Textbook-Math121 OqD PDF

12280_FM_ptg01_hr_i-xxii.qk_12280_FM_ptg01_hr_i-xxii.qk 12/15/11 3:32 PM Page iii CONTENT S Preface ix To the Student xvi...

12280_FM_ptg01_hr_i-xxii.qk_12280_FM_ptg01_hr_i-xxii.qk 12/15/11 3:32 PM Page iii CONTENT S Preface ix To the Student xvi Diagnostic Tests xvii 1 FUNCTIONS AND LIMITS 1 1.1 Functions and Their Representations 1 1.2 A Catalog of Essential Functions 11 1.3 The Limit of a Function 24 1.4 Calculating Limits 35 1.5 Continuity 46 1.6 Limits Involving Infinity 56 Review 70 2 DERIVATIVES 73 2.1 Derivatives and Rates of Change 73 2.2 The Derivative as a Function 84 2.3 Basic Differentiation Formulas 95 2.4 The Product and Quotient Rules 107 2.5 The Chain Rule 114 2.6 Implicit Differentiation 123 2.7 Related Rates 128 2.8 Linear Approximations and Differentials 135 Review 140 3 INVERSE FUNCTIONS: Exponential, Logarithmic, and Inverse Trigonometric Functions 145 3.1 Exponential Functions 145 3.2 Inverse Functions and Logarithms 151 3.3 Derivatives of Logarithmic and Exponential Functions 163 12280_FM_ptg01_hr_i-xxii.qk_12280_FM_ptg01_hr_i-xxii.qk 12/15/11 3:32 PM Page iv iv CONTENTS 3.5 Inverse Trigonometric Functions 179 3.6 Hyperbolic Functions 184 3.7 Indeterminate Forms and l’Hospital’s Rule 191 Review 199 4 APPLICATIONS OF DIFFERENTIATION 203 4.1 Maximum and Minimum Values 203 4.2 The Mean Value Theorem 210 4.3 Derivatives and the Shapes of Graphs 216 4.4 Curve Sketching 225 4.5 Optimization Problems 231 4.6 Newton’s Method 242 4.7 Antiderivatives 247 Review 253 5 INTEGRALS 257 5.1 Areas and Distances 257 5.2 The Definite Integral 268 5.3 Evaluating Definite Integrals 281 5.4 The Fundamental Theorem of Calculus 291 5.5 The Substitution Rule 300 Review 308 6 TECHNIQUES OF INTEGRATION 311 6.1 Integration by Parts 311 6.2 Trigonometric Integrals and Substitutions 317 6.3 Partial Fractions 327 6.6 Improper Integrals 353 Review 362 12280_FM_ptg01_hr_i-xxii.qk_12280_FM_ptg01_hr_i-xxii.qk 12/15/11 3:32 PM Page v CONTENTS v 7 APPLICATIONS OF INTEGRATION 365 7.1 Areas Between Curves 365 7.2 Volumes 370 7.3 Volumes by Cylindrical Shells 381 7.4 Arc Length 386 7.5 Area of a Surface of Revolution 393 Review 421 12280_ch01_ptg01_hr_001-011.qk_12280_ch01_ptg01_hr_001-011 11/16/11 11:56 AM Page 1 1 FUNCTIONS AND LIMIT S Calculus is fundamentally different from the mathematics that you have studied previously. Calculus is less static and more dynamic. It is concerned with change and motion; it deals with quantities that approach other quantities. So in this first chapter we begin our study of calculus by investigating how the values of functions change and approach limits. 1.1 FUNCTIONS AND THEIR REPRESENTATIONS Functions arise whenever one quantity depends on another. Consider the following four situations. A. The area A of a circle depends on the radius r of the circle. The rule that connects r and A is given by the equation A ! ! r 2. With each positive number r there is associated one value of A, and we say that A is a function of r. B. The human population of the world P depends on the time t. The table gives esti- Population mates of the world population P!t" at time t, for certain years. For instance, Year (millions) 1900 1650 P!1950" # 2,560,000,000 1910 1750 But for each value of the time t there is a corresponding value of P, and we say that 1920 1860 P is a function of t. 1930 2070 1940 2300 C. The cost C of mailing an envelope depends on its weight w. Although there is no 1950 2560 simple formula that connects w and C, the post office has a rule for determining C 1960 3040 when w is known. 1970 3710 D. The vertical acceleration a of the ground as measured by a seismograph during an 1980 4450 earthquake is a function of the elapsed time t. Figure 1 shows a graph generated by 1990 5280 seismic activity during the Northridge earthquake that shook Los Angeles in 1994. 2000 6080 For a given value of t, the graph provides a corresponding value of a. 2010 6870 a {cm/s@} 100 50 5 10 15 20 25 30 t (seconds) FIGURE 1 _50 Vertical ground acceleration during the Northridge earthquake Calif. Dept. of Mines and Geology Each of these examples describes a rule whereby, given a number (r, t, w, or t), another number ( A, P, C, or a) is assigned. In each case we say that the second num- ber is a function of the first number. Unless otherwise noted, all content on this page is © Cengage Learning. 