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All Textbook Solutions for Calculus: Early Transcendentals

Find the area between a large loop and the enclosed small loop of the curve r = 1 + 2 cos 3.Find all points of intersection of the given curves. 37. r =sin , r = 1 sinFind all points of intersection of the given curves. 38. r = 1 + cos , r = 1 sinFind all points of intersection of the given curves. 39. r = 2 sin 2, r = 140EFind all points of intersection of the given curves. 41. r = sin , r = sin 242EThe points of intersection of the cardioid r = 1 + sin and the spiral loop r = 2 , /2 /2, cant be found exactly. Use a graphing device to find the approximate values of at which they intersect. Then use these values to estimate the area that lies inside both curves.When recording live performances, sound engineers often use a microphone with a cardioid pickup pattern because it suppresses noise from the audience. Suppose the microphone is placed 4 m from the front of the stage (as in the figure) and the boundary of the optimal pickup region is given by the cardioid r = 8 + 8 sin , where r is measured in meters and the microphone is at the pole. The musicians want to know the area they will have on stage within the optimal pickup range of the microphone. Answer their question.Find the exact length of the polar curve. 45. r = 2 cos , 0Find the exact length of the polar curve. 46. r = 5, 0 2Find the exact length of the polar curve. 47. r = 2, 0 2Find the exact length of the polar curve. 48. r = 2(1 + cos )Find the exact length of the curve. Use a graph to determine the parameter interval. 49. r = cos4(/4)Find the exact length of the curve. Use a graph to determine the parameter interval. 50. r = cos2(/2)Use a calculator to find the length of the curve correct to four decimal places. If necessary, graph the curve to determine the parameter interval. 51. One loop of the curve r = cos 252E53E54E(a) Use Formula 10.2.6 to show that the area of the surface generated by rotating the polar curve r = f() a b (where f is continuous and 0 a b ) about the polar axis is S=ab2rsinr2+(drd)2d (b) Use the formula in part (a) to find the surface area generated by rotating the lemniscate r2 = cos 2 about the polar axis.56EFind the vertex, focus, and directrix of the parabola and sketch its graph. 1. x2 = 6yFind the vertex, focus, and directrix of the parabola and sketch its graph. 2. 2y2 = 5xFind the vertex, focus, and directrix of the parabola and sketch its graph. 3. 2x = y24EFind the vertex, focus, and directrix of the parabola and sketch its graph. 5. (x + 2)2 = 8(y 3)6EFind the vertex, focus, and directrix of the parabola and sketch its graph. 7. y2 + 6y + 2x + 1 = 0Find the vertex, focus, and directrix of the parabola and sketch its graph. 8. 2x2 16x 3y + 38 = 0Find an equation of the parabola. Then find the focus and directrix. 9.Find an equation of the parabola. Then find the focus and directrix. 10.Find the vertices and foci of the ellipse and sketch its graph. 11 .x22+y24=112EFind the vertices and foci of the ellipse and sketch its graph. 13. x2 + 9y2 = 914EFind the vertices and foci of the ellipse and sketch its graph. 15. 9x2 18x + 4y2 = 27Find the vertices and foci of the ellipse and sketch its graph. 16. x2 + 3y2 + 2x 12y + 10 = 017E18E19E20E21E22E23E24EIdentify the type of conic section whose equation is given and find the vertices and foci. 25. 4x2 = y2 + 4Identify the type of conic section whose equation is given and find the vertices and foci. 26. 4x2 = y + 427E28E29EIdentify the type of conic section whose equation is given and find the vertices and foci. 30. x2 2x + 2y2 8y + 7 = 031EFind an equation for the conic that satisfies the given conditions. 32. Parabola, focus (0, 0), directrix y = 633E34E35E36E37EFind an equation for the conic that satisfies the given conditions. 