MATH 152: Calculus 2, SET8 EXAMPLES [Belmonte, 2018] 6 Applications of Integration. 5.5 The Substitution Rule. 6.2 Volumes. 6.1 Areas Between Curves

Transcription

1 MATH 5: Calculus, SET8 EXAMPLES [Belmote, 8] 6 Applicatios of Itegratio 55 The Substitutio Rule [48/6] Evaluate the itegral t + dt by makig the substitutio u = t + [48/8] Evaluate x e x3 dx via the Substitutio Rule 3 [49/] Same as precedig for sec x 4 [49/3] Evaluate ta x dx 5 [49/38] Evaluate dt cos t + ta t z z 3 + dz 6 [49/48] Evaluate x 3 x + dx 3 dx 7 [49/63] Evaluate the defiite itegral 3 ( + x) 8 [49/68] Evaluate 4 x +x dx 9 [49/77] Evaluate (x + 3) 4 x dx by writig it as the sum of two itegrals ad the usig symmetry [49/78] Evaluate x x 4 dx by makig a substitutio ad iterpretig the resultig itegral i terms of a area 6 Areas Betwee Curves [434/6] Fid the area bouded by the curve y = six ad the lies y = x, x = π/, ad x = π [434/] Fid the area bouded by 4x + y = ad x = y 3 [434/8] Fid the area bouded by y = x ad x y = 4 [434/4] Fid the area bouded by y = cos x ad y = cos x for x π 5 [435/3] Fid the area of the regio eclosed by the curves x y = ad y = x, where x + x 9 x 6 [435/36] Evaluate the itegral 3x x dx ad iterpret it as the area of a regio 9 [436/5] The rates at which rai fell, i iches per hour, i two differet locatios t hours after the start of a storm are f (t) = 73t 3 t +t + 6 ad g(t) = 7t 5t + Compute the area betwee their graphs for t ad iterpret your result i this cotext [436/56] Fid the area of the regio bouded by the parabola y = x, its taget lie at (,), ad the x-axis 6 Volumes [446/3] Fid the volume of the solid obtaied by rotatig the regio bouded by y = x, y =, x = 5, about the x-axis [446/6] Fid the volume of the solid obtaied by rotatig the regio bouded by x = y, x =, y = 4, about the y-axis 3 [446/9] Fid the volume of the solid obtaied by rotatig the regio bouded by y = x, x = y, about the y-axis 4 [446/] Fid the volume of the solid obtaied by rotatig the regio bouded by y = x 3, y =, x =, about y = 3 5 [446/5] Fid the volume of the solid obtaied by rotatig the regio bouded by y = x 3, y =, x =, about x = 6 [447/35] Fid approximate x-coordiates of the poits of itersectio of the curves y = l ( x 6 + ) ad y = 3 x 3 The fid the approximate volume of the solid obtaied by rotatig about the x-axis the regio boud by the curves 7 [447/38] Use a CAS to fid the exact volume of the solid obtaied by rotatig the regio boud by the curves y = x, y = xe x/ about y = 3 8 [447/46] Cosider a profile view of a bird s egg (a) A model for its shape is obtaied by rotatig about the x-axis the regio uder the graph of f (x) = ( ax 3 + bx + cx + d ) x, x Use a CAS to fid the volume of such a egg (b) For a red-throated loo, a = 6, b = 4, c =, ad d = 54 Graph f ad fid the volume of a egg of this species 9 [448/55] The base of a solid S is a elliptical regio with boudary curve 9x + 4y = 36 Cross-sectios perpedicular to the x-axis are isosceles right triagles with hypoteuse i the base Fid the volume of S [448/6] The base of a solid S is the regio eclosed by y = x ad the x-axis Cross-sectios perpedicular to the y-axis are quarter-circles Fid the volume of S 7 [435/4] Graph the regio betwee the curves y = e x ad y = x 4 Compute its area correct to 5 decimal places 8 [435/48] The widths w (i meters) of a kidey-shaped swimmig pool were measured at -meter itervals as idicated i the table below Use the Midpoit Rule to estimate the area of the pool x w

