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Calculus early transcendentals 6th edition answers pdf free pdfWhat happens to the rate of cooling as time goes by? f !x" ! 51. Its graph is shown in Figure 22. Evaluate those that are convergent. C 0 1 2 x 21. We also saw that it arises when we try to find the distance traveled by an object. (b) A solid is generated by rotating about the x-axis the region under the curve y 苷 f 共x兲, where f is a positive function and x 0. b FIGURE FOR PROBLEM 5 (a) tan sin cos 苷 sec (b) 2 tan x 苷 sin 2x 1 tan 2x 8. SOLUTION The curves y 苷 x and y 苷 x 2 intersect at the points 共0, 0兲 and 共1, 1兲. M V EXAMPLE 2 Draw the graph of the function FIGURE 4 f !x" ! 10 3 2 y f ·(x) x 2 % 7x % 3 x2 in a viewing rectangle that contains all the important features of the function.What do all members of the family of linear functions f !x" ! 1 % m!x % 3" have in common? Find the volume of the resulting solid. We all have an intuitive idea of what the area of a region is. (b) Determine a way to “slice” the water into parallel cross-sections that are trapezoids and then set up a definite integral for the volume of the water. dy byinterpreting the integrals as areas. f !x" ! tan x 5 cos x 7. y sec t !sec t ' tan t" dt 17. Prove that xl0 2 1 68. y 4 2 1 dx ln x 64. From the graph in Problem 1, it appears that the tangent at P0 passes through P1 and the tangent at P3 passes through P2. What happens if you try to use l’Hospital’s Rule to evaluate lim xl! x sx 2 1 Evaluate the limit usinganother method. y 3 dt t4 26. h#x" ! 4 2 32. For instance, the function f 共x兲 苷 冑 x2 1 ln共cosh x兲 xe sin 2x x 3 2x 1 is an elementary function. (a) If y共0兲 苷 2 10 7 kg, find the biomass a year later. 1 tan y 1 tan d x 2 y 共x 2兲 3 兾3 y 兾4 dx stan d sin 2 60. 6 ft 8 ft m1 m2 r2 3 ft 10 ft 30. "2 ( x ( 2 Therefore the domain of f is the interval "2, 2,.Sketch the curve 共 x 2 y 2 兲3 苷 4 x 2 y 2. y 3 0 2 x 5 dx 30. y 9 30. The ends of the trough are equilateral triangles with sides 8 m long and vertex at the bottom. This is not surprising; we expect the spring to oscillate about its equilibrium position and so it is natural to think that trigonometric functions are involved. 484. Show that the equation 3x ! 2cos x ! 5 ! 0 has exactly one real root. y 苷 2 x 9 15. SOLUTION Here we integrate by parts with u 苷 sec x du 苷 sec x tan x dx dv 苷 sec 2x dx v 苷 tan x SECTION 7.2 TRIGONOMETRIC INTEGRALS 465 y sec x dx 苷 sec x tan x y sec x tan x dx 3 Then 2 苷 sec x tan x y sec x 共sec 2x 1兲 dx 苷 sec x tan x y sec 3x dx y sec x dx Using Formula 1 andsolving for the required integral, we get y sec x dx 苷 (sec x tan x ln sec x tan x ) C 1 2 3 M Integrals such as the one in the preceding example may seem very special but they occur frequently in applications of integration, as we will see in Chapter 8. The graph was the final object that we produced. We have found four points of 1 1 1 1 intersection: (2, 兾6), ( 2, 5 兾6), ( 2, 7 兾6), and ( 2, 11 兾6). y 苷 ln x, 4. If we know what the curve looks like for x , 0, then we need only reflect about the y-axis to obtain the complete curve [see Figure 3(a)]. N N 3 N xvi 4 N PREFACE Applications of Differentiation The basic facts concerning extreme values and shapes of curves are deduced from the Mean ValueTheorem. (c) Find f ! t. It is undefined whenever cos x ! 