1 Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 12280_ch01_ptg01_hr_001-011.qk_12280_ch01_ptg01_hr_001-011 11/16/11 11:56 AM Page 2 2 CHAPTER 1 FUNCTIONS AND LIMITS A function f is a rule that assigns to each element x in a set D exactly one element, called f !x", in a set E. We usually consider functions for which the sets D and E are sets of real numbers. The set D is called the domain of the function. The number f !x" is the value of f at x and is read “ f of x.” The range of f is the set of all possible values of f !x" as x varies throughout the domain. A symbol that represents an arbitrary number in the domain of a function f is called an independent variable. A symbol that represents a number in the range of f is called a dependent variable. In Example A, for instance, r is the independent variable and A is the dependent variable. x f ƒ It’s helpful to think of a function as a machine (see Figure 2). If x is in the domain (input) (output) of the function f, then when x enters the machine, it’s accepted as an input and the machine produces an output f !x" according to the rule of the function. Thus we can FIGURE 2 think of the domain as the set of all possible inputs and the range as the set of all pos- Machine diagram for a function ƒ sible outputs. Another way to picture a function is by an arrow diagram as in Figure 3. Each x ƒ arrow connects an element of D to an element of E. The arrow indicates that f !x" is associated with x, f !a" is associated with a, and so on. a f(a) The most common method for visualizing a function is its graph. If f is a function with domain D, then its graph is the set of ordered pairs f #!x, f !x"" $ x ! D% D E (Notice that these are input-output pairs.) In other words, the graph of f consists of all FIGURE 3 points !x, y" in the coordinate plane such that y ! f !x" and x is in the domain of f. Arrow diagram for ƒ The graph of a function f gives us a useful picture of the behavior or “life history” of a function. Since the y-coordinate of any point !x, y" on the graph is y ! f !x", we can read the value of f !x" from the graph as being the height of the graph above the point x. (See Figure 4.) The graph of f also allows us to picture the domain of f on the x-axis and its range on the y-axis as in Figure 5. y { x, ƒ} y range y ! ƒ(x) ƒ f (2) f (1) 0 1 2 x x 0 x domain FIGURE 4 FIGURE 5 y EXAMPLE 1 The graph of a function f is shown in Figure 6. (a) Find the values of f !1" and f !5". (b) What are the domain and range of f ? 1 SOLUTION 0 1 x (a) We see from Figure 6 that the point !1, 3" lies on the graph of f , so the value of f at 1 is f !1" ! 3. (In other words, the point on the graph that lies above x ! 1 is 3 units above the x-axis.) When x ! 5, the graph lies about 0.7 unit below the x-axis, so we estimate that FIGURE 6 f !5" & !0.7. Unless otherwise noted, all content on this page is © Cengage Learning. Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 12280_ch01_ptg01_hr_001-011.qk_12280_ch01_ptg01_hr_001-011 11/16/11 11:56 AM Page 3 SECTION 1.1 FUNCTIONS AND THEIR REPRESENTATIONS 3 The notation for intervals is given (b) We see that f !x" is defined when 0 ! x ! 7, so the domain of f is the closed on Reference Page 3. The Reference interval #0, 7$. Notice that f takes on all values from "2 to 4, so the range of f is Pages are located at the back of the book. %y & "2 ! y ! 4' ! #"2, 4$ www.stewartcalculus.com REPRESENTATIONS OF FUNCTIONS See Additional Examples A, B. There are four possible ways to represent a function: verbally (by a description in words) visually (by a graph) numerically (by a table of values) algebraically (by an explicit formula) If a single function can be represented in all four ways, it is often useful to go from one representation to another to gain additional insight into the function. But certain functions are described more naturally by one method than by another. With this in mind, let’s reexamine the four situations that we considered at the beginning of this section. A. The most useful representation of the area of a circle as a function of its radius Population is probably the algebraic formula A!r" ! # r 2, though it is possible to compile a t (millions) table of values or to sketch a graph (half a parabola). Because a circle has to have 0 1650 & a positive radius, the domain is %r r $ 0' ! !0, %", and the range is also !0, %". 