2 63 Volumes by Cylidrical Shells [453/] Let S be the solid obtaied by rotatig the regio bouded by y = x(x ) ad y = about the y-axis Fid the volume of S via cylidrical shells [459/6] The table shows values of a force fuctio f (x), where x is measured i meters ad f (x) i ewtos Use the Midpoit Rule to estimate the work doe by the force i movig a object from x = 4 to x = x f (x) [459/7] A force of lb is required to hold a sprig stretched 4i beyod its atural legth How much work (i ft-lb) is doe i stretchig it from it atural legth to 6i beyod its atural legth? 4 [459/9] Suppose that J of work is eeded to stretch a sprig from its atural legth of 3cm to a legth of 4cm [454/6] Use the method of cylidrical shells to fid the volume geerated by rotatig the regio bouded by y = 4x x, y = x, about the y-axis 3 [454/3] Use the method of cylidrical shells to fid the volume geerated by rotatig the regio bouded by x = + (y ), x =, about the x-axis 4 [454/8] Use the method of cylidrical shells to fid the volume geerated by rotatig the regio bouded by y = x, x = y, about x = 5 5 [454/3] Fid the volume of the solid obtaied by rotatig the regio bouded by the curves y = ±cos 4 x, π x π, about x = π 6 [454/7] Use the Midpoit Rule with = 5 to estimate the volume obtaied by rotatig about the y-axis the regio uder the curve y = + x 3, x (a) How much work is eeded to stetch the sprig from 35cm to 4cm? (b) How far beyoud its atural legth will a force of 3N keep the sprig stretched? 5 [459/] If 6J of work is eeded to stetch a sprig from cm to cm ad aother J is eeded to stretch it from cm to 4cm, what is the atural legth of the sprig? 6 [459/5] A cable that weighs lb/ft is used to lift 8lb of coal up a mie shaft 5ft deep Fid the total work doe 7 [459/] A circular swimmig pool has a diameter of 4ft, the sides are 5ft high, ad the depth of the water is 4ft How much work is required to pump all of the water out over the side? (The weight desity of water is δ = 65lb/ft 3 ) 8 [459/3] A tak is full of water, the mass desity of which is ρ = kg/m 3 Fid the work required to pump the water out of the spout (Gravitatioal acceleratio is g = 98m/s ) 7 [454/34] Fid the approximate x-coordiates of the poits of itersectio of the curves y = e six ad y = x 4x + 5 The estimate the volume of the solid obtaied by rotatig about the y-axis the regio eclosed by these curves 8 [454/36] Use a CAS to fid the exact volume of the solid obtaied by rotatig the regio bouded by y = x 3 six, y =, x π, about x = 9 [455/39] The regio bouded by y x = ad y = is rotated about the x-axis Fid the volume of the resultig solid by ay method you wish 9 [459/4] Same as #8 with the spherical tak depicted below [455/4] The regio bouded by x + (y ) = is rotated about the y-axis Fid the volume of the resultig solid by ay method you wish 64 Work [458/3] A variable force of F (x) = 5x pouds moves a object alog a straight lie whe it is x feet from the origi Calculate the work doe i movig the object from x = ft to x = ft [46/8] Same as #9, but with the tak half full of oil that has mass desity ρ = 9kg/m 3