0, that is, when x ! &' 2, &3' 2, . (In a sense this is the inverse problem of the velocity problem that we discussed in Section 2.1.) If the velocity remains constant, then the distance problem is easy to solve by means of the formula distance ! velocity ( time But if the velocity varies, it’s not so easyto find the distance traveled. If we were to zoom in toward the point !a, 0", the graphs would start to look almost linear. If the fish are swimming against a current u !u ( v", then the time required to swim a distance L is L#!v ! u" and the total energy E required to swim the distance is given by L E!v" ! av 3 ! v!u where a is the proportionality constant.Since f &!x" 0 when x ' 0 !x " !1" and f &!x" ' 0 when x 0 !x " 1", f is increasing on !!", !1" and !!1, 0" and decreasing on !0, 1" and !1, "". But when the can is small or joining is costly, h#r should be substantially larger. y 苷 x2 4 x 2 2x 11. y 29. (a) Express the monthly cost C as a function of the distance driven d, assuming that a linearrelationship gives a suitable model. Estimate this value of d by graphical methods. (e) If the initial population is 1000, write a formula for the population after t years. If x*i is a point in the ith such interval, then all points in the interval are lifted by approximately the same amount, namely x*i . Illustrate the various possible shapes with graphs. (d) Thetrip from one station to the next takes 37.5 minutes. Use the Midpoint Rule with n 苷 5 to estimate the volume of the log. Computers are particularly good at matching patterns. f !x" ! x " 2, 1 x"4 t!x" ! sx 2 " 1 30. (b) Draw a direction field (either by hand or with a computer algebra system). You measure and find that a football is 28 cm long.Multiplying by the integrating factor we get or Therefore e x 2x dx 苷 e x 2 e x y 2xe x y 苷 e x 2 (e x y) 苷 e x 2 2 2 2 e x y 苷 y e x dx C 2 2 SECTION 9.5 LINEAR EQUATIONS Even though the solutions of the differential equation in Example 3 are expressed in terms of an integral, they can still be graphed by a computer algebra system (Figure 3). Abetter linear model is obtained by a procedure from statistics SECTION 1.2 MATHEMATICAL MODELS: A CATALOG OF ESSENTIAL FUNCTIONS A computer or graphing calculator finds the regression line by the method of least squares, which is to minimize the sum of the squares of the vertical distances between the data points and the line. y 苷 1x x 63. N xl0 SOLUTION Notice that this limit is indeterminate since 0 x ! 0 for any x ' 0 but x 0 ! 1 for any x " 0. 2 2 dx 0.7468241328 0.8820813908 0.8862073483 0.8862269118 0.8862269255 0.8862269255 EXAMPLE 10 The integral TA B L E 2 t 1 x t 2 5 10 100 1000 10000 7.8 because x 关共1 e 兲兾x兴 dx y 1 1 e x dx is divergent by theComparison Theorem x 1 e x 1 x x 0.8636306042 1.8276735512 2.5219648704 4.8245541204 7.1271392134 9.4297243064 and x1 共1兾x兲 dx is divergent by Example 1 [or by (2) with p 苷 1]. We will see later why the graphs have these shapes. r 苷 3共1 cos 兲 11. Even if no substitution is obvious (Step 2), some inspiration or ingenuity (or evendesperation) may suggest an appropriate substitution. A tank with a capacity of 400 L is full of a mixture of water and chlorine with a concentration of 0.05 g of chlorine per 608 CHAPTER 9 DIFFERENTIAL EQUATIONS liter. If a supply curve is modeled by the equation p 苷 200 0.2x 3 / 2, find the producer surplus when the selling price is 400.2 4 6 10 x 8 45. From this graph it appears that there is an absolute minimum value of about !15.33 when x !1.62 (by using the cursor) and f is decreasing on !!", !1.62" and increasing on !!1.62, "". Let A", # be the area between P B( ) A( ) R the chord PQ and the arc PQ. y 5 33. (b) Sketch the graph of f . Explain why Newton’s method doesn’t workfor finding the root of the equation x 3 ! 3x " 6 ! 0 if the initial approximation is chosen to be x 1 ! 1. A subtangent is a portion of the x-axis that lies directly beneath the segment of a tangent line from the point of contact to the x-axis. We will also learn how to find numerical approximations to solutions. The machine connects each point to thepreceding plotted point to form a representation of the graph of f . Evaluate lim y sin{ } 1 cos x dt , where t"x# ! y 1 " sin"t 2 #% dt, find f #"&!2#. 