38. Ellipse, foci (0, 2), vertices (0, 2)39E40E41E42E43E44E45E46E47E48EThe point in a lunar orbit nearest the surface of the moon is called perilune and the point farthest from the surface is called apolune. The Apollo 11 spacecraft was placed in an elliptical lunar orbit with perilune altitude 110 km and apolune altitude 314 km (above the moon). Find an equation of this ellipse if the radius of the moon is 1728 km and the center of the moon is at one focus.A cross-section of a parabolic reflector is shown in the figure. The bulb is located at the focus and the opening at the focus is 10cm. (a) Find an equation of the parabola. (b) Find the diameter of the opening |CD|, 11 cm from the vertex.The LORAN (LOng RAnge Navigation) radio navigation system was widely used until the 1990s when it was superseded by the GPS system. In the LORAN system, two radio stations located at A and B transmit simultaneous signals to a ship or an aircraft located at P. The on board computer converts the time difference in receiving these signals into a distance difference |PA| |PB|, and this, according to the definition of a hyperbola, locates the ship or aircraft on one branch of a hyperbola (see the figure). Suppose that station B is located 400 mi due east of station A on a coastline. A ship received the signal from B 1200 microseconds (s) before it received the signal from A. (a) Assuming that radio signals travel at a speed of 980 ft/s, find an equation of the hyperbola on which the ship lies. (b) If the ship is due north of B, how far off the coastline is the ship?Use the definition of a hyperbola to derive Equation 6 for a hyperbola with foci (c, 0) and vertices (a, 0).Show that the function defined by the upper branch of the hyperbola y2/a2 x2/b2 = 1 is concave upward.Find an equation for the ellipse with foci (1, 1) and (1, 1) and major axis of length 4.Determine the type of curve represented by the equation x2k+y2k16=1 in each of the following cases: (a) k 16 (b) 0 k 16 (c) k 0 (d) Show that all the curves in parts (a) and (b) have the same foci, no matter what the value of k is.56E57E58E59E60EFind the area of the region enclosed by the hyperbola x2/a2 y2/b2 = 1 and the vertical line through a focus.62EFind the centroid of the region enclosed by the x-axis and the Lop half of the ellipse 9x2 + 4y2 = 36.64E65ELet P(x1, y1) be a point on the hyperbola x2/a2 y2/b2 = 1 with foci F1 and F2 and let and be the angles between the lines PF1, PF2 and the hyperbola as shown in the figure. Prove that = . (This is the reflection property of the hyperbola. It shows that light aimed at a focus F2 of a hyperbolic mirror is reflected toward the other focus F1.)1E2E3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25EJupiter's orbit has eccentricity 0.048 and the length of the major axis is 1.56 109 km. Find a polar equation for the orbit.The orbit of Halleys comet, last seen in 1986 and due to return in 2061, is an ellipse with eccentricity 0.97 and one focus at the sun. The length of its major axis is 36.18 AU. [An astronomical unit (AU) is the mean distance between the earth and the sun, about 93 million miles.] Find a polar equation for the orbit of Halleys comet. What is the maximum distance from the comet to the sun?28E29E30E31E(a) What is a parametric curve? (b) How do you sketch a parametric curve?2RCC3RCC4RCC5RCC6RCC7RCC8RCC(a) What is the eccentricity of a conic section? (b) What can you say about the eccentricity if the conic section is an ellipse? A hyperbola? A parabola? (c) Write a polar equation for a conic section with eccentricity e and directrix x = d. What if the directrix is x = d? y = d? y = d?Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If the parametric curve x = f(t), y = g(t) satisfies g(1) = 0, then it has a horizontal tangent when t = 1.Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If x = f(t) and y = g(t) are twice differentiable, then d2ydx2=d2y/dt2d2x/dt2Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The length of the curve x = f(t), y = g(t), a t b, is ab[f(t)]2+[g(t)]2dtDetermine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If a point is represented by (x, y) in Cartesian coordinates (where x 0) and (r, ) in polar coordinates, then = tanl(y/x).5RQDetermine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The equations r = 2, x2 + y2 = 4, and x = 2 sin 3t, y = 2 cos 3t (0 r 2) all have the same graph.Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The parametric equations x = t2, y = t4 have the same graph as x= t3, y = t6.Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The graph of y2 = 2y + 3x is a parabola.Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. A tangent line to a parabola intersects the parabola only once.Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. A hyperbola never intersects its directrix.1RE2RE3RE4RE5RE6RE7RE8RE9RE10RE11RESketch the polar curve. 12. r = 3 + cos 313RE14RE15RE16RE17RE18REThe curve with polar equation r = (sin )/ is called a cochleoid. Use a graph of r as a function of in Cartesian coordinates to sketch the cochleoid by hand. Then graph it with a machine to check your sketch.20RE21RE22RE23RE24RE25RE26RE27RE28RE29RE30REFind the area enclosed by the curve r2 = 9 cos 5.32RE33RE34REFind the area of the region that lies inside both of the circles r = 2 sin and r = sin + cos .Find the area of the region that lies inside the curve r = 2 + cos 2 but outside the curve r = 2 + sin .37RE38RE39RE40RE41RE42RE43RE44RE45RE46RE47RE48RE49RE50RE51RE52RE53RE54RE55RE56REIn the figure the circle of radius a is stationary, and for every , the point P is the midpoint of the segment QR. The curve traced out by P for 0 is called the long-bow curve. Find parametric equations for this curve.A curve called the folium of Descartes is defined by the parametric equations x=3t1+t3y=3t21+t3 (a) Show that if (a, b) lies on the curve, then so does (b, a); that is, the curve is symmetric with respect to the line y = x. Where does the curve intersect this line? (b) Find the points on the curve where the tangent lines are horizontal or vertical. (c) Show that the line y = x 1 is a slant asymptote. (d) Sketch the curve. (e) Show that a Cartesian equation of this curve is x3 + y3 = 3xy. (f) Show that the polar equation can be written in the form r=3sectan1+tan3 (g) Find the area enclosed by the loop of this curve. (h) Show that the area of the loop is the same as the area that lies between the asymptote and the infinite branches of the curve. (Use a computer algebra system to evaluate the integral.)The outer circle in the figure has radius 1 and the centers of the interior circular arcs lie on the outer circle. Find the area of the shaded region.2P3PFour bugs are placed at the four corners of a square with side length a. The bugs crawl counterclockwise at the same speed and each bug crawls directly toward the next bug at all times. They approach the center of the square along spiral paths. (a) Find the polar equation of a bugs path assuming the pole is at the center of the square. (Use the fact that the line joining one bug to the next is tangent to the bugs path.) (b) Find the distance traveled by a bug by the time it meets the other bugs at the center.5PA circle C of radius 2r has its center at the origin. A circle of radius r rolls without slipping in the counterclockwise direction around C. A point P is located on a fixed radius of the rolling circle at a distance b from its center, 0 b r. [See parts (i) and (ii) of the figure.] Let L be the line from the center of C to the center of the rolling circle and let be the angle that L makes with the positive x-axis. (a) Using as a parameter, show that parametric equations of the path traced out by P are x=bcos3+3rcosy=bsin3+3rsin Note: If b = 0, the path is a circle of radius 3r; if b = r, the path is an epicycloid. The path traced out by P for 0 b r is called an epitrochoid. (b) Graph the curve for various values of b between 0 and r. (c) Show that an equilateral triangle can be inscribed in the epitrochoid and that its centroid is on the circle of radius b centered at the origin. Note: This is the principle of the Wankel rotary engine. When the equilateral triangle rotates with its vertices on the epitrochoid, its centroid sweeps out a circle whose center is at the center of the curve. (d) In most rotary engines the sides of the equilateral triangles are replaced by arcs of circles centered at the opposite vertices as in part (iii) of the figure. (Then the diameter of the rotor is constant.) Show that the rotor will fit in the epitrochoid if b32(23)r. (i) (ii) (iii)(a) What is a sequence? (b) What does it mean to say that limn an. = 8? (c) What does it mean to say that limn an = ?