Bartleby Sitemap - Textbook Solutions
All Textbook Solutions for Calculus (MindTap Course List)
20E21E22E23E24E25E2530 Identify the type of conic section whose equation is given and find the vertices and foci. 4x2=y+427E28E29E30E31E32E33E3148 Find an equation for the conic that satisfies the given conditions. Parabola, focus (2,1), vertex (2,3)35E36E37E38E39E40E41E42E43E44E45E46E47E48EThe 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 blub is located at the focus and the opening at the focus is 10 cm. 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cm from the vertex.The LORAN LOng RAnge Navigation radio navigation system was widely used until the 1990s when it was superseded by the GPS system.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 onboard 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).53E54E55E56E57E58E59E60E61E62E63Ea Calculate the surface area of the ellipsoid that is generated by rotating an ellipse about its major axis. b What is the surface area if the ellipse is rotated about its minor axis?Let P(x1,y1) be a point on the ellipse x2/a2+y2/b2=1 with foci F1 and F2 and let and be the angles between the lines PF1, PF2 and the ellipse as shown in the figure. Prove that =. This explains how whispering galleries and lithotripsy work. Sound coming from one focus is reflected and passes through the other focus. Hint: Use the formula in Problem 17 on page 201 to show that tan =tan.Let P(x1,y1) be a point on the hyperbola x2/a2y2/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 F1.1E2E3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26EThe 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?28E29E30E31Ea What is a parametric curve? b How do you sketch a parametric curve?2CC3CC4CC5CC6CC7CCa Give a definition of a hyperbola in terms of foci. b Write an equation for the hyperbola with foci (c,0) and vertices (a,0). c Write equations for the asymptotes of the hyperbola in part b.9CC1TFQ2TFQ3TFQ4TFQ5TFQ6TFQ7TFQ8TFQDetermine 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.10TFQ1E2E3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28EAt what points does the curve x=2acostacos2ty=2asintasin2t have vertical or horizontal tangents? Use this information to help sketch the curve.30EFind the area enclosed by the curve r2=9cos5.32E33E34E35EFind the area of the region that lies inside the curve r=2+cos2 but outside the curve r=2+sin.3740 Find the length of the curve. x=3t2,y=2t3,0t238E3740 Find the length of the curve. r=1/,240E4142 Find the area of the surface obtained by rotating the given curve about the x-axis. x=4t,y=t33+12t2,1t442E43E44E45E46E47E48E49EFind an equation of the parabola with focus (2,1) and directrix x=4.51E52E53E54E55E56EIn 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 and vertical. c Show that the line y=x1 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.a Find the highest and lowest points on the curve x4+y4=x2+y2. b Sketch the curve. Notice that it is symmetric with respect to both axes and both of the lines y=x, so it suffices to consider yx0 initially. c Use polar coordinates and a computer algebra system to find the area enclosed by the curve.What is the smallest viewing rectangle that contains every member of the family of polar curves r=1+csin,where0c1? Illustrate your answer by graphing several members of the family in this viewing rectangle.Four 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.Show that any tangent line to a hyperbola touches the hyperbola halfway between the points of intersection of the tangent and the asymptotes.A 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, 0br. 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+3rcos y=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 0br 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.a What is a sequence? b What does it mean to say that limnan=8? c What does it mean to say that limnan=?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. an=2n2n+1List the first five terms of the sequence. an=n21n2+1List the first five terms of the sequence. an=(1)n15nList the first five terms of the sequence. an=cosn2List the first five terms of the sequence. an=1(n+1)!List the first five terms of the sequence. an=(1)nnn!+1List the first five terms of the sequence. a1=1,an+1=5a3List the first five terms of the sequence. a1=6,an+1=annList the first five terms of the sequence. a1=2,an+1=an2+anList the first five terms of the sequence. a1=2,a2=1,an+1=anan1Find a formula for the general term an of the sequence, assuming that the pattern of the first few terms continues. {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. {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. {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. {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. {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. {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. 