兾 2 55. In general, when we write n lim & f !x *" (x ! y n l ' i!1 i we replace lim , by x, x*i by x, and (x by dx. Vector fields are introduced in Section 16.1 by depictions of actual velocity vector fieldsshowing San Francisco Bay wind patterns. (a) Evaluate the Riemann sum for f "x# ! x ! x 2 4. (a) Sketch the graph of the ramp function y ! tH!t". In this section our point of view is completely different. If the region shown in the figure is rotated about the y-axis to form a solid, use the Midpoint Rule with n 苷 5 to estimate the volume of the solid. (b)Sketch the graph of the voltage V!t" in a circuit if the switch is turned on at time t ! 0 and the voltage is gradually increased to 120 volts over a 60-second time interval. 1 " cos x SOLUTION If we blindly attempted to use l’Hospital’s Rule, we would get lim " xl% sin x cos x ! lim " ! "! xl% 1 " cos x sin x This is wrong! Although the numerator sin x l 0 asx l % ", notice that the denominator !1 " cos x" does not approach 0, so l’Hospital’s Rule can’t be applied here. y 苷 tan 1 x 1 x 1 1. A cubic Bézier curve is determined by four control points, P0共x 0 , y0 兲, P1共x 1, y1 兲, P2共x 2 , y 2 兲, and P3共x 3 , y 3 兲, and is defined by the parametric equations x 苷 x0 共1 t兲3 3x1 t共1 t兲2 3x 2 t 2共1 t兲 x 3t 3 y 苷 y0 共1 t兲3 3y1 t共1 t兲2 3y 2 t 2共1 t兲 y 3 t 3 where 0 t 1. (a) f !x" ! sin x (b) f !x" ! 1 x (c) f !x" ! x n, n " "1 SOLUTION (a) If F!x" ! "cos x, then F#!x" ! sin x, so an antiderivative of sin x is "cos x. (ii) The rabbit runs up the y-axis and the dog always runs straight for the rabbit. Multiplying numerator and denominator by s1 x , we have y 冑 1 x1 x dx 苷 y dx 1 x s1 x 2 苷y 1 x dx y dx s1 x 2 s1 x 2 苷 sin 1x s1 x 2 C M CAN WE INTEGRATE ALL CONTINUOUS FUNCTIONS? Suppose that the position of one particle at time t is given by x 1 苷 3 sin t y1 苷 2 cos t 0 艋 t 艋 2 ; 47. (c) Solve the equation in part (a) using x 1 ! 0.57. (a) Explain why x0t f !s" ds represents the loss in value of themachine over the period of time t since the last overhaul. The graph of the velocity function of a particle is shown in f !0" ! 8 29. 2 2 25. Thus x 2x y 苷 2400 FIGURE 2 From this equation we have y 苷 2400 2x, which gives A 苷 x共2400 2x兲 苷 2400x 2x 2 Note that x 艌 0 and x 艋 1200 (otherwise A 0). Barbara weighs 60 kg and is on a diet of1600 calories per day, of which 850 are used automatically by basal metabolism. How large should n be to guarantee that the Simpson’s Rule 2 approximation to x01 e x dx is accurate to within 0.00001? (b) Find new estimates using ten rectangles in each case. The following figure shows the graphs of f, f #, and x y y 1"x 35. Find the correspondingamplitude A needed for the voltage E共t兲 苷 A sin共120 t兲. If we allow for this we would increase h#r. 3 DEFINITION OF AN IMPROPER INTEGRAL OF TYPE 2 (a) If f is continuous on 关a, b兲 and is discontinuous at b, then Parts (b) and (c) of Definition 3 are illustrated in Figures 8 and 9 for the case where f 共x兲 0 and f has vertical asymptotes at aand c, respectively. Show that a cubic function (a third-degree polynomial) always has exactly one point of inflection. xl0 SOLUTION First notice that as x l 0 , we have 1 sin 4x l 1 and cot x l !, so the given limit is indeterminate. 550 CHAPTER 8 FURTHER APPLICATIONS OF INTEGRATION D I S COV E RY PROJECT COMPLEMENTARY COFFEECUPS Suppose you have a choice of two coffee cups of the type shown, one that bends outward and one inward, and you notice that they have the same height and their shapes fit together snugly. y s25 t 9. Compare the average velocity with the maximum velocity. is a decreasing function on &0, 1', its absolute maximum value is M ! f !0" ! 