www.mhhe.com/hoffmann.
Questions
1. What weight does the allometric equation w 440L3 predict for a western hognose snake 0.7 meters long? If this snake is male, what does the equation w 446L2.99 predict for its weight? If it is female, what does the equation w 429L2.90 predict for its weight? What basal metabolic rates are predicted by the equation M with weights of 50 kg, 100 kg, and 350 kg? 70w3/4 for animals
2.
*Edward Batschelet, Introduction to Mathematics for Life Scientists, 3rd ed., New York: Springer-Verlag, 1979, p. 178.
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Think About It 95
3.
Observations show that the brain weight b, measured in grams, of adult female primates is given by the allometric equation b 0.064w0.822, where w is the weight in grams (g) of the primate. What are the predicted brain weights of female primates weighing 5 kg, 10 kg, 25 kg, 50 kg, and 100 kg? (Recall that 1 kg 1,000 g). If y Cx k where C and k are constants, then y and x are said to be related by a positive allometry if k 1 and a negative allometry if k 1. The weight-specic metabolic rate of an animal is dened to be the basal metabolic rate M of the animal divided by its weight w; that is, M/w. Show that if the basal metabolic rate is the positive allometry of the weight in Exercise 2, then the weight-specic metabolic rate is a negative allometry equation of the weight. Show that if we assume all animal bodies are shaped like cubes, then the surface area S of an animal body is proportional to V 2/3, where V is the volume of the body. By combining this fact with the assumption that the weight w of an animal is proportional to its volume, show that S is proportional to w 2/3. Show that if we assume all animal bodies are shaped like spheres, then the surface area S of an animal body is proportional to V 2/3, where V is the volume of the body. By combining this fact with the assumption that the weight w of an animal is proportional to its volume, show that S is proportional to w 2/3. [Hint: Recall that a 4 sphere of radius r has surface area 4 r2 and volume r 3.] 3
4.
5.
6.
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