(a) What is a convergent sequence? Give two examples. (b) What is a divergent sequence? Give two examples.List the first five terms of the sequence. 3. an=2n2n+1List the first five terms of the sequence. 4. an=n21n2+1List the first five terms of the sequence. 5. an=(1)n15nList the first five terms of the sequence. 6. an=cosn2List the first five terms of the sequence. 7. an=1(n+1)!List the first five terms of the sequence. 8. an=(1)nnn!+1List the first five terms of the sequence. 9. a1 = 1, an+1 = 5an 3List the first five terms of the sequence. 10. a1 = 6, an+1=annList the first five terms of the sequence. 11. a1 = 2, an+1=an1+anList the first five terms of the sequence. 12. a1 = 2, a2 = 1, an + 1 = an an 1Find a formula for the general term an of the sequence, assuming that the pattern of the first few terms continues. 13. {12,14,16,18,110,...}Find a formula for the general term an of the sequence, assuming that the pattern of the first few terms continues. 14. {4,1,14,116,164,...}Find a formula for the general term an of the sequence, assuming that the pattern of the first few terms continues. 15. {3,2,43,89,1627,...}Find a formula for the general term an of the sequence, assuming that the pattern of the first few terms continues. 16. {5, 8, 11, 14, 17, . . .}Find a formula for the general term an of the sequence, assuming that the pattern of the first few terms continues. 17. {12,43,94,165,256,...}Find a formula for the general term an of the sequence, assuming that the pattern of the first few terms continues. 18. {1, 0, 1, 0, 1, 0, 1, 0, . . .}Calculate, to four decimal places, the first ten terms of the sequence and use them to plot the graph of the sequence by hand. Does the sequence appear to have a limit? If so, calculate it. If not, explain why. 19. an=3n1+6nCalculate, to four decimal places, the first ten terms of the sequence and use them to plot the graph of the sequence by hand. Does the sequence appear to have a limit? If so, calculate it. If not, explain why. 20. an=2+(1)nnCalculate, to four decimal places, the first ten terms of the sequence and use them to plot the graph of the sequence by hand. Does the sequence appear to have a limit? If so, calculate it. If not, explain why. 21. an=1+(12)nCalculate, to four decimal places, the first ten terms of the sequence and use them to plot the graph of the sequence by hand. Does the sequence appear to have a limit? If so, calculate it. If not, explain why. 22. an=1+10n9nDetermine whether sequence converges or diverges. If it converges, find the limit. 23. an=3+5n2n+n2Determine whether sequence converges or diverges. If it converges, find the limit. 24. an=3+5n21+nDetermine whether sequence converges or diverges. If it converges, find the limit. 25. an=n4n32nDetermine whether sequence converges or diverges. If it converges, find the limit. 26. an = 2 + (0.86)nDetermine whether sequence converges or diverges. If it converges, find the limit. 27. an = 3n7nDetermine whether sequence converges or diverges. If it converges, find the limit. 28. an=3nn+2Determine whether sequence converges or diverges. If it converges, find the limit. 29. an=e1/nDetermine whether sequence converges or diverges. If it converges, find the limit. 30. an=4n1+9nDetermine whether sequence converges or diverges. If it converges, find the limit. 31. an=1+4n21+n2Determine whether sequence converges or diverges. If it converges, find the limit. 32. an=cos(nn+1)Determine whether sequence converges or diverges. If it converges, find the limit. 33. an=n2n3+4nDetermine whether sequence converges or diverges. If it converges, find the limit. 34. an=e2n/(n+2)Determine whether sequence converges or diverges. If it converges, find the limit. 35. an=(1)n2nDetermine whether sequence converges or diverges. If it converges, find the limit. 