3 MATH 5: Calculus, SET8 EXAMPLES [Belmote, 8] 7 Techiques of Itegratio 7 Itegratio by Parts [476/] Evaluate the itegral x lxdx usig itegratio by parts with u = lx ad dv = xdx [476/8] Evaluate the itegral t siβt dt 3 [476/] Evaluate the itegral x ta xdx 4 [476/4] Evaluate the defiite itegral ( x + ) e x dx 5 [476/3] Evaluate the defiite itegral 3 arcta(/x) dx r 3 6 [477/34] Evaluate dr 4 + r 7 [477/4] First make a substitutio ad the use itegratio by arcsi(lx) parts to evaluate the itegral dx x 8 [477/46] Evaluate the idefiite itegral x sixdx, the graph both the itegrad ad its atiderivative (take C = ) 9 [477/58] Fid the area of the regio bouded by y = x e x ad y = xe x [477/64] Use the method of cylidrical shells to fid the volume geerated by rotatig the regio bouded by the curves y = e x, x =, ad y = 3 about the x-axis 7 Trigoometric Itegrals 73 Trigoometric Substitutio [49/] Use the trigoometric substitutio x = taθ x 3 to evaluate the itegral x + 4 dx 3 x [49/6] Evaluate the itegral dx 36 x /3 dx 3 [49/6] Evaluate /3 x 5 9x 4 [49/8] Evaluate 5 [49/8] Evaluate dx ((ax) b ) 3/ x + (x x + ) dx π/ cost 6 [49/3] Evaluate + si t dt 7 [49/34] Fid the area of the regio bouded by the hyperbola 9x 4y = 36 ad the lie x = 3 8 [49/38] Fid the volume of the solid obtaied by rotatig about the lie x = the regio uder the curve y = x x, x 9 [49/4] The parabola y = x divides the circular disk x + y 8 ito two parts Fid the areas of both parts [49/4] A charged rod of legth L produces a electric field L a λb at poit P(a,b) give by E (P) = dx a 4πε (x + b 3/ ) where λ is the charge desity per uit legth o the rod ad ε is the free space permittivity (see figure) Evaluate the itegral to obtai a expressio for the electric field E (P) [484/] Evaluate the itegral si 3 θ cos 4 θ dθ [484/] Evaluate the itegral π si t cos 4 t dt 3 [484/6] Evaluate the itegral π/4 sec 6 θ ta 6 θ dθ 4 [484/8] Evaluate the itegral ta 5 x sec 3 xdx 5 [484/3] Evaluate π/4 ta 4 xdx 6 [485/36] Evaluate π/ π/4 cot3 xdx 7 [485/4] Evaluate siθ si6θ dθ 8 [485/46] Evaluate π/4 cos4θ dθ 9 [485/58] Fid the area of the regio bouded by the curves y = tax, y = ta x, x π/4 [485/64] Fid the volume obtaied by rotatig the regio bouded by the curves y = secx, y = cosx, x π/3, about the lie y = 74 Itegratio of Ratioal Fuctios by Partial Fractios [5/4] Give the form of the partial fractio decompositio (PFD) of these expressios [I MATLAB, automatically determie the PFDs usig the partfrac commad!] (a) x4 x 3 + x + x x x + (b) x x 3 + x + x

4 [5/] Evaluate the itegral x 4 x 5x + 6 dx x 3 + 4x + x 3 [5/6] Evaluate x 3 + x dx x 4 [5/3] Evaluate x + 4x + 3 dx 5 [5/4] Make a substitutio to express the itegrad as a ratioal fuctio the evaluate + 3 x dx 6 [5/48] Same as #5 for 7 [5/5] Same as #5 for six cos x 3cosx dx e x (e x )(e x + ) dx 8 [5/6] Use t = ta ( ) x, cosx = t, six = t +t dx = +t dt to evaluate π/ π/3 + six cosx dx +t ad 9 [5/64] Fid the area of the regio uder the curve y = / ( x 3 + x ), x [5/66] Fid the volume of the resultig solid if the regio uder the curve y = / ( x + 3x + ) from x = to x = is rotated about (a) the x-axis ad (b) the y-axis 78 Improper Itegrals [534/] Which of the followig itegrals are improper? Why? (a) π/4 taxdx (b) π taxdx (c) dx (d) e x3 dx x x [534/6] Determie whether the itegral 4 dx is + x coverget or diverget If coverget, compute its value 3 [534/] Same as # for 4 [534/6] Same as # for 5 [534/8] Same as # for 5 6 [535/34] Same as # for r dr siθe cosθ dθ dv v + v 3 w w dw π/ cosθ 7 [535/38] Same as # for dθ siθ 8 [535/56] Evaluate x dx by usig a trigoometric x 4 substitutio, the splittig the iterval of itegratio i the resultig itegral 9 [535/58] Fid the values of p for which e x(lx) p dx coverges ad evaluate the itegral for those values of p [536/8] Fid the value of C for which the improper itegral x x + C dx coverges The evaluate the itegral 3x + for this value of C