10 1750 B. We are given a description of the function in words: P!t" is the human population 20 1860 of the world at time t. Let’s measure t so that t ! 0 corresponds to the year 1900. 30 2070 The table of values of world population provides a convenient representation of this 40 2300 function. If we plot these values, we get the graph (called a scatter plot) in Figure 50 2560 7. It too is a useful representation; the graph allows us to absorb all the data at once. 60 3040 What about a formula? Of course, it’s impossible to devise an explicit formula that 70 3710 gives the exact human population P!t" at any time t. But it is possible to find an 80 4450 expression for a function that approximates P!t". In fact, we could use a graphing 90 5280 calculator with exponential regression capabilities to obtain the approximation 100 6080 110 6870 P!t" ( f !t" ! !1.43653 & 10 9 " ' !1.01395"t Figure 8 shows that it is a reasonably good “fit.” The function f is called a mathe- matical model for population growth. In other words, it is a function with an explicit formula that approximates the behavior of our given function. We will see, however, that the ideas of calculus can be applied to a table of values; an explicit formula is not necessary. P P 5x10' 5x10' 0 20 40 60 80 100 120 t 0 20 40 60 80 100 120 t FIGURE 7 Scatter plot of data points for population growth FIGURE 8 Graph of a mathematical model for population growth Unless otherwise noted, all content on this page is © Cengage Learning. Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 12280_ch01_ptg01_hr_001-011.qk_12280_ch01_ptg01_hr_001-011 11/16/11 11:56 AM Page 4 4 CHAPTER 1 FUNCTIONS AND LIMITS The function P is typical of the functions that arise whenever we attempt to apply calculus to the real world. We start with a verbal description of a function. A function defined by a table of values Then we may be able to construct a table of values of the function, perhaps from is called a tabular function. instrument readings in a scientific experiment. Even though we don’t have com- plete knowledge of the values of the function, we will see throughout the book that it is still possible to perform the operations of calculus on such a function. C. Again the function is described in words: Let C!w" be the cost of mailing a large w (ounces) C!w" (dollars) envelope with weight w. The rule that the US Postal Service used as of 2011 is as 0&w'1 0.88 follows: The cost is 88 cents for up to one ounce, plus 20 cents for each successive 1&w'2 1.08 ounce (or less) up to 13 ounces. The table of values shown in the margin is the most 2&w'3 1.28 convenient representation for this function, though it is possible to sketch a graph 3&w'4 1.48 (see Example 6). 4&w'5 1.68 D. The graph shown in Figure 1 is the most natural representation of the vertical accel- % % eration function a!t". It’s true that a table of values could be compiled, and it is even % % % % possible to devise an approximate formula. But everything a geologist needs to know—amplitudes and patterns—can be seen easily from the graph. (The same is true for the patterns seen in electrocardiograms of heart patients and polygraphs for lie-detection.) In the next example we sketch the graph of a function that is defined verbally. EXAMPLE 2 When you turn on a hot-water faucet, the temperature T of the water T depends on how long the water has been running. Draw a rough graph of T as a function of the time t that has elapsed since the faucet was turned on. SOLUTION The initial temperature of the running water is close to room tempera- ture because the water has been sitting in the pipes. When the water from the hot- water tank starts flowing from the faucet, T increases quickly. In the next phase, 0 t T is constant at the temperature of the heated water in the tank. When the tank is drained, T decreases to the temperature of the water supply. This enables us to make FIGURE 9 the rough sketch of T as a function of t in Figure 9. EXAMPLE 3 Find the domain of each function. 