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. 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. an=1+(12)n22EDetermine whether the sequence converges or diverges. If it converges, find the limit. an=3+5n2n+n2Determine whether the sequence converges or diverges. If it converges, find the limit. an=3+5n21+nDetermine whether the sequence converges or diverges. If it converges, find the limit. an=n4n32nDetermine whether the sequence converges or diverges. If it converges, find the limit. an=2+(0.86)nDetermine whether the sequence converges or diverges. If it converges, find the limit. an=3n7nDetermine whether the sequence converges or diverges. If it converges, find the limit. an=3nn+2Determine whether the sequence converges or diverges. If it converges, find the limit. an=e1/nDetermine whether the sequence converges or diverges. If it converges, find the limit. an=4n1+9nDetermine whether the sequence converges or diverges. If it converges, find the limit. an=1+4n21+n2Determine whether the sequence converges or diverges. If it converges, find the limit. an=cos(nn+1)Determine whether the sequence converges or diverges. If it converges, find the limit. an=n2n3+4nDetermine whether the sequence converges or diverges. If it converges, find the limit. an=e2n/(n+2)Determine whether the sequence converges or diverges. If it converges, find the limit. an=(1)n2nDetermine whether the sequence converges or diverges. If it converges, find the limit. an=(1)n+1nn+nDetermine whether the sequence converges or diverges. If it converges, find the limit. {(2n1)!(2n+1)!}Determine whether the sequence converges or diverges. If it converges, find the limit. {lnnln2n}Determine whether the sequence converges or diverges. If it converges, find the limit. {sinn}Determine whether the sequence converges or diverges. If it converges, find the limit. an=tan1nnDetermine whether the sequence converges or diverges. If it converges, find the limit. {n2en}Determine whether the sequence converges or diverges. If it converges, find the limit. an=ln(n+1)lnnDetermine whether the sequence converges or diverges. If it converges, find the limit. an=cos2n2nDetermine whether the sequence converges or diverges. If it converges, find the limit. an=21+3nnDetermine whether the sequence converges or diverges. If it converges, find the limit. an=nsin(1/n)Determine whether the sequence converges or diverges. If it converges, find the limit. an=2ncosnDetermine whether the sequence converges or diverges. If it converges, find the limit. an=(1+2n)nDetermine whether the sequence converges or diverges. If it converges, find the limit. an=nnDetermine whether the sequence converges or diverges. If it converges, find the limit. an=ln(2n2+1)ln(n2+1)Determine whether the sequence converges or diverges. If it converges, find the limit. an=(lnn)2nDetermine whether the sequence converges or diverges. If it converges, find the limit. an=arctan(lnn)Determine whether the sequence converges or diverges. If it converges, find the limit. an=nn+1n+3Determine whether the sequence converges or diverges. If it converges, find the limit. {0,1,0,0,1,0,0,0,1,...}Determine whether the sequence converges or diverges. If it converges, find the limit. {11,13,12,14,13,15,14,16,...}Determine whether the sequence converges or diverges. If it converges, find the limit. an=n!2n56E57EUse 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 739 for advice on graphing sequences. an=sinnn59E60E61E62E63Ea Determine whether the sequence defined as follows is convergent or divergent: a1=1an+1=4an for n1 b What happens if the first term is a1=2?65E66EA fish farmer has 5000 catfish in his pond. The number of catfish increases by 8 per month and the farmer harvests 300 catfish per month. a Show that the catfish population Pn after n months is given recursively by Pn=1.08Pn1300P0=5000 b How many catfish are in the pond after six months?Find the first 40 terms of the sequence defined by an+1={12an3an+1ifanisanevennumberifanisanoddnumber and a1=11. Do the same if a1=25. Make a conjecture about this type of sequence.For what values of r is the sequence {nrn} convergent?70E71E72E73E74EDetermine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? an=n(1)n76EDetermine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? an=32nen78EFind 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 limnan exists. b Find limnan.Show that the sequence defined by a1=1an+1=31an is increasing and an3 for all n. Deduce that {an} is convergent and find its limit.Show that the sequence defined by a1=2an+1=13an satisfies 0an2 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 is the Fibonacci sequence defined in Example 3c. b Let an=fn+1/fn and show that an1=1+1/an2. 