5 MATH 5: Calculus, SET8 EXAMPLES [Belmote, 8] Parametric Equatios Curves Defied by Parametric Equatios [645/4] Sketch the curve x = e t +t, y = e t t, t State how the curve is traversed as t icreases [645/6] Sketch the curve x = 3t +, y = t + 3 State how the curve is traversed as t icreases Elimiate the parameter to fid a Cartesia equatio of the curve 3 [645/8] Sketch the curve x = sit, y = cost, t π State how the curve is traversed as t icreases Elimiate the parameter to fid a Cartesia equatio of the curve 4 [645/5] Sketch the curve x = t, y = lt State how the curve is traversed as t icreases Elimiate the parameter to fid a Cartesia equatio of the curve 5 [645/8c] Sketch the curve x = sit, y = si(t + sit) 6 [647/3] Graph the curves y = x 3 4x ad x = y 3 4y ad fid their poits of itersectio to two decimal places 7 [647/34] Cosider the ellipse x a + y b = (a) Give parametric equatios for x ad y i terms of t (b) Graph the ellipse for a = 3 ad b =,, 4, 8 (c) How does the ellipse s shape chage as b icreases? 8 [648/45] The positio of oe particle at time t is give by x = 3sit, y = cost, t π ad the positio of a secod particle is give by x = 3 + cost, y = + sit, t π (a) Graph the paths of both particles How may poits of itersectio are there? Fid ad plot them (b) Are ay of these poits of itersectio collisio poits? I other words, are the particles ever at the same place at the same time? If so, fid ay collisio poits (c) Repeat (a) ad (b) if the path of the secod particle is x 3 = 3 + cost, y 3 = + sit, t π 9 [648/47] Ivestigate the family of curves defied by the parametric equatios x = t, y = t 3 ct How does the shape chage as c icreases? Illustrate by graphig members of the family for c =,,, [648/5] Graph several members of the family of curves x = sit + sit, y = cost + cost, t π, where is a positive iteger What features do they have i commo? What happes as icreases? Use =,, 3, 4, 5 Calculus with Parametric Curves [655/3] Fid a equatio of the taget lie to the curve x = t 3 +, y = t 4 +t, for t = [655/7] Fid parametric ad Cartesia equatios for the taget lie to the curve x = + lt, y = t +, at (,3) 3 [655//] Fid dy/dx ad d y/dx give x = t +, y = t +t For which values of t is the curve cocave upward? 4 [655/7] Fid the poits o the curve x = t 3 3t, y = t 3, where the taget is horizotal or vertical Illustrate 5 [655/6] Graph the curve x = cost, y = sit + si t to discover where the curve crosses itself The fid equatios of both tagets at that poit Illustrate with a graph 6 [655/3] Fid the area eclosed by the curve x = t t, y = t ad the y-axis 7 [656/46] Graph the curve x = cost + l ( ta t), y = sit, π 4 t 3π 4 ad fid its arc legth 8 [656/48] Fid the legth of the loop of the curve x = 3t t 3, y = 3t 9 [656/6] Fid the surface area obtaied by rotatig the curve x = t + t, y = 8 t, t 3, about the x-axis [657/66] Fid the surface area obtaied by rotatig the curve x = e t t, y = 4e t/, t, about the y-axis 3 Polar Coordiates [666/] Sketch the regio i the plae whose polar coordiates satisfy r 3, π 6 < θ < 5π 6 [666/6] Fid a Cartesia equatio for the polar curve r = 4secθ Illustrate 3 [667//6] Fid a polar equatio for the Cartesia equatio x y = 4 Illustrate 4 [667/5] Fid a polar equatio for the Cartesia equatio ( x + y ) 3 = 4x y, the sketch the curve 5 [667/6] Fid the taget lie to the curve r = + cosθ at θ = π/3 Illustrate 6 [667/6] Fid the poits o the curve r = 3cosθ where the taget lie is horizotal or vertical Sketch 7 [668/68] Graph the hippopede r = 8si θ 8 [668/7] Graph the valetie curve r = taθ cotθ 9 [668/7] Graph the curve r = + cos(9θ/4) Beauty, eh? [668/74] Graph the curve r = siθ Fid the y-coordiate of the highest poits o the curve