1 and itsabsolute minimum value is m ! f !1" ! e%1. What is the largest possible total area of the four pens? y sin x cos x ln共sin x兲 dx 21. y ! ) * 2 x 1 22. (b) Prove that this resistance is minimized when cos ! 5 km r24 r14 (c) Find the optimal branching angle (correct to the nearest degree) when the radius of the smaller blood vessel is two-thirds the radiusof the larger vessel. Integration is not as straightforward as differentiation; there are no rules that absolutely guarantee obtaining an indefinite integral of a function. The base of S is the region enclosed by the parabola y 苷 1 x and the x-axis. I have also included a section on probability. f !x" ! cos x, f !0" ! 1, f #!0" ! 2, f )!0" ! 3 47. y 4 x 10 x dx 2x74. Likewise, we can take a circular cone with base radius r and slant height l , cut it along the dashed line in Figure 2, and flatten it to form a sector of a circle with radius l and central angle 苷 2 r兾l. Use Newton’s method to find the slope of that line correct to six decimal places. r 苷 3 cos 33. 404 CHAPTER 5 INTEGRALS This rule says thatwhen using a substitution in a definite integral, we must put everything in terms of the new variable u, not only x and dx but also the limits of integration. charge: I!t" ! Q#!t". But if six or eight decimal places are required, then repeated zooming becomes tiresome. (c) What is the y-intercept of the graph and what does it represent? For that reason itmay be useful to have an overview of the subject before beginning its intensive study. y 共x 5兲共x 2兲 dx 2 4 2 2 2 7–38 Evaluate the integral. Suppose we consider the cholesterol level of a person chosen at random from a certain age group, or the height of an adult female chosen at random, or the lifetime of a randomly chosen battery of a certaintype. r 苷 cos 2共 兾2兲 55. can be put in the form 2 . In particular, we would like to know the value of the limit 1 lim x l1 ln x x%1 In computing this limit we can’t apply Law 5 of limits (the limit of a quotient is the quotient of the limits, see Section 2.3) because the limit of the denominator is 0. As increases from to 3 兾2, r decreases from 1 to 0 asshown in part (c). This gives N This equation in x and y describes where the particle has been, but it doesn’t tell us when the particle was at a particular point. f "!x" # 0 and f !!x" # 0 for all x 1 30. This means that the graph of f lies entirely outside the viewing rectangle "2, 2, by "2, 2,. In Exercises 23 and 24 use the fact that water weighs 62.5 lb兾ft3. Also, 0 ( s8 " 2x 2 ( s8 ! 2s2 / 2.83 3 3 1 FIGURE 3 so the range of f is the interval [0, 2s2 ]. (a) Write an expression for a Riemann sum of a function f . The domain of t is !. 关0, 兾2兴 7. 200 cm 44. The geometry behind Newton’s method is shown in Figure 2, where the root that we are trying to find is labeled r. An arc PQ of a circle subtends acentral angle , as in the figure. Then we have a place to start, the point !0, 1", and the direction in which we move our pencil is given at each stage by the derivative F#!x" ! f !x". It is perhaps easier to remember in the following notation. 13–20 Show how to approximate the required work by a Riemann sum. f 共x兲 苷 sin 4x cos 6x, 48. N A!y 1 0 x3 x dx! 3 2 ' 1 0 ! 13 03 1 ! ! 3 3 3 M If you compare the calculation in Example 6 with the one in Example 2 in Section 5.1, you will see that the Fundamental Theorem gives a much shorter method. So for each additional unit sold, the decrease in price will be 201 10 and the demand function is 1 p!x" ! 350 ! 10 20 !x ! 200" ! 450 ! 2 x The revenue functionis R!x" ! xp!x" ! 450x ! 12 x 2 Since R*!x" ! 450 ! x, we see that R*!x" ! 0 when x ! 