1 (a) f !x" ! sx " 2 (b) t!x" ! 2 x !x SOLUTION If a function is given by a formula (a) Because the square root of a negative number is not defined (as a real number), and the domain is not stated explicitly, the domain of f consists of all values of x such that x " 2 # 0. This is equivalent to the convention is that the domain is the x # !2, so the domain is the interval #!2, $". set of all numbers for which the formula makes sense and defines a real number. 1 1 (b) Since t!x" ! ! x2 ! x x!x ! 1" and division by 0 is not allowed, we see that t!x" is not defined when x ! 0 or % x ! 1. Thus the domain of t is $x x " 0, x " 1&, which could also be written in interval notation as !!$, 0" ! !0, 1" ! !1, $". The graph of a function is a curve in the xy-plane. But the question arises: Which curves in the xy-plane are graphs of functions? This is answered by the following test. THE VERTICAL LINE TEST A curve in the xy-plane is the graph of a function of x if and only if no vertical line intersects the curve more than once. Unless otherwise noted, all content on this page is © Cengage Learning. Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 12280_ch01_ptg01_hr_001-011.qk_12280_ch01_ptg01_hr_001-011 11/16/11 11:56 AM Page 5 SECTION 1.1 FUNCTIONS AND THEIR REPRESENTATIONS 5 The reason for the truth of the Vertical Line Test can be seen in Figure 10. If each vertical line x ! a intersects a curve only once, at !a, b", then exactly one functional value is defined by f !a" ! b. But if a line x ! a intersects the curve twice, at !a, b" and !a, c", then the curve can’t represent a function because a function can’t assign two different values to a. y y x=a x=a (a, c) (a, b) (a, b) 0 a x 0 a x FIGURE 10 PIECEWISE DEFINED FUNCTIONS The functions in the following three examples are defined by different formulas in dif- ferent parts of their domains. V EXAMPLE 4 A function f is defined by f !x" ! # 1 ! x if x " 1 x2 if x # 1 Evaluate f !0", f !1", and f !2" and sketch the graph. SOLUTION Remember that a function is a rule. For this particular function the rule is the following: First look at the value of the input x. If it happens that x " 1, then the value of f !x" is 1 ! x. On the other hand, if x # 1, then the value of f !x" is x 2. Since 0 " 1, we have f !0" ! 1 ! 0 ! 1. y Since 1 " 1, we have f !1" ! 1 ! 1 ! 0. Since 2 # 1, we have f !2" ! 2 2 ! 4. How do we draw the graph of f ? We observe that if x " 1, then f !x" ! 1 ! x, 1 so the part of the graph of f that lies to the left of the vertical line x ! 1 must coin- cide with the line y ! 1 ! x, which has slope !1 and y-intercept 1. If x # 1, then 1 x f !x" ! x 2, so the part of the graph of f that lies to the right of the line x ! 1 must coincide with the graph of y ! x 2, which is a parabola. This enables us to sketch the FIGURE 11 graph in Figure l1. The solid dot indicates that the point !1, 0" is included on the graph; the open dot indicates that the point !1, 1" is excluded from the graph. The next example of a piecewise defined function is the absolute value function. Recall that the absolute value of a number a, denoted by a , is the distance from a $ $ to 0 on the real number line. Distances are always positive or 0, so we have www.stewartcalculus.com $a$ $ 0 for every number a For a more extensive review of For example, absolute values, click on Review of Algebra. $3$ ! 3 $ !3 $ ! 3 $0$ ! 0 $ s2 ! 1 $ ! s2 ! 1 $3 ! %$ ! % ! 3 Unless otherwise noted, all content on this page is © Cengage Learning. Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 12280_ch01_ptg01_hr_001-011.qk_12280_ch01_ptg01_hr_001-011 11/16/11 11:56 AM Page 6 6 CHAPTER 1 FUNCTIONS AND LIMITS In general, we have !a! ! a if a ! 0 ! a ! ! "a if a # 0 y (Remember that if a is negative, then "a is positive.) y=| x | EXAMPLE 5 Sketch the graph of the absolute value function f "x# ! x. ! ! SOLUTION From the preceding discussion we know that 0 x !x! ! $ x "x if x ! 