6 4 Areas ad Legths i Polar Coordiates [673/8] Fid the area of the regio betwee the curve r = lθ, θ π ad the origi [673/] Sketch the curve r = cosθ ad fid the area it ecloses 9 [68/5] A cross-sectio of a parabolic reflector is show i the figure The bulb is located at the focus F ad the opeig at the focus is cm (a) Fid a equatio of the parabola (b) Fid CD, the diameter of the opeig that is cm from the vertex 3 [673//] Fid the area of the regio eclosed by oe loop of the curve r = si5θ 4 [673/4] Fid the area of the regio that lies iside the curve r = si θ ad outside the curve r = Illustrate 5 [673/6] Fid the area of the regio that lies iside the curve r = + cosθ ad outside the curve r = cosθ Illustrate 6 [673/3] Fid the area of the regio that lies iside both r = 3 + cosθ ad r = 3 + siθ Illustrate 7 [673/36] Fid the area of betwee a large loop ad the eclosed small loop of r = + cos 3θ Illustrate 8 [673/4] Fid all poits of itersectio of the curves r = cos3θ ad r = si3θ Illustrate 9 [673/46] Fid the legth of r = 5 θ, θ π Illustrate [673/49] Fid the legth of the curve r = cos 4 (θ/4) Graph 5 Coic Sectios [68/8] Fid the vertex, focus, ad directrix of the parabola x 6x 3y + 38 = ad sketch its graph [68/4] Fid the vertices ad foci of the ellipse x + 36y = 5 ad sketch its graph 3 [68//8] Fid a equatio of the ellipse depicted The fid its foci 4 [68/4] Fid the vertices, foci, ad asymptotes of the hyperbola 9y 4x 36y 8x = 4 ad sketch its graph 5 [68/3] Idetify the type of coic sectio whose equatio is x x + y 8y + 7 = Fid vertices ad foci; graph 6 [68/3] Fid a equatio for the parabola with focus (,) ad directrix y = 6 Sketch 7 [68/4] Fid a equatio for the ellipse with foci (, ) ad (8, ) ad vertex (9, ) Sketch 8 [68/44] Fid a equatio for the hyperbola with foci (,±5) ad vertices (,±) Sketch [68/54] Fid a equatio for the slated ellipse with foci (,) ad (,) ad major axis of legth 4 Graph 6 Coic Sectios i Polar Coordiates [688/] Write a polar equatio of a parabola with the focus at the origi ad directrix x = 3 Sketch the curve [688/4] Write a polar equatio of a hyperbola with the focus at the origi, eccetricity 3 ad directrix x = 3 Illustrate 3 [688/6] Write a polar equatio of a ellipse with the focus at the origi, eccetricity 3/5 ad directrix r = 4cscθ Graph 4 [688/] Let r = + siθ (a) Fid the eccetricity (b) Idetify the coic (c) Give a equatio of the directrix (d) Sketch the coic 5 [688/] Same as #4 for r = 6 [688/4] Same as #4 for r = 7 [688/6] Same as #4 for r = 5 4cosθ 3 3siθ 4 + 3cosθ 8 [688/8] Graph the coic r = 4/(5 + 6cosθ) alog with its directrix Additioally graph the coic obtaied by rotatig this curve about the origi through a agle π/3 9 [688/5] The orbit of Mars aroud the su is a ellipse with eccetricity 93 ad the semimajor axis 8 8 km Fid a polar equatio for the orbit [688/3] Jupiter s orbit has eccetricity 48 ad the legth of the major axis is 56 9 km Fid a polar equatio for the orbit