450. Comparison with Figure 11 in Section 9.2 shows that we were able to draw a fairly accurate solution curve from the direction field. So instead of x, let’s use t to represent time, in minutes. It too is a useful representation; the graph allows us to absorb all the data atonce. (b) The probability that X lies between 4 and 8 is 8 8 P共4 艋 X 艋 8兲 苷 y f 共x兲 dx 苷 0.006 y 共10x x 2 兲 dx 4 4 [ 苷 0.006 5x x 1 3 2 ] 3 8 4 苷 0.544 M V EXAMPLE 2 Phenomena such as waiting times and equipment failure times are commonly modeled by exponentially decreasing probability density functions. If, for a particular drug, A !0.01, p ! 4, k ! 0.07, and t is measured in minutes, estimate the times corresponding to the inflection points and explain their significance. Discuss the asymptotic behavior of f !x" ! !x 4 % 1"#x in the same manner as in Exercise 70. Use Newton’s method to find x accurate to four decimal places. 共0, 0兲 y 苷 sin t sin 2t ; 共 1, 1兲 2 27. If L is thelargest length for a species, then the hypothesis is that the rate of growth in length is proportional to L L , the length yet to be achieved. If water drains through a hole with area a at the bottom of the tank, then Torricelli’s Law says that 1 dV 苷 a s2th dt where t is the acceleration due to gravity. 2 N when x ! 4, u ! 2"4# " 1 ! 9 ! 13 "9 3!2 ! 13!2 #! 263 y FIGURE 2 9 1 2 1 N 3 and 4 x EXAMPLE 8 Evaluate y 0 y 1 2 1 œ„ u 2 9 u dx . Find all curves with the property that if the normal line is drawn at any point P on the curve, then the part of the normal line between P and the x-axis is bisected by the y-axis. y& sin x s2 dx ' x 2 54. y * sx ! 1 * dx &!4 0 4 dx "1 " tan t#3 sec2t dt 4 0 cos x y s1 " sin xdx 40. We highly encourage our visitors to purchase original books from the respected publishers. We illustrate the method on the initial-value problem that we used to introduce direction fields: 1.5 0.5 1 x FIGURE 13 Euler approximation with step size 0.5 y共0兲 苷 1 The differential equation tells us that y 共0兲 苷 0 1 苷 1, so the solution curve hasslope 1 at the point 共0, 1兲. V EXAMPLE 2 Evaluate y x 2 2x 1 dx. 8.3 APPLICATIONS TO PHYSICS AND ENGINEERING Among the many applications of integral calculus to physics and engineering, we consider two here: force due to water pressure and centers of mass. If we look more closely at some real cans, we see that the lid and the base areformed from discs with radius larger than r that are bent over the ends of the can. (a) (b) y (c) y y 3 1 2 2. If we use the Quotient Rule again and simplify, we get f #!x" ! ! 2 cos x !1 ! sin x" !2 % sin x" 3 Because !2 % sin x" 3 0 and 1 ! sin x ) 0 for all x , we know that f #!x" 0 when cos x ' 0, that is, (#2 ' x ' 3(#2. y 苷 x y 1 1. Note that the curvehas vertical tangents at M !0, 0" and !6, 0" because f "!x" l & as x l 0 and as x l 6. (See Example 7.) 3 4 (c) Graph the functions y ! sx , y ! s x, y ! s x , and 5 y ! sx on the same screen using the viewing rectangle !!1, 3" by !!1, 2". N 500 The expression for s!t" is valid until the ball hits the ground. In much the same way, this chapter starts with the areaand distance problems and uses them to formulate the idea of a definite integral, which is the basic concept of integral calculus. y 苷 2x, y 苷 x 2; 27. y 苷 x共 y 2 4兲 17. (c) On what intervals is t concave downward? A heavy rope, 50 ft long, weighs 0.5 lb兾ft and hangs over the 30 20 10 0 9. 3 t!x" dx with six sub6. (b) Shift 3 units downward.State the domain and range of f . x 0 e sx 29. evaluate x13 !2e x % 1" dx. It’s possible to express f as a sum of simpler fractions provided that the degree of P is less than the degree of Q. If the integrand contains even powers of both sine and cosine, this strategy fails. In the following table we list trigonometric substitutions that are effective for thegiven radical expressions because of the specified trigonometric identities. (a) On what intervals is C increasing? In Section 5.1 we guessed that this was true for the case where the object moves in the positive direction, but now we have proved that it is always true. Directional derivatives are introduced in Section 14.6 by using a temperature contourmap to estimate the rate of change of temperature at Reno in the direction of Las Vegas. sin x ' ! 