SELF-CHECK DIAGNOSTIC TEST - Soo t tan calculus- 123docz.net

a. a linear function

b. a polynomial function of degree 4 c. a rational function

d. a power function e. an algebraic function f. a trigonometric function

2. The book value of an asset at time (measured in years) being depreciated linearly over a period of years is given by

where and (in dollars) give the initial and scrap value of the asset, respectively.

a. What is the -intercept? Interpret your result.

b. By how much is the asset being depreciated annually?

3. By cutting away identical squares from each corner of a square piece of cardboard with sides 12 in. long and then folding up the resulting flaps, an open box can be made. If the square cutaways have dimensions in.

by in., find a function giving the volume of the resulting box.

Answers to Self-Check Diagnostic Test 0.6 can be found on page ANS 6.

12 2x 12

x x

x V

S C

V(t)C CS n t n

1. Formulate.Given a real-world problem, our first task is to formulate the problem using the language of mathematics. This mathematical description of the real-world phenomenon is called a mathematical model.The many techniques that are used in constructing mathematical models range from theoretical consid- eration of the problem on the one extreme to an interpretation of data associated with the problem on the other. For example, the mathematical model that gives the accumulated amount at any time after a certain sum of money has been deposited in the bank can be derived theoretically (see Section 0.8, pp. 87–

89). On the other hand, the mathematical models in Examples 2 and 3 of this section are constructed by requiring that they fit the data associated with the problem “best” according to some specified criterion. In calculus we are pri- marily concerned with how one (dependent) variable depends on one or more (independent) variables. Consequently, most of our mathematical models will involve functions of one or more variables or equations defining these functions (implicitly).

2. Solve.Once a mathematical model has been constructed, we can use the appropriate mathematical techniques, which we will develop throughout this text, to solve the problem.

3. Interpret.Bearing in mind that the solution obtained in Step 2 is just the solution of the mathematical model, we need to interpret these results in the context of the original real-world problem.

4. Test.Some mathematical models of real-world applications describe the situa- tions with complete accuracy. For example, the model describing a deposit in a bank account gives the exact accumulated amount in the account at any time.

But other mathematical models give, at best, an approximate description of the real-world problem. In such cases we need to test the accuracy of the model by observing how well it describes the original real-world problem and how well it predicts past and/or future behavior. If the results are unsatisfactory, then we might have to reconsider the assumptions that were made in the construction of the model or, in the worst case, return to Step 1.

Modeling with Functions

Many real-world phenomena, such as the speed at which a screwdriver falls after being accidentally dropped from a building under construction, the speed of a chemical reac- tion, the population of a certain strain of bacteria, the life expectancy of a female infant at birth in a certain country, and the demand for a product, can be modeled by an appropriate function.

Mathematical modeling is a process that enables us to use mathematics as a tool to analyze and understand real-world phenomena. The four steps in this process are illus- trated in Figure 1.

FIGURE 1

Solution of mathematical model

Mathematical model

Solve Solution of

real-world problem Real-world

problem Test

Formulate

Interpret

In what follows, we will recall some familiar functions and give examples of real- world phenomena that are modeled by using these functions.

Polynomial Functions

A polynomial function of degree is a function of the form

where is a nonnegative integer and the numbers , , , are constants called the coefficientsof the polynomial function. For example, the functions

are polynomial functions of degree 5 and 3, respectively. Observe that a polynomial function is defined for every value of , so its domain is .

A polynomial function of degree 1 has the form

and is an equation of a straight line in the slope-intercept form with slope and -intercept (see Section 0.1). For this reason a polynomial function of degree 1 is called a linear function.

Linear functions are used extensively in mathematical modeling for two important reasons. First, some models are linear by nature. For example, the formula for convert- ing temperature from Celsius (°C) to Fahrenheit (°F) is , and is a linear function of for in any feasible prescribed domain (see Figure 2a). Second, some natural phenomena exhibit linear characteristics over a small range of values and can therefore be modeled by a linear function that is restricted to a small interval. For example, according to Hooke’s Law, the magnitude of a force required to stretch a spring by an elongation beyond its unstretched length is given by , provided that the elongation is not too great. If stretched beyond a certain point, called the elastic limit, the spring will become permanently deformed and will not return to its natural length when the force is removed. The constant is called the spring constant or the stiffnessof the spring. In this instance we have to restrict our interest to the portion of the graph that is linear (see Figure 2b).

k x

Fkx x

F C

F F95 C32 ba0

ma1

a10 yf(x)a1xa0

(n1)

(⬁, ⬁) x

t(x)0.001x30.2x210x200 f(x)2x53x4 1

2 x3 12x26 an

a1 p a0

an0 f(x)anxnan1xn1p a2x2a1xa0

FIGURE 2 The graph of a linear function and the graph of a function that is linear over a small interval

In the following example we assume that Hooke’s Law applies.