7 MATH 5: Calculus, SET8 EXAMPLES [Belmote, 8] Ifiite Sequeces & Series Sequeces [74/8] List the first 5 terms of the sequece a = ( )! + [74/4] Fid a formula for the geeral term a of the sequece { 4,, 4, 6, 64,}, assumig that the patter for the first few terms cotiues 3 [74/6] Determie whether the sequece a = + (86) coverges or diverges If it coverges, fid the limit 4 [74/36] Same as #3 for a = ( ) + 5 [74/5] Same as #3 for a = ( ) 6 [74/6] Graph the sequece a =! to decide whether it is coverget or diverget 7 [75/64] Defie a sequece by a = c, a + = 4 a, where c is a costat Determie if the sequece is coverget or diverget i each of the two cases (a) a = ad (b) a = 8 [75/77] Determie whether the sequece a = 3 e is icreasig, decreasig, or ot mootoic Is it bouded? 9 [75/8] A recursive sequece is give by a =, a + = + a (a) Show that {a } is a icreasig sequece ad bouded above by 3 Apply the Mootoic Sequece Theorem (MST) to show that lim a exists (b) Fid the limit [75/8] Show that the recursive sequece defied by a =, a + = 3 a satisfies < a ad is decreasig Deduce that it coverges ad fid its limit Series [75/4] Calculate the sum of the series a whose partial sums are s = 4 + [75/8] Calculate the first eight partial sums of the series ( ) to four decimal places Does it appear that the! series is coverget or diverget? 3 [75/] Fid the first te partial sums of cos to four decimal places Graph both the sequece of terms {a } ad the sequece of partial sums {s } o the same plot Does it appear that the series is coverget or diverget? If it is coverget, fid the sum If it is diverget, explai why 4 [76/4] Same as #3 for ( si ) si + 5 [76/8] Determie whether the geometric series is coverget or diverget If it is coverget, fid its sum 6 [76/4] Same as #5 for 7 [76/6] Same as #5 for = 3 + ( ) [76/8] Is the series coverget or diverget? If it is coverget, fid its sum 9 [76/35] Same as #8 for k= (si)k [76/36] Same as #8 for +(/3) [76/44] Determie whether the series l + is coverget or diverget by expressig it as a telescopig sum If it is coverget, fid its sum [76/45] Same as # for 3 ( + 3) 3 [76/56] Express the umber as a ratio of itegers (The digits uder the bar repeat) 4 [76/6] Fid the values of x for which = ( 4) (x 5) coverges Fid the sum of the series for those values of x 5 [77/64] We have see that the harmoic series is a diverget series whose terms approach Show that ( l + ) is aother series with this property 6 [77/66] Use MATLAB s partial fractio commad partfrac to fid a coveiet expressio for the th partial sum of the series The fid the sum of the series + 4 =3 7 [77/75] Fid the value of c if = ( + c) = 8 [77/76] Fid the value of c such that = ec = 9 [78/8] Suppose that a with a is kow to be coverget Show that (/a ) diverges [78/85] If a is coverget ad b is diverget, show that the series (a + b ) is diverget (Argue by cotradictio) 3 The Itegral Test ad Estimates of Sums [75/4] Use the Itegral Test (IT) to determie whether the series 3/ is coverget or diverget

8 [75/8] Same as # for e 3 3 [76/4] Determie if this p-series coverves or diverges 4 [76/] Same as # for 5 [76/6] Same as # for = [76/8] Explai why the Itegral Test ca t be used to cos determie whether the series is coverget + 7 [76/3] Fid the values of p for which the series is coverget 8 [76/36] Cosider the series ζ (4) = Exercise 76/35 i the textbook l p 4 = π4 give i 9 (a) Fid the th partial sum s of the series Estimate the error i usig it to approximate the sum of the series (b) Use iformatio from the Remaider Estimate for the Itegral Test i a maer similar to Example 6 i the text to give a improved estimate of the sum (c) Compare your estimate i part (b) with ζ (4) (d) Fid a value of so that s is withi 5 of the sum 9 [76/4] Cosider