12 &? y cos x dx 4. y ! x s2 ! x 3 28. Use this fact to help sketch the conchoid. At another point B on the ceiling, at a distance d from C (where d * r), a rope of length ! is attached and passed through the pulley at F and connected to a weight W. ye 2.Unfortunately, it is usually impossible to find explicit formulas for R and W as functions of t. 160 hare 120 9 lynx Thousands 80 of hares 6 Thousands of lynx 40 3 FIGURE 6 Relative abundance of hare and lynx from Hudson’s Bay Company records 0 1850 1875 1900 1925 Although the relatively simple Lotka-Volterra model has had some success inexplaining and predicting coupled populations, more sophisticated models have also been proposed. Which would you expect to give the best estimate? 1 5 7. Use the given graphs of f and t to estimate the value of f ! t!x"" for x ! "5, "4, "3, . , x n and equal width x. Comment on any discrepancy. Year Height (ft) Year Height (ft) 1900 1904 1908 19121920 1924 1928 1932 1936 1948 1952 10.83 11.48 12.17 12.96 13.42 12.96 13.77 14.15 14.27 14.10 14.92 1956 1960 1964 1968 1972 1976 1980 1984 1988 1992 1996 14.96 15.42 16.73 17.71 18.04 18.04 18.96 18.85 19.77 19.02 19.42 (a) Make a scatter plot and decide whether a linear model is appropriate. If f is even, then f # is even. Use thedata in the table to model the population of the world in the 20th century by a cubic function. Two new workers were hired for an assembly line. Then from Figure 14 we see that Q yy 苷 TC QC 苷 r r cos 苷 r 共1 cos 兲 x T O x 苷 OT PQ 苷 r r sin 苷 r共 sin 兲 x r FIGURE 14 Therefore parametric equations of the cycloid are 1 x 苷 r共 sin 兲 y 苷 r共1 cos 兲 僆 One arch of the cycloid comes from one rotation of the circle and so is described by 0 艋 艋 2. y ( x ! 2 ( x () dx 44. Round the answer to four decimal places. Find the point on the hyperbola x y ! 8 that is closest to the point !3, 0". Illustrate by sketching the supply and demand curves and identifying the surpluses asareas. A crosssection perpendicular to the x-axis at a distance x from the origin is a triangle ABC, as shown in Figure 17, whose base is y 苷 s16 x 2 and whose height is BC 苷 y tan 30 苷 s16 x 2 兾s3 . y x共x 1兲 dx 48. Sketch the graph and the rectangles. (a) Use Exercise 83 to show that the angle between the tan- ; 78. Let I 苷 (Round youranswers to six decimal places.) Compare your results to the actual value to determine the error in each approximation. (a) Find the values of f !1" and f !5". Also, there appears to be a horizontal tangent at the origin and inflection points when x ! 0 and when x is somewhere between !2 and !1. If the circle C rolls on the outside of the fixed circle, thecurve traced out by P is called an epicycloid. (b) We use the Chain Rule to eliminate t: dW dW dR 苷 dt dR dt so dW dW dt 0.02W 0.00002RW 苷 苷 dR dR 0.08R 0.001RW dt 610 CHAPTER 9 DIFFERENTIAL EQUATIONS (c) If we think of W as a function of R, we have the differential equation dW 0.02W 0.00002RW 苷 dR 0.08R 0.001RWWe draw the direction field for this differential equation in Figure 1 and we use it to sketch several solution curves in Figure 2. In Exercise 14 in Section 9.1 we considered a 95 C cup of cof- fee in a 20 C room. lim xl0 x2 e 4x " 1 " 4x 10. We computed that the surface area was 苷 N (17 s17 5 s5 ) 30.85 6 4 y 17 5 su du 苷 4 [ 2 3 u 3兾2 ] 17 5 (17s17 5s5 ) 6 SOLUTION 2 Using x 苷 sy we have which seems reasonable. Or, since 共d兾dx兲 sec x 苷 sec x tan x, we can separate a sec x tan x factor and convert the remaining (even) power of tangent to secant. If a decimal approximation is desired, we can use a calculator to approximate tan!1 2. The material for the cans is cut from sheets ofmetal. The positive root of sin x ! x 2 16. Use Newton’s method to find all roots of the equation sin x ! x 2 " 3x ! 