36. an=(1)n+1nn+nDetermine whether sequence converges or diverges. If it converges, find the limit. 37. {(2n1)!(2n+1)!}Determine whether sequence converges or diverges. If it converges, find the limit. 38. {1nn1n2n}Determine whether sequence converges or diverges. If it converges, find the limit. 39. {sin n}Determine whether sequence converges or diverges. If it converges, find the limit. 40. an=tan1nnDetermine whether sequence converges or diverges. If it converges, find the limit. 41. {n2en}Determine whether sequence converges or diverges. If it converges, find the limit. 42. an = 1n(n + 1) 1n nDetermine whether sequence converges or diverges. If it converges, find the limit. 43. an=cos2n2nDetermine whether sequence converges or diverges. If it converges, find the limit. 44. an=21+3nnDetermine whether sequence converges or diverges. If it converges, find the limit. 45. an = n sin(1/n)Determine whether sequence converges or diverges. If it converges, find the limit. 46. an = 2ncos nDetermine whether sequence converges or diverges. If it converges, find the limit. 47. an=(1+2n)nDetermine whether sequence converges or diverges. If it converges, find the limit. 48. an=nnDetermine whether sequence converges or diverges. If it converges, find the limit. 49. an = 1n(2n2 + 1) 1n(n2 + 1)Determine whether sequence converges or diverges. If it converges, find the limit. 50. an=(1nn)2nDetermine whether sequence converges or diverges. If it converges, find the limit. 51. an = arctan(1n n)Determine whether sequence converges or diverges. If it converges, find the limit. 52. an=nn+1n+3Determine whether sequence converges or diverges. If it converges, find the limit. 53. {0, 1, 0, 0, 1, 0, 0, 0, 1, . . .}Determine whether sequence converges or diverges. If it converges, find the limit. 54. {11,13,12,14,13,15,14,16,...}Determine whether sequence converges or diverges. If it converges, find the limit. 55. an=n!2nDetermine whether sequence converges or diverges. If it converges, find the limit. 56. an=(3)nn!57E58E59E60E61EUse a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the Limit from the graph and then prove your guess. (See the margin note on page 699 for advice on graphing sequences.) 62. an=135(2n1)n!Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the Limit from the graph and then prove your guess. (See the margin note on page 699 for advice on graphing sequences.) 63. an=135(2n1)(2n)n(a) Determine whether the sequence defined as follows is convergent or divergent: a1 = 1 an + 1 = 4 an for n 1 (b) What happens if the first term is a1 = 2?If 1000 is invested at 6% interest, compounded annually, then after n years the investment is worth an = 1000(1.06)n dollars. (a) Find the first five terms of the sequence {an}. (b) Is the sequence convergent or divergent? Explain. 41550-11.1-66EIf you deposit 100 at the end of every month into an account that pays 3% interest per year compounded monthly, the amount of interest accumulated after n months is given by the sequence In=100(1.0025n10.0025n) (a) Find the first six terms of the sequence. (b) How much interest will you have earned after two years?67E68EFor what values of r is the sequence {nrn} convergent?(a) If {an} is convergent, show that limnan+1=limnan (b) A sequence {an} is defined by a1 = 1 and an+1 = 1/(1 + an) for n 1. Assuming that {an} is convergent, find its limit.Suppose you know that {an} is a decreasing sequence and all its terms lie between the numbers 5 and 8. Explain why the sequence has a limit. What can you say about the value of the limit?Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? 72. an = cos nDetermine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? 73. an=12n+3Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? 74. an=1n2+nDetermine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? 75. an = n(1)nDetermine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? 76. an=2+(1)nnDetermine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? 77. an = 3 2nenDetermine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? 