the series (a) Show that it is coverget (l ) (b) Fid a upper boud for the error i the approximatio s s (c) What is the smallest value of such that this upper boud is less tha 5? (d) Fid s for this value of [77/46] Fid all values of c for which the series coverges ( c ) + 4 The Compariso Tests (CT) [73/] Suppose that a ad b are series with positive terms ad b is kow to be diverget (a) If a > b for all, what ca you say about a? Why? (b) If a < b for all, what ca you say about a? Why? [73/6] Determie whether the series diverges Explai why 3 [73/] Same as # for 4 [73/8] Same as # for 5 [73/4] Same as # for 6 [73/3] Same as # for k= (k ) ( k ) (k + )(k + 4) ! 3 coverges or + 7 [73/36] Use the first terms to approximate the sum of the series 3 Estimate the error [73/38] For what values of p does the series coverge? = 9 [73/44] Show that if a > ad a is coverget, the l( + a ) is coverget p l [73/46] If a ad b are both coverget series with positive terms, is it true that a b is also coverget? Why? 5 Alteratig Series [736/4] Test the series l3 l4 + l5 l6 + l7 for covergece or divergece [736/8] Same as # for 3 [736/4] Same as # for 4 [736/6] Same as # for 5 [736/] Same as # for 6 [736/] For the series ( ) + + ( ) arcta cos π ( ) ( + ) ( ) do the followig 8 (a) Graph both the sequece of terms ad the sequece of partial sums o the same plot (b) Use the graph to make a rough estimate of the sum of the series (c) The use the Alteratig Series Estimatio Theorem (ASET) to estimate the sum correct to 4 decimal places

9 7 [736/6] Cosider the series ( ) (a) Show that the series is coverget (b) How may terms do we eed to add i order to fid the sum of the series with a error < 5? (c) Compute this estimate of the sum 8 [736/8] Approximate the sum of the series correct to 4 decimal places 9 [736/3] For what values of p is the series coverget? [736/34] Same as #9 for = ( ) (l) p ( ) + 6 ( ) 6 Absolute Covergece ad the Ratio ad Root Tests ( ) [74/4] Determie whether the series 3 is absolutely + coverget (AC) or coditioally coverget (CC) [743/6] Same as # for ( ) [743/4] Use the Ratio Test to determie whether the series! is coverget or diverget 4 [743/] Same as #4 for ()! (!) 5 [743/6] Use the Root Test to determie whether the series ( ) is coverget or diverget 6 [743/34] Use ay test(s) to determie whether the series 5 is absolutely coverget (AC), coditioally + coverget (CC), or diverget (D) 7 [743/38] Same as #6 for = ( ) l 8 [743/44] For which positive itegers k is the series coverget? p (!) (k)! 9 [744/48] Compute the th partial sum as well as the actual sum of the series (Use MATLAB) [744/5] Aroud 9, the Idia mathematicia Sriivasa Ramauja discovered the formula π = (4)!( ) 98 = (!) William Gosper used this series i 985 to compute the first 7 millio digits of π (a) Verify that the series is coverget (b) How may correct decimal places of π do you get if you use just the first term of the series? What if you use the first two terms? 8 Power Series [75/6] Fid the radius of covergece R ad iterval of ( ) x covergece I of the power series [75/8] Same as # for 3 [75/] Same as # for 4 [75/] Same as # for 5 [75/8] Same as # for 6 [75/] Same as # for 7 [75/] Same as # for 8 [75/4] Same as # for 9 [75/6] Same as # for = = x x ( ) 5 x 8 (x + 6) (x ) 5 b l (x a) where b > x 4 6 () x (l) [75/34] Graph the first several partial sums s (x) of the series = x, together with the fuctio f (x) = /( x) o the same plot O what iterval do these partial sums appear to be covergig to f (x)? 9 Represetatio of Fuctios as Power Series [757/4] Fid a power series represetatio of f (x) = 5 4x ad determie its radius ad iterval of covergece [757/8] Same as # for f (x) = x x + 3