C (C) F (F)

20 60 80

0 (a) F is linear in C.

20 40 x (ft)

Elastic limit F (lb)

(b) F is linear for small values of x.

EXAMPLE 1 Force Required to Stretch a Spring A force of 3.18 lb is required to stretch a spring by 2.4 in. beyond its unstretched length (see Figure 3).

a. Use Hooke’s Law to find a mathematical model that describes the force required to stretch the spring by feet beyond its unstretched length.

b. What is the spring constant?

c. Find the force required to stretch the spring by 1.8 in. beyond its unstretched length.

Solution

a. By Hooke’s Law, , where is the spring constant. Next, using the given data, we find

2.4 in. is equal to 0.2 ft

from which we deduce that . Therefore, the required mathematical

model is .

b. From the result of part (a) we see that the spring constant is 15.9 lb/ft.

c. We first note that 1.8 in. is equal to 0.15 ft. Then, using the model obtained in part (a), we see that the required force is

or approximately 2.39 lb.

In Example 1 the model was constructed by using the data obtained from one meas- urement. In practice, one normally takes a set of measurements and then uses these data to construct a mathematical model. This practice generally results in a more accurate model.

F(15.9)(0.15)2.385 F15.9x

k15.90

3.180.2k k

Fkx

FIGURE 3

The spring in part (a) is stretched by an elongation of feet beyond its natural length by a weight in part (b).

EXAMPLE 2 Force Required to Stretch a Spring Table 1 gives the force required to stretch the spring (Example 1) by an elongation ft beyond its unstretched length.

As Hooke’s Law predicts, the data points in the scatter plotassociated with these data appear to lie close to a straight line passing through the origin (see Figure 4).

(ft)

x 0 0.1 0.2 0.3 0.4 0.5

(lb)

F 0 1.68 3.18 4.84 6.36 8.02

TABLE 1

FIGURE 4 The data points are scattered about a line through the origin.

(b) (a)

x (ft) F (lb)

2 6 10 14

4 8 12

0 0.2 0.4 0.6 0.8 1

To find a mathematical model based on these data, we use the method of least squaresto find a function of the form (as suggested by Hooke’s Law) that fits the data “best” in the sense of least squares. (See Exercises 3.7,Problems 71 and 72.) We obtain the function

as the required model. Incidentally, this model also tells us that the spring constant is approximately 16.02 lb/ft.

Notes

1. If you use the linear least-squares regression program that is built into most graphing calculators and computers to find a mathematical model using the data in Example 2, you will obtain a different model, namely, . This occurs because the program finds the “best” fit for the data (in the sense of least squares) using the most general linear function, that is, one having the form

2. Since must be equal to zero if is equal to zero, we see that the class of functions chosen to fit the data should have the form , that is, with . Therefore, the model that we found in Example 2 should be regarded as being a more accurate mathematical model than the model suggested by

, in which . As a consequence, we should accept the spring constant to be 16.02 lb/ft found in Example 2 rather than the figure of 15.94 that is found by using the function as the model.

A polynomial function of degree 2 has the form

or, more simply, and is called a quadratic function.The graph of a quadratic function is a parabola (see Figure 5). The parabola opens upward if and downward if . To see this, we rewrite

x0 f(x)ax2bxcx2aa b

x c x2b a0

a0 yax2bxc

a20 yf(x)a2x2a1xa0

t t(0)0.0280 t(x)15.94x0.028

F16.02x

b0 f(x)ax

x F

f(x)axb

t(x)15.94x0.028 f(x)16.02x

f(x)kx

FIGURE 5 The graph of a quadratic

function is a parabola.

x y

(a) If a > 0, the parabola opens upward.

x y

(b) If a < 0, the parabola opens downward.