78. an = n3 3n + 3Find the limit of the sequence {2,22,222,...}A sequence {an} is given by a1=2,an+1=2+an. (a) By induction or otherwise, show that {an} is increasing and bounded above by 3. Apply the Monotonic Sequence Theorem to show that limn an. exists. (b) Find limn an.Show that the sequence defined by a1=1an+1=31an is increasing and an 3 for all n. Deduce that {an} is convergent and find its limit.Show that the sequence defined by a1=2an+1=13an satisfies 0 an 2 and is decreasing. Deduce that the sequence is convergent and find its limit.(a) Fibonacci posed the following problem: Suppose that rabbits live forever and that every month each pair produces a new pair which becomes productive at age 2 months. If we start with one newborn pair, how many pairs of rabbits will we have in the nth month? Show that the answer is fn, where {fn} is the Fibonacci sequence defined in Example 3(c). (b) Let an = fn+1/fn and show that an 1 = 1 + 1/an 2. Assuming that {an} is convergent, find its limit.(a) Let a1 = a, a2 = f(a), a3 = f(a2) = f(f(a)), . . . , an + 1 = f(an), where f is a continuous function. If limn an = L, show that f(L) = L. (b) Illustrate part (a) by taking f(x) = cos x, a = 1, and estimating the value of L to five decimal places.(a) Use a graph to guess the value of the limit limnn5n! (b) Use a graph of the sequence in part (a) to find the smallest values of N that correspond to = 0.1 and = 0.001 in Definition 2.Use Definition 2 directly to prove that limn rn = 0 when |r| 1.Prove Theorem 6. [Hint: Use either Definition 2 or the Squeeze Theorem.]88EProve that if limn an = 0 and {bn} is bounded, then limn (anbn) = 0.Let an=(1+1n)n. (a) Show that if 0 a b, then bn+1an+1ba(n+1)bn (b) Deduce that bn[(n + 1)a nb] an+1 (c) Use a =1 + 1/(n + 1) and b = 1 + 1/n in part (b) to show that {an} is increasing (d) Use a = 1 and b = 1+1/(2n) in part (b) to show that a2n 4. (e) Use parts (c) and (d) to show that an 4 for all n. (f) Use Theorem 12 to show that limn (1 + 1/n)n exists. (The Limit is e. See Equation 3.6.6.)Let a and b be positive numbers with a b. Let a1 be their arithmetic mean and b1 their geometric mean: a1=a+b2b1=ab Repeat this process so that, in general, an+1=an+bn2bn+1=anbn (a) Use mathematical induction to show that anan+1bn+1bn (b) Deduce that both {an} and {bn} are convergent. (c) Show that limn an = limn bn. Gauss called the common value of these limits the arithmetic-geometric mean of the numbers a and b.(a) Show that if limn a2n = L and limn a2n+1 = L, then {an} is convergent and limn an = L. (b) If a1= 1 and an+1=1+11+an find the first eight terms of the sequence {an}. Then use part (a) to show that limnan=2. This gives the continued fraction expansion 2=1+12+12+The size of an undisturbed fish population has been modeled by the formula pn+1=bpna+pn where pn is the fish population after n years and a and b are positive constants that depend on the species and its environment. Suppose that the population in year 0 is P0 0. (a) Show that if {pn} is convergent, then the only possible values for its Limit are 0 and b a. (b) Show that Pn+1 (b/a)pn. (c) Use part (b) to show that if a b, then limn pn = 0; in other words, the population dies out. (d) Now assume that a b. Show that if P0 b a, then {pn} is increasing and 0 pn b a. Show also that if P0 b a, then {pn} is decreasing and pn b a. Deduce that if a b, then limn pn = b a.(a) What is the difference between a sequence and a series? (b) What is a convergent series? What is a divergent series?Explain what it means to say that n=1an=5.Calculate the sum of the seriesn=1an whose partial sums are given. 3. sn = 2 3(0.8)nCalculate the sum of the seriesn=1an whose partial sums are given. 4. sn=n214n2+1Calculate the first eight terms of the sequence of partial sums correct to four decimal places. Does it appear that the series is convergent or divergent? 5. n=11n4+n2Calculate the first eight terms of the sequence of partial sums correct to four decimal places. Does it appear that the series is convergent or divergent? 6. n=11n3
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