0 if x # 0 FIGURE 12 Using the same method as in Example 4, we see that the graph of f coincides with the line y ! x to the right of the y-axis and coincides with the line y ! "x to the left of the y-axis (see Figure 12). C 1.50 EXAMPLE 6 In Example C at the beginning of this section we considered the cost C"w# of mailing a large envelope with weight w. In effect, this is a piecewise defined function because, from the table of values on page 4, we have 1.00 0.88 if 0 # w $ 1 0.50 1.08 if 1 # w $ 2 C"w# ! 1.28 if 2 # w $ 3 1.48 if 3 # w $ 4 0 1 2 3 4 5 w The graph is shown in Figure 13. You can see why functions similar to this one are FIGURE 13 called step functions—they jump from one value to the next. SYMMETRY www.stewartcalculus.com If a function f satisfies f ""x# ! f "x# for every number x in its domain, then f is See Additional Examples C, D. called an even function. For instance, the function f "x# ! x 2 is even because f ""x# ! ""x#2 ! ""1#2 x 2 ! x 2 ! f "x# The geometric significance of an even function is that its graph is symmetric with respect to the y-axis (see Figure 14). This means that if we have plotted the graph of y y f(_x) ƒ _x ƒ 0 _x 0 x x x x FIGURE 14 An even function FIGURE 15 An odd function Unless otherwise noted, all content on this page is © Cengage Learning. Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 12280_ch01_ptg01_hr_001-011.qk_12280_ch01_ptg01_hr_001-011 11/16/11 11:56 AM Page 7 SECTION 1.1 FUNCTIONS AND THEIR REPRESENTATIONS 7 y f for x # 0, we obtain the entire graph simply by reflecting this portion about the 1 f y-axis. If f satisfies f !"x" ! "f !x" for every number x in its domain, then f is called an odd function. For example, the function f !x" ! x 3 is odd because _1 1 x f !"x" ! !"x"3 ! !"1"3 x 3 ! "x 3 ! "f !x" _1 The graph of an odd function is symmetric about the origin (see Figure 15 on page 6). If we already have the graph of f for x # 0, we can obtain the entire graph by rotat- (a) ing this portion through 180$ about the origin. y V EXAMPLE 7 Determine whether each of the following functions is even, odd, or 1 g neither even nor odd. (a) f !x" ! x 5 ! x (b) t!x" ! 1 " x 4 (c) h!x" ! 2x " x 2 1 x SOLUTION (a) f !"x" ! !"x"5 ! !"x" ! !"1"5x 5 ! !"x" ! "x 5 " x ! "!x 5 ! x" ! "f !x" (b) Therefore f is an odd function. y 1 h (b) t!"x" ! 1 " !"x"4 ! 1 " x 4 ! t!x" So t is even. 1 x (c) h!"x" ! 2!"x" " !"x"2 ! "2x " x 2 Since h!"x" " h!x" and h!"x" " "h!x", we conclude that h is neither even nor odd. (c) The graphs of the functions in Example 7 are shown in Figure 16. Notice that the FIGURE 16 graph of h is symmetric neither about the y-axis nor about the origin. INCREASING AND DECREASING FUNCTIONS y The graph shown in Figure 17 rises from A to B, falls from B to C, and rises again B D from C to D. The function f is said to be increasing on the interval #a, b$, decreasing y=ƒ on #b, c$, and increasing again on #c, d$. Notice that if x 1 and x 2 are any two numbers between a and b with x 1 % x 2, then f !x 1 " % f !x 2 ". We use this as the defining prop- C erty of an increasing function. f(x™) A f(x¡) A function f is called increasing on an interval I if 0 a x¡ x™ b c d x f !x 1 " % f !x 2 " whenever x 1 % x 2 in I FIGURE 17 It is called decreasing on I if f !x 1 " & f !x 2 " whenever x 1 % x 2 in I Unless otherwise noted, all content on this page is © Cengage Learning. Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 12280_ch01_ptg01_hr_001-011.qk_12280_ch01_ptg01_hr_001-011 11/16/11 11:56 AM Page 8 8 CHAPTER 1 FUNCTIONS AND LIMITS In the definition of an increasing function it is important to realize that the inequal- ity f !x 1 $ # f !x 2 $ must be satisfied for every pair of numbers x 1 and x 2 in I with x 1 # x 2. You can see from Figure 18 that the function f !x$ ! x 2 is decreasing on the inter- val !!", 0" and increasing on the interval #0, "$. y y=≈ 0 x FIGURE 18 1.1 EXERCISES 1. If f !x$ ! x $ s2 ! x and t!u$ ! u $ s2 ! u , is it true (f) State the domain and range of t. that f ! t? y 2. If x2 ! x g f !x$ ! and t!x$ ! x f x!1 2 is it true that f ! t? 0 2 x 3. The graph of a function f is given. (a) State the value of f !1$. (b) Estimate the value of f !!1$. (c) For what values of x is f !x$ ! 1? (d) Estimate the value of x such that f !x$ ! 0. 5–8 Determine whether the curve is the graph of a function (e) State the domain and range of f. of x. If it is, state the domain and range of the function. (f) On what interval is f increasing? 