Observe that if is large in absolute value, then the expression inside the parenthe- ses is close to , so behaves like for large values of . Therefore, for large values of , is large and positive if (the parabola opens upward) and is large in magnitude and negative if (the parabola opens downward). The high- est point on a parabola that opens downward or the lowest point on a parabola that opens upward is called the vertexof the parabola. The vertex of the parabola with

a0 yf(x)

x ax2

f(x) a

equation , where , is , since . You can verify this fact by using the method of completing the square (see Exer-

cise 30).

Quadratic functions serve as mathematical models for many phenomena. For example, Newton’s Second Law of Motion can be used to show that the distance covered by a falling object dropped near the surface of the earth is given by D12tt2where yf(x) f(b>(2a)))

(b>(2a) a0

yax2bxc

, the gravitational constant at sea level at the equator, is approximately 32.088 ft/sec2. In fact, a model for this motion can be found, experimentally, as the following example shows.

EXAMPLE 3 A steel ball is dropped from a height of 10 ft. The distance covered by the ball at intervals of one tenth of a second is measured and recorded in Table 2. A scatter plot of the data is shown in Figure 6. You can see from the figure that the points associated with the data do lie close to a parabola with equation for some constant , as was suggested earlier.a

yat2

Time (sec) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Distance (ft) 0 0.1608 0.6416 1.4444 2.5672 4.0108 5.7760 7.8614 TABLE 2

FIGURE 6

To find a mathematical model to describe this motion, we use the method of least squares to find a function of the form that fits the data “best.” We obtain the function

(See Exercises 3.7,Problems 73 and 74.) On the basis of this model, the ball will hit

the ground when . Solving the equation gives .

Rejecting the negative root, we conclude that the ball will hit the ground approximately 0.79 sec after it is dropped. Thus, a complete description of the mathematical model for this motion is

where Dis the distance covered by the ball after sec.t

0 t 0.79

D16.044t2

t0.7895 16.044t210

y10

y16.044t2 yat2

t (sec) y (ft)

2 4 8 6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Notes

1. Observe that even though the function is defined on , we need to restrict its domain to the interval to obtain a mathematical model for the motion of the ball. Once the ball reaches the ground, the function no longer describes its motion.

2. If you use the quadratic regression program that is found in most graphing calculators and in computers, you will find the quadratic model

which is not very satisfactory, since we know that when . Besides, as you will be able to confirm later, this model implies that the ball started out with an initial velocity of 0.00075 ft/sec. But we know that the steel ball had an initial velocity of 0 ft/sec.

A polynomial of degree three is called a cubic polynomial,one of degree four is called a quartic polynomial,and one of degree five is called a quinticpolynomial. In general, the higher the degree of the polynomial function, the more its graph wiggles.

Figure 7a–c shows the graph of a cubic, a quartic, and a quintic, respectively.

t0 D0

D16.0425t20.00075t0.000075

f [0, 0.79]

(⬁, ⬁) f(t)16.044t2

FIGURE 7

Cubic polynomials lend themselves to modeling some phenomena in business and economics. For example, let denote the total cost incurred when units of a certain commodity are produced. A typical graph of the function is shown in Figure 8.

As the level of production increases, the cost per unit drops, so increases but at a slower pace. However, a level of production is soon reached at which the cost per unit begins to increase dramatically (because of overtime, a shortage of raw materials, and breakdown of machinery due to excessive stress and strain), so continues to increase at a faster pace. The graph of a cubic polynomial can exhibit precisely the characteristics just described.

The following example shows how we can use a quartic function to describe the assets of the Social Security system.

C C x

C x C(x)

FIGURE 8

A total cost function is often modeled by using a cubic function.

x y

2 4 6

(a) y x3 x2 2x 2 (a cubic)

2 4

4 2 2 4 6

x y

5 10

(b) y x4 6x2 x 2 (a quartic)

2 4

4 2 5 10

x y

1000 2000

2 4 6

6 4 1000 2000

x C (x)

EXAMPLE 4 Social Security Trust Fund Assets The projected assets of the Social Security trust fund (in trillions of dollars) from 2008 through 2040 are given in Table 3. The scatter plot associated with these data is shown in Figure 9a, where corresponds to 2008. A mathematical model giving the approximate value of the assets in the trust fund (in trillions of dollars) in year is

A(t) 0.00000268t40.000356t30.00393t20.2514t2.4094 t

A(t)

a. The first baby boomers will turn 65 in 2011. What will the assets of the Social Security system trust fund be at that time? The last of the baby boomers will turn 65 in 2029. What will the assets of the trust fund be at that time?

b. Unless payroll taxes are increased significantly and/or benefits are scaled back dramatically, it is only a matter of time before the assets of the current system are depleted. Use the graph of the function to estimate the year in which the current Social Security system is projected to go broke.