1 correct to six decimal places. We see that the logistic model provides a very good fit. The circuit also contains a resistor with a resistance of R ohms ( ) and an inductor with an inductance of L henries (H). (a) Show that cos!x 2" & cos x for0 % x % 1. A ball with diameter 10 cm is placed in the bowl and water is poured into the bowl to a depth of h centimeters. In order to define its surface area, we divide the interval 关a, b兴 into n subintervals with endpoints x0, x1, . y 苷 x 2 y 2 1 3 0 1 0 3 4 1 0 1 4 . Again we can obtain the complete curve if we know what it looks like for x , 0.(b) What role does - play in the shape of the curve? (See Figure 3.) y y y ƒ y ƒ 0 FIGURE 3 a b x 0 a b x 434 CHAPTER 6 APPLICATIONS OF INTEGRATION y We divide the interval 关a, b兴 into n subintervals 关x i 1, x i 兴 of equal width x and let x i be the midpoint of the ith subinterval. y x cos 5x dx y re 2 r兾2 dr sin x dx 7. If we were to zoom intoo far, however, we would get an inaccurate graph because tan x is close to x when x is small. y y y 2x ; 47–50 Evaluate the indefinite integral. What nitrogen level gives the best yield? [Recall that f 共x兲 measures how fast the slope of y 苷 f 共x兲 changes.] 3 ERROR BOUNDS Suppose f 共x兲 K for a x b. Use the properties of integrals and theresult of Example 3 to 63. r 2 r 2 2 1 0 13. rectangles to find a lower estimate and an upper estimate for the area under the given graph of f from x ! 0 to x ! 10. From Figure 14(a) we see that there is only one solution and it lies between 0 and 1. Comment on how well your models fit the data. If f is a function with domain D, then its graph is the setof ordered pairs % !x, f !x"" x ! D& E (Notice that these are input-output pairs.) In other words, the graph of f consists of all points !x, y" in the coordinate plane such that y ! f !x" and x is in the domain of f . How does the graph change when P0 varies? If f and t are increasing on an interval I , then f " t is increasing on I . [Hint: The TI-83’s graphingwindow is 95 pixels wide. Most of them should be memorized. The graph of a function f is given at the left. Concavity and Points of Inflection Compute f )!x" and use the Concavity Test. y ! sin x 36. Sketch the graph of a continuous function on 关0, 2兴 for which the Trapezoidal Rule with n 苷 2 is more accurate than the Midpoint Rule. P 10,100 10,0009,900 9,800 P 9840 350 1 2.05e 0.48(t-1990) FIGURE 5 Logistic model for the population of Belgium 0 1980 1984 1988 1992 1996 2000 t OTHER MODELS FOR POPULATION GROWTH The Law of Natural Growth and the logistic differential equation are not the only equations that have been proposed to model population growth. Let’s modifythose equations as follows: dR 苷 0.08R共1 0.0002R兲 0.001RW dt CAS with a Lotka-Volterra system. r 苷 tan sec 21–26 Find a polar equation for the curve represented by the given Cartesian equation. (Think of slicing a loaf of bread.) If we choose sample points x*i in 关x i 1, x i 兴, we can approximate the ith slab Si (the part of S that liesbetween the planes Px i 1 and Px i ) by a cylinder with base area A共x*i 兲 and “height” x. For motion close to the ground we may assume that t is constant, its value being about 9.8 m s2 (or 32 ft s2 ). Euler’s idea was to improve on this approximation by proceeding only a short distance along this tangent line and then making a midcoursecorrection by changing direction as indicated by the direction field. 