5. y 6. y y 1 1 0 1 x 0 1 x 1 0 1 x 7. y 8. y 4. The graphs of f and t are given. 1 1 (a) State the values of f !!4$ and t!3$. 0 0 1 x 1 x (b) For what values of x is f !x$ ! t!x$? (c) Estimate the solution of the equation f !x$ ! !1. (d) On what interval is f decreasing? (e) State the domain and range of f. Unless otherwise noted, all content on this page is © Cengage Learning. ; Graphing calculator or computer required CAS Computer algebra system required 1 Homework Hints at stewartcalculus.com Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 12280_ch01_ptg01_hr_001-011.qk_12280_ch01_ptg01_hr_001-011 11/16/11 11:56 AM Page 9 SECTION 1.1 FUNCTIONS AND THEIR REPRESENTATIONS 9 9. The graph shown gives the weight of a certain person as a in minutes since the plane has left the terminal, let x!t" be function of age. Describe in words how this person’s weight the horizontal distance traveled and y!t" be the altitude of varies over time. What do you think happened when this the plane. person was 30 years old? (a) Sketch a possible graph of x!t". (b) Sketch a possible graph of y!t". (c) Sketch a possible graph of the ground speed. 200 (d) Sketch a possible graph of the vertical velocity. Weight 150 19. If f !x" ! 3x 2 ! x " 2, find f !2", f !!2", f !a", f !!a", (pounds) 100 f !a " 1", 2 f !a", f !2a", f !a 2 ", [ f !a"] 2, and f !a " h". 50 20. A spherical balloon with radius r inches has volume V!r" ! 43 # r 3. Find a function that represents the amount of 0 10 20 30 40 50 60 70 Age air required to inflate the balloon from a radius of r inches (years) to a radius of r " 1 inches. 10. The graph shows the height of the water in a bathtub as a 21–24 Evaluate the difference quotient for the given function. function of time. Give a verbal description of what you Simplify your answer. think happened. f !3 " h" ! f !3" Height 21. f !x" ! 4 " 3x ! x 2, h (inches) f !a " h" ! f !a" 22. f !x" ! x 3, 15 h 10 1 f !x" ! f !a" 23. f !x" ! , 5 x x!a x"3 f !x" ! f !1" 0 24. f !x" ! , 5 10 15 Time x"1 x!1 (min) 11. You put some ice cubes in a glass, fill the glass with cold 25–29 Find the domain of the function. water, and then let the glass sit on a table. Describe how the temperature of the water changes as time passes. Then x"4 2x 3 ! 5 sketch a rough graph of the temperature of the water as a 25. f !x" ! 26. f !x" ! x2 ! 9 2 x "x!6 function of the elapsed time. 27. F! p" ! s2 ! sp 12. Sketch a rough graph of the number of hours of daylight as a function of the time of year. 28. t!t" ! s3 ! t ! s2 " t 13. Sketch a rough graph of the outdoor temperature as a func- 1 tion of time during a typical spring day. 29. h!x" ! s4 x 2 ! 5x 14. Sketch a rough graph of the market value of a new car as a function of time for a period of 20 years. Assume the car is 30. Find the domain and range and sketch the graph of the well maintained. function h!x" ! s4 ! x 2. 15. Sketch the graph of the amount of a particular brand of cof- fee sold by a store as a function of the price of the coffee. 31–42 Find the domain and sketch the graph of the function. 16. You place a frozen pie in an oven and bake it for an 31. f !x" ! 2 ! 0.4x 32. F!x" ! x 2 ! 2x " 1 hour. Then you take it out and let it cool before eating it. 4 ! t2 Describe how the temperature of the pie changes as time 33. f !t" ! 2t " t 2 34. H!t" ! passes. Then sketch a rough graph of the temperature of the 2!t pie as a function of time. 35. t!x" ! sx ! 5 36. F!x" ! 2x " 1 # # 17. A homeowner mows the lawn every Wednesday afternoon. Sketch a rough graph of the height of the grass as a function 37. G!x" ! 3x " x # # 38. t!x" ! x ! x # # of time over the course of a four-week period. x 18. An airplane takes off from an airport and lands an hour later at another airport, 400 miles away. If t represents the time 39. f !x" ! $ x"2 1!x if x $ 0 if x % 0 Unless otherwise noted, all content on this page is © Cengage Learning. Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 12280_ch01_ptg01_hr_001-011.qk_12280_ch01_ptg01_hr_001-011 11/16/11 11:56 AM Page 10 10 CHAPTER 1 FUNCTIONS AND LIMITS 40. f !x" ! & 3 ! 12 x 2x ! 