Solution

a. The assets of the Social Security trust fund in 2011 will be

or approximately $3.19 trillion. The assets of the trust fund in 2029 will be

or approximately $5.60 trillion.

b. From Figure 9b we see that the graph of crosses the -axis at approximately . So unless the current system is changed, it is projected to go broke in 2040. (At this time the first of the baby boomers will be 94, and the last of the baby boomers will be 76.)

Note Observe that the model in Example 4 utilizes only a small portion of the graph of , as is often the case in practice. A more complete picture of the graph of is shown in Figure 10.

f f

t32

t A

0.00393(21)20.2514(21)2.40945.60 A(21) 0.00000268(21)40.000356(21)3

(t21) 0.00393(3)20.2514(3) 2.40943.19

A(3) 0.00000268(3)40.000356(3)3 (t3) A(t)

Year 2008 2011 2014 2017 2020 2023 2026 2029 2032 2035 2038 2040 Assets $2.4 $3.2 $4.0 $4.7 $5.3 $5.7 $5.9 $5.6 $4.9 $3.6 $1.7 0 TABLE 3

FIGURE 9

Source:Social Security Administration.

FIGURE 10

The graph of in the viewing window [40, 40][10, 10]

t (years)

(a) Scatter plot A(t) ($ trillion)

2 2 4 6

0 5 10 15 20 25 30 t (years)

(b) Graph of A A(t) ($ trillion)

2 2 4 6

0 5 10 15 20 25 30

40 40 10

The graph of Ais shown in Figure 9b.

Power Functions

A power functionis a function of the form , where is a real number. If is a nonnegative integer, then is just a polynomial function of degree with one term (a monomial). Examples of other power functions are

, , , and

whose graphs are shown in Figure 11.

f(x)x1>3 13x f(x)x1>2 1x

f(x)x1 1 f(x)x2 1 x

a f

a a

f(x)xa

FIGURE 11

The graphs of some power functions

Power functions serve as mathematical models in many fields of study. For example, according to Newton’s Law of Gravitation, the force exerted by a particle of mass on another particle of mass a distance away is directed toward and has magnitude

where is the universal gravitational constant. The graph of is similar to that of for (see Figure 12).

Rational Functions

A rational functionis a quotient of two polynomials. Examples of rational functions are

and In general, a rational function has the form

where and are polynomial functions. The domain of a rational function is the set of all real numbers except the zeros of , that is, the roots of the equation . Thus, the domain of is , and the domain of is . A mathematical model involving a rational function is suggested by the experiments conducted by A.J. Clark on the response R(x)of a frog’s heart muscle to the injection of units ofx

{xx 1}

{xx2} t f

Q(x)0 Q

Q P

f(x) P(x) Q(x)

t(x) x21 x21 f(x) 3x3x2x1

x2 x0

f(x)x2

F G

FGm1m2 r2

r m2

FIGURE 12

The magnitude of a gravitational force F

x y

1 2 3

(a) f(x) x2

x y

1 2 3 4

(b) f(x) x1 2

2 1

4 3

2 4 6 4

2 2 4

4 2 4

x y

1 2 3 4 5 6

5 10 15

x y

2 4 6

(d) f(x) x1/3

10 10 20 20

r F

acetylcholine (as a percentage of the maximum possible effect of the drug). His results show that has the form

where is a positive constant that depends on the particular frog (see Figure 13).