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 an+1=f(an), show that f(L)=L. b Illustrate part a by taking f(x)=cosx, 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 the 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 limnrn=0 when |r|1.Prove Theorem 6. Hint: Use either Definition 2 or the Squeeze Theorem.88E89ELet an=(1+1n)n. a Show that if 0ab, then bn+1an+1ba(n+1)bn b Deduce that bn[(n+1)anb]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 a2n4. e Use parts c and d to show that an4 for all n. f Use Theorem 12 to show that limn(1+1/n)n exists. The limit is e. See Equation 6.4.9 or 6.4*.9.Let a and b be positive numbers with ab. 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 limnan=limnbn Gauss called the common value of these limits the arithmetic-geometric mean of the numbers a and b.a Show that if limna2n=L and limna2n+1=L, then {an} is convergent and limnan=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 is pn 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 p00. a Show that if {pn} is convergent, then the only possible values for its limit are 0 and ba. b Show that pn+1(b/a)pn. c Use part b to show that if ab, then limnpn=0; in other words, the population dies out. d Now assume that ab Show that if p0ba, then {pn} is increasing and 0pnba. Show also that if p0ba, then {pn} is decreasing and pnba. Deduce that if ab then limnpn=ba.a What is the difference between a sequence and a series? b What is the convergent series? What is a divergent series?Explain what it means to say that n=1an=5.Calculate the sum of the series n=1an whose partial sums are given. sn=23(0.8)nCalculate the sum of the series n=1an whose partial sums are given. sn=n214n2+15ECalculate 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? n=11n37ECalculate 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? n=1(1)n1n!Find at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial same screen. Does it appear that the series is convergent or divergent? If it is convergent, find the sum. If it is divergent, explain why. n=112(5)nFind at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial same screen. Does it appear that the series is convergent or divergent? If it is convergent, find the sum. If it is divergent, explain why. n=1cosnFind at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial same screen. Does it appear that the series is convergent or divergent? If it is convergent, find the sum. If it is divergent, explain why. n=1nn2+4Find at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial same screen. Does it appear that the series is convergent or divergent? If it is convergent, find the sum. If it is divergent, explain why. n=17n+110nFind at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial same screen. Does it appear that the series is convergent or divergent? If it is convergent, find the sum. If it is divergent, explain why. n=11n2+114ELet an=2n3n+1. a Determine whether {an} is convergent. b Determine whether n=1an is convergent.16EDetermine whether the geometric series is convergent or divergent. If it is convergent, find its sum. 34+163649+...Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. 4+3+942716+...Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. 102+0.40.08+...Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. 2+0.5+0.125+0.03125+...Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. n=112(0.73)n1Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. n=15nDetermine whether the geometric series is convergent or divergent. If it is convergent, find its sum. n=1(3)n14n24E25E26EDetermine whether the series is convergent or divergent. If it is convergent, find its sum. 13+16+19+112+115+...28EDetermine whether the series is convergent or divergent. If it is convergent, find its sum. n=12+n12n30EDetermine whether the series is convergent or divergent. If it is convergent, find its sum. n=13n+14n32EDetermine whether the series is convergent or divergent. If it is convergent, find its sum. n=114+enDetermine whether the series is convergent or divergent. If it is convergent, find its sum. n=12n+4nenDetermine whether the series is convergent or divergent. If it is convergent, find its sum. k=1(sin100)k36EDetermine whether the series is convergent or divergent. If it is convergent, find its sum. n=1lnn2+12n2+138E39E40EDetermine whether the series is convergent or divergent. If it is convergent, find its sum. n=1(1en+1n(n+1))42EDetermine whether the series is convergent or divergent by expressing sn as a telescoping sum as in Example 8. If it is convergent, find its sum. n=22n2144EDetermine whether the series is convergent or divergent by expressing sn as a telescoping sum as in Example 8. If it is convergent, find its sum. n=13n(n+3)46E