5 if x # 2 if x $ 2 54. The functions in Example 6 and Exercises 52 and 53(a) are called step functions because their graphs look like stairs. Give two other examples of step functions that arise 41. f !x" ! & x"2 x2 if x # !1 if x $ !1 in everyday life. & 55–56 Graphs of f and t are shown. Decide whether each !1 if x # !1 function is even, odd, or neither. Explain your reasoning. 42. f !x" ! 3x " 2 if x % 1 % % 55. y 56. y 7 ! 2x if x & 1 g f f 43–46 Find an expression for the function whose graph is the given curve. x x g 43. The line segment joining the points !1, !3" and !5, 7" 44. The line segment joining the points !!5, 10" and !7, !10" 45. The bottom half of the parabola x " ! y ! 1"2 ! 0 46. The top half of the circle x 2 " ! y ! 2" 2 ! 4 57. (a) If the point !5, 3" is on the graph of an even function, what other point must also be on the graph? 47–51 Find a formula for the described function and state its (b) If the point !5, 3" is on the graph of an odd function, domain. what other point must also be on the graph? 47. A rectangle has perimeter 20 m. Express the area of the 58. A function f has domain #!5, 5$ and a portion of its graph rectangle as a function of the length of one of its sides. is shown. 48. A rectangle has area 16 m2. Express the perimeter of the (a) Complete the graph of f if it is known that f is even. rectangle as a function of the length of one of its sides. (b) Complete the graph of f if it is known that f is odd. 49. Express the area of an equilateral triangle as a function of y the length of a side. 50. Express the surface area of a cube as a function of its volume. 51. An open rectangular box with volume 2 m3 has a square base. Express the surface area of the box as a function of _5 0 5 x the length of a side of the base. 59–64 Determine whether f is even, odd, or neither. If you 52. A cell phone plan has a basic charge of $35 a month. The have a graphing calculator, use it to check your answer visually. plan includes 400 free minutes and charges 10 cents for x x2 59. f !x" ! 60. f !x" ! each additional minute of usage. Write the monthly cost C 2 x "1 4 x "1 as a function of the number x of minutes used and graph C x as a function of x for 0 # x # 600. 61. f !x" ! x"1 62. f !x" ! x x % % 53. In a certain country, income tax is assessed as follows. There is no tax on income up to $10,000. Any income over 63. f !x" ! 1 " 3x 2 ! x 4 64. f !x" ! 1 " 3x 3 ! x 5 $10,000 is taxed at a rate of 10%, up to an income of $20,000. Any income over $20,000 is taxed at 15%. 65. If f and t are both even functions, is f " t even? If f and t (a) Sketch the graph of the tax rate R as a function of the are both odd functions, is f " t odd? What if f is even and income I. t is odd? Justify your answers. (b) How much tax is assessed on an income of $14,000? On $26,000? 66. If f and t are both even functions, is the product ft even? If (c) Sketch the graph of the total assessed tax T as a function f and t are both odd functions, is ft odd? What if f is even of the income I. and t is odd? Justify your answers. Unless otherwise noted, all content on this page is © Cengage Learning. Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 12280_ch01_ptg01_hr_001-011.qk_12280_ch01_ptg01_hr_001-011 11/16/11 11:56 AM Page 11 SECTION 1.2 A CATALOG OF ESSENTIAL FUNCTIONS 11 1.2 A CATALOG OF ESSENTIAL FUNCTIONS In solving calculus problems you will find that it is helpful to be familiar with the graphs of some commonly occurring functions. These same basic functions are often used to model real-world phenomena, so we begin with a discussion of mathematical modeling. We also review briefly how to transform these functions by shifting, stretch- ing, and reflecting their graphs as well as how to combine pairs of functions by the standard arithmetic operations and by composition. MATHEMATICAL MODELING A mathematical model is a mathematical description (often by means of a function or an equation) of a real-world phenomenon such as the size of a population, the demand for a product, the speed of a falling object, the concentration of a product in a chemical reaction, the life expectancy of a person at birth, or the cost of emission reductions. The purpose of the model is to understand the phenomenon and perhaps to make predictions about future behavior. Figure 1 illustrates the process of mathematical modeling. Given a real-world prob- lem, our first task