Algebraic Functions

Algebraic functions are functions that can be expressed as sums, differences, prod- ucts, quotients, or roots of polynomial functions. By definition, rational functions are algebraic functions. The function

is another example of an algebraic function. The following example from the special theory of relativity involves an algebraic function.

f(x)2x331x x23 x21 x(x 1x) b

x0 R(x) 100x

bx R

FIGURE 13

The graph of R(x) 100x bx

EXAMPLE 5 Special Theory of Relativity According to the special theory of relativity, the relativistic mass of a particle moving with a speed is

where is the rest mass (the mass at zero speed) and m/sec is the speed of light in a vacuum. What is the speed of a particle whose relativistic mass is twice that of its rest mass?

Solution We solve the equation

for , obtaining

or approximately 0.866 times the speed of light (approximately 2.596108m/sec).

√ 13 2 c

√2 c2 3

4 1 √2

c2 1 4 B1 √2

c2 1 2

2 1

B1 √2 c2

√

2m0 m0

B1 √2 c2

c2.9979108 m0

mf(√) m0 B1 √2

√

x R(x)

Trigonometric Functions

Trigonometric functions were reviewed in Section 0.3. The characteristics of the trigonometric functions make them suitable for modeling phenomena that exhibit cyclical, or almost cyclical, behavior such as the motion of sound waves, the vibration of strings, and the motion of a simple pendulum.

EXAMPLE 6 Average Temperature Table 4 gives the average monthly temperature in degrees Fahrenheit recorded in Boston.

Month Jan. Feb. March April May June July Aug. Sept. Oct. Nov. Dec.

Temp (°F) 28.6 30.3 38.6 48.1 58.2 67.7 73.5 71.9 64.8 54.8 45.3 33.6

TABLE 4

To find a model describing the average temperature in month , we assume that is a sine function with period 12 and amplitude given by . A possible model is

where t1corresponds to January. The graph of is shown in Figure 14.T T51.0522.45 sinCp6(t4.3)D

2(73.528.6)22.45 T t

Source: The Boston Globe.

FIGURE 14 A model of the average temperature in Boston is

T51.0522.45 sinCp6(t4.3)D t (months)

T (F)

20 40 60 80

10 30 50 70

0 2 4 6 8 10 12 14

Other functions, such as exponentialand logarithmicfunctions, also play an important role in modeling and will be studied in later chapters.

Constructing Mathematical Models

We close this section by showing how some mathematical models can be constructed by using elementary geometric and algebraic arguments.

The following guidelines can be used to construct mathematical models.

Guidelines for Constructing Mathematical Models

1. Assign a letter to each variable mentioned in the problem. If appropriate, draw and label a figure.

2. Find an expression for the quantity that is being sought.

3. Use the conditions given in the problem to write the quantity being sought as a function of one variable. Note any restrictions to be placed on the domain of from physical considerations of the problem.f

EXAMPLE 7 Enclosing an Area The owner of Rancho Los Feliz has 3000 yd of fencing with which to enclose a rectangular piece of grazing land along the straight portion of a river. Fencing is not required along the river. Letting denote the width of the rectangle, find a function in the variable giving the area of the grazing land if she uses all of the fencing (see Figure 15).

x f

FIGURE 15 The rectangular grazing land has width and length .x y

Solution

1. This information is given in the statement of the problem.

2. The area of the rectangular grazing land is . Next, observe that the amount of fencing is and that this must be equal to 3000, since all the fencing is to be used; that is,

3. From the equation we see that . Substituting this value of into the expression for gives

Finally, observe that both and must be positive, since they represent the width and length of a rectangle, respectively. Thus, and , but the lat- ter is equivalent to , or . So the required function is

with domain 0x1500. f(x)3000x2x2

x1500 30002x0

y0 x0

y x

Axyx(30002x)3000x2x2 A

y y30002x

2xy3000 2xy

Axy

EXAMPLE 8 Charter Flight Revenue If exactly 200 people sign up for a charter flight, Leisure World Travel Agency charges $300 per person. However, if more than 200 people sign up for the flight (assume that this is the case), then each fare is reduced by $1 for each additional person. Letting denote the number of passengers above 200, find a function giving the revenue realized by the company.

Solution

1. This information is given.

2. If there are passengers above 200, then the number of passengers signing up for the flight is . Furthermore, the fare will be dollars per passenger.

3. The revenue will be

number of passengers fare per passenger

Clearly, must be positive, and , or . So the required function is f(x) x2100x60,000with domain (0, 300).

x300 300x0

x2100x60,000

ⴢ

R(200x)(300x)

(300x) 200x