10 3 [757/] Same as # for f (x) = x + 3 x Start by + 3x + fidig the partial fractio decompositio of f (x) 4 [757/6] Same as # for f (x) = x ta ( x 3), but oly give the radius of covergece 5 [757/] Same as #4 for f (x) = x + x ( x) 3 6 [758/4] Cosider the fuctio f (x) = ta (x) (a) Fid a power series represetatio for f (x) (b) Graph f ad several partial sums s (x) together (c) What happes as icreases? ta x 7 [758/8] Evaluate the idefiite itegral dx as a x power series What is the radius of covergece? 8 [758/3] Use a power series to approximate the defiite itegral / arcta(x/) dx to six decimal places = ( ) x 9 [758/34] Show that the fuctio f (x) = is a ()! solutio of the differetial equatio f (x) + f (x) = [758/4] Cosider the geometric series = x (a) Fid the sum of the series (b) Fid sums of the series (c) Fid the sums of the series =, ad x, x < x for x < ad = ( )x for x <, Taylor & Maclauri Series [77/8] Fid the first four ozero terms of the Taylor series for f (x) = lx cetered at a = usig the defiitio [77/6] Fid the Maclauri series for f (x) = xcosx ad its radius of covergece 3 [77/4] Fid the Taylor series for f (x) = cosx cetered at a = π/ ad its radius of covergece 4 [77/4] Fid the Maclauri series for f (x) = x l ( + x 3) x six 5 [77/44] Same as #4 for f (x) = x 3 if x 6 if x = 6 [77/48] Fid the Maclauri series for f (x) = ta ( x 3) ad its radius of covergece The graph both f ad its first few Taylor polyomials o the same plot Commet 7 [77/56] Evaluate the idefiite itegral arcta ( x ) dx as a ifiite series 8 [77/58] Use series to approximate the defiite itegral si ( x 4) dx to four decimal places cosx 9 [77/6] Use series to evaluate the limit lim x + x e x [77/78] Fid the sum l + (l)! (l)3 3! Applicatios of Taylor Polyomials + [78/4] Fid the Taylor polyomial T 3 (x) for f (x) = six cetered at a = π/6 Graph them together [78/8] Same as #4 for f (x) = xcosx at a = 3 [78/] Same as #4 for f (x) = ta x at a = 4 [78/] Use a CAS to fid Taylor polyomials T (x) for f (x) = 3 + x cetered at a = for =,3,4,5 Graph them together 5 [78/6] Let f (x) = six, a = π/6, = 4, ad I = [, 3 π] (a) Approximate f by T (x) (b) Use Taylor s Iequality to estimate the approximatio f (x) T (x) o I (c) Check part (b): graph R (x) = f (x) T (x) o I 6 [78/] Same as #5 for f (x) = xlx, a =, = 3, ad I = [5, 5] 7 [78/4] Use iformatio from #5 to esimate si38 correct to 5 decimal places 8 [78/8] Estimate the rage of x for which the approximatio cosx x + 4 x4 is accurate to withi 5 9 [78/36] A uiformly charged disk has radius R ad surface charge desity σ The electric potetial V at poit P at a distace d alog ( the perpedicular cetral axis of the disk is d ) V = πk e σ + R d where k e is Coulomb s costat Show that V πk e R σ/d for large d (See diagram i text) [78/37] If a surveyor measures differeces i elevatio whe makig plas for a highway across a desert, correctios must be made for the curvature of the earth (See diagram i text) (a) If R is the radius of the earth ad L is the legth of the highway, show the correctio is C = R sec(l/r) R (b) Use a Taylor polyomial to show that C L R + 5L4 4R 3 (c) Compare the correctios give by the formulas i parts (a) ad (b) for a highway that is km log (Take the radius of the earth to be 637km) 4