is to formulate a mathematical model by identifying and naming the independent and dependent variables and making assumptions that simplify the phe- nomenon enough to make it mathematically tractable. We use our knowledge of the physical situation and our mathematical skills to obtain equations that relate the vari- ables. In situations where there is no physical law to guide us, we may need to collect data (either from a library or the Internet or by conducting our own experiments) and examine the data in the form of a table in order to discern patterns. From this numeri- cal representation of a function we may wish to obtain a graphical representation by plotting the data. The graph might even suggest a suitable algebraic formula in some cases. Real-world Formulate Mathematical Solve Mathematical Interpret Real-world problem model conclusions predictions Test FIGURE 1 The modeling process The second stage is to apply the mathematics that we know (such as the calculus that will be developed throughout this book) to the mathematical model that we have formulated in order to derive mathematical conclusions. Then, in the third stage, we take those mathematical conclusions and interpret them as information about the origi- nal real-world phenomenon by way of offering explanations or making predictions. The final step is to test our predictions by checking against new real data. If the pre- dictions don’t compare well with reality, we need to refine our model or to formulate a new model and start the cycle again. A mathematical model is never a completely accurate representation of a physical situation—it is an idealization. A good model simplifies reality enough to permit mathematical calculations but is accurate enough to provide valuable conclusions. It is important to realize the limitations of the model. In the end, Mother Nature has the final say. There are many different types of functions that can be used to model relationships observed in the real world. In what follows, we discuss the behavior and graphs Unless otherwise noted, all content on this page is © Cengage Learning. Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 12280_ch01_ptg01_hr_012-021.qk_12280_ch01_ptg01_hr_012-021 11/16/11 12:01 PM Page 12 12 CHAPTER 1 FUNCTIONS AND LIMITS of these functions and give examples of situations appropriately modeled by such functions. Linear Models www.stewartcalculus.com When we say that y is a linear function of x, we mean that the graph of the function To review the coordinate geometry is a line, so we can use the slope-intercept form of the equation of a line to write a for- of lines, click on Review of Analytic mula for the function as Geometry. y ! f !x" ! mx ! b where m is the slope of the line and b is the y-intercept. A characteristic feature of linear functions is that they grow at a constant rate. For instance, Figure 2 shows a graph of the linear function f !x" ! 3x " 2 and a table of sample values. Notice that whenever x increases by 0.1, the value of f !x" increases by 0.3. So f !x" increases three times as fast as x. Thus the slope of the graph y ! 3x " 2, namely 3, can be interpreted as the rate of change of y with respect to x. y x f !x" ! 3x " 2 y=3x-2 1.0 1.0 1.1 1.3 1.2 1.6 0 x 1.3 1.9 1.4 2.2 _2 1.5 2.5 FIGURE 2 V EXAMPLE 1 (a) As dry air moves upward, it expands and cools. If the ground temperature is 20#C and the temperature at a height of 1 km is 10#C, express the temperature T (in °C) as a function of the height h (in kilometers), assuming that a linear model is appropriate. (b) Draw the graph of the function in part (a). What does the slope represent? (c) What is the temperature at a height of 2.5 km? SOLUTION (a) Because we are assuming that T is a linear function of h, we can write T ! mh ! b www.stewartcalculus.com We are given that T ! 20 when h ! 0, so See Additional Examples A, B. 20 ! m ! 0 ! b ! b In other words, the y-intercept is b ! 20. We are also given that T ! 10 when h ! 1, so 10 ! m ! 1 ! 20 The slope of the line is therefore m ! 10 " 20 ! "10 and the required linear func- tion is T ! "10h ! 20 Unless otherwise noted, all content on this page is © Cengage Learning. Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 12280_ch01_ptg01_h