2 about BC 27. (a) What is the average value of a function f on an 3. y 36. Therefore we define the area A of the region S in the following way. (e) If the initial charge is Q共0兲 苷 0 C, use Euler’s method with step size 0.1 to estimate the charge after half a second. f !x" ! 1 !1 ! x 2 "2 cx 2 2 cx 1 c 2x 2 33. y (sx 3 ! 12 x 3 ' 14 x ! 2) dx 8. (a) Show that 1 % s1 " x 3 % 1 " x 3 for x & 0. x 苷 5 y , x 苷 0, 41. (b) Express the distance s between the plane and the radar station as a function of d. m¡ 40 m 30 2 5 21. For instance, in Example 6 we need only have plotted points for 0 兾2 and then reflected about the polar axis to obtainthe complete circle. You can see why functions similar to this one are called step functions—they jump from one value to the next. (See also Exercise 35 in Section 7.3.) Let be the region, illustrated in Figure 2, bounded by the polar curve r 苷 f 共 兲 and by the rays 苷 a and 苷 b, where f is a positive continuous function and where 0 b a 2 . y sx dx 2 16 11. 共1 tan y兲y 苷 x 2 1 1 sr du 6. What features do the curves have in common? When x 苷 0, u 苷 a; when x 苷 a, u 苷 0. 1 1.7 1 1 1.9 1.9 1 1.7 FIGURE 16 FIGURE 17 [email protected](2.4 ) cos (2.4 ) [email protected](1.2 ) cos#(6 ) Some graphing devices have commands that enable us to graph polar curves directly. We conclude,from (3) and the Squeeze Theorem, that SECTION 5.3 THE FUNDAMENTAL THEOREM OF CALCULUS 4 t !x" ! lim hl0 383 t!x # h" % t!x" ! f !x" h If x ! a or b, then Equation 4 can be interpreted as a one-sided limit. (b) Use calculus to find the production level for maximum profit. y 苷 1兾x, y 苷 0, x 苷 1, x 苷 3; about y 苷 1 15. % % EXAMPLE 8Use the first and second derivatives of f !x" ! e 1#x, together with asymp- totes, to sketch its graph. The base of S is the same base as in Exercise 60, but cross- sections perpendicular to the x-axis are isosceles triangles with height equal to the base. If a constant load W is distributed evenly along its length, the beam takes the shape of the deflectioncurve 20. (b) Give another estimate using the velocities at the end of the time periods. r 苷 8 cos 2 , r苷2 2 r 苷 1 cos 28. N AREAS AND DISTANCES In this section we discover that in trying to find the area under a curve or the distance traveled by a car, we end up with the same special type of limit. derive the following reciprocal algorithm: x n"1 ! 2xn ! ax n2 (This algorithm enables a computer to find reciprocals without actually dividing.) (b) Use part (a) to compute 1#1.6984 correct to six decimal places. Let b be the number of hours before noon that it began to snow. 53. Find the exact values. (b) What is the meaning of the area of the shaded region? y x sin 1 e 2x 2 x arcsin x dx s1 x 2 dx x ln xdx sx 2 1 y dx x 1 xe x x 2 4兲 dx dx 1 68. y (mg/ L) 6 4 2 Ax 1 k dx 0 2 4 6 8 10 12 14 t (seconds) SECTION 8.5 PROBABILITY 8.5 555 PROBABILITY Calculus plays a role in the analysis of random behavior. (This means that the values of ln x become very large negative as x approaches 0.) y x y ln x 0 ln 5 ( 0.773976 ln 8 x EXAMPLE 11Sketch the graph of the function y ! ln!x ! 2" ! 1. We do not use the Quotient Rule. y 苷 x 5兾3 5x 2兾3 3 x2 1 29. But the sums in (2) are Riemann sums for the function t共 兲 苷 12 关 f 共 兲兴 2, so O n lim FIGURE 3 兺 n l i苷1 1 2 关 f 共 i*兲兴 2

Calculus early transcendentals 6th edition answers pdf free pdf What happens to the rate of cooling as time goes by? f !x" ! 51. Its graph is shown in Figure 22. Evaluate those that are convergent. C 0 1 2 x 21. We also saw that it arises when we try to find the distance traveled by an object. (b) A solid is generated by rotating about the x .