Math Exam Practice Exam 4 will cover.-., 0. ad 0.. Note that eve though. was tested i exam, questios from that sectios may also be o this exam. For practice problems o., refer to the last review. This sheet has three sectios. The first sectio will remid you about techiques ad formulas that you should kow. The secod gives umber of practice questios for you to work o. The third sectio give the aswers of the questios i sectio. Review Tests for Covergece We leared about the followig tests for covergece: Divergece Test If 0 the diverges. This is a excellet test to start with because the limit is ofte easy to calculate. Keep i mid, however, if the limit is 0, the the Divergece test tells you othig. You must try some other test. p series If you recogize a series as a p series, p the you ca use the fact that a p series coverges whe (ad oly whe p>. Geometric series We discussed this i the last subsectio. Compariso Test To use the compariso test, we eed to have a large group of test series available. We also eed to kow if these test series coverge or ot. The most commo test series for the compariso test are the p series ad the geometric series. If the series acts like a p series, or acts like a geometric series, the you may wish to use the compariso test. Remember, if 0 b ad b coverges, the coverges. diverges, the b diverges. Limit Compariso Test This test works well for the type of problems that also work with the compariso test, but is somewhat easier. You still eed the test series, but you do t eed to work to make the terms of the series greater tha or less tha some kow series. You oly eed to check the limit lim. b If it is fiite ad positive, the both series coverge or both diverge. Sice you already kow about oe of them, you the kow about the other. Itegral Test If we are tryig to determie whether coverges, ad there is a fuctio f(x with f( =, the the sum coverges iff a f(x dx coverges. (We assume that both the series { } ad f(x are positive. So the itegral test is hady if the associated fuctio ca be itegrated without too much difficulty. Alteratig Series Test To use the alteratig series test, you eed to verify three thigs: The series is alteratig. (This ca usually be doe by ispectio. The terms of the series coverge to 0. (Hopefully you did this whe you applied the Divergece test. Fially, the terms of the absolute values are decreasig. The secod statemet does ot ecessarily imply the third. If this is true, the the alteratig series test tells us the series coverges. Ratio Test If lim + = L, the the series is absolutely coverget if L< ad diverget if L>. If L =, the test fails. This test works really well whe a factorial is preset i.
Root Test If lim a = L, the the series is absolutely coverget if L< ad diverget if L>. If L =, the test fails. This test works really well whe there are powers of i. Remember, the Itegral test ad the compariso tests oly work whe the series has o-egative terms. If you have a series where the terms are both positive ad egative, the you must be able to say whether the series coverges absolutely, coverges coditioally, or diverges. It is oe of these. These are mutually exclusive coditios. Estimatig the tail I a ifiite series, the tail is a term usually used to idicate the last part of the series. For example, if we wish to approximate the sum of the followig coverget series, the we ca write it as k =0 =0 ( +, ( + + =k+ ( + The part that is still a ifiite sum is called the tail. The sum of the tail is called the error of our approximatio. If we ca test covergece of a series by the itegral test, the there is a easy way to fid a estimate of the tail: Assume f(x is defied o [b, for some b, ad f( =. The k+ f(x dx =k+ k f(x dx. For example suppose that we sum the first 5 terms of the above series: How close is this? We fid that ad =0 4 ( + ( + =0 = + 7 + 5 + 4 + 79 =.0494 5 4 (x + dx = 484 =0.00066570, (x + dx = 4 =0.000864975. Thus, the error is betwee these two umbers. If we ca use the compariso test to fid covergece, the we ca sometimes still use the above formula, but oly for upper bouds. For example, if I am tryig to estimate =k+ +, the fact that meas that =k+ + < + < k x dx.
Power Series Recall that a power series is a series of the form (x c. =0 The value c is called the ceter of the power series, ad the values are called the coefficiets. A power series is a way to represet a fuctio. However, the power series may have a differet domai tha the fuctio does. To fid the domai of the power series, (called the iterval of covergece, we do the followig:. Apply the ratio or root test to the power series. If the limit is 0, the power series coverges everywhere ad the radius of covergece is. If the limit is, the power series coverges oly at the ceter, ad the radius of covergece is 0. Otherwise, set the limit to be less tha, ad rework the iequality so it says x c <R. R is the radius of covergece.. The power series is ow guarateed to coverge absolutely o (c R, c + R, ad diverge o (, c R (c + R,. We ow test the power series at the edpoits. Plug the edpoits c R ad c + R ito the power series ad use oe of the other 5 tests (ot Ratio, ot Root to determie whether they coverge. State the iterval of covergece usig paretheses to idicate the power series does ot coverge at a edpoit, ad a bracket to idicate it does. Fidig sums of series Fidig a power series that represets a specific fuctio is the ext topic. The first oe we leared was the geometric series: x = x iff x (,. We the foud the sum of several series by differetiatig, itegratig, multiplyig by x, etc. The Taylor series of a fuctio is f ( (c (x c! =0 =0 ad ca also be used to fid the power series of a fuctio. Notice that the iterval of covergece of these series is still very importat. We eed to kow whe we ca trust them. I additio to the geometric series above, the followig Maclauri series (with iterval of covergece are importat: ta x = l( + x = e x = =0 =0 ( x +,[, ] + ( x +,(, ] + =0 x,(,! si x = cos x = sih x = =0 =0 =0 ( x +,(, ( +! ( x,(, (! x +,(, ( +! ( r ( + x r = x x,(, where the biomial cosh x =,(, =0 coefficiets are ( (! =0 r = r(r (r +! If you eed to costruct a Maclauri series of a fuctio ad some of the above fuctios are icluded, it is almost always easier to maipulate the Maclauri series istead of costructig the series by scratch. Approximatig sums of series I additio to fidig whether sums of series coverge or ot, we also were able to fid approximatios to the error. There were basic approximatios to the error give by the Itegral test, Alteratig Series test, ad the Taylor Series.
. If a k is coverget with sum s ad f(k =a k where f is a cotiuous, positive, ad decreasig fuctio for x, the the remaider R = s s = k=+ a k satisfies the iequality f(x dx R + f(x dx. If {a k } is a positive decreasig sequece with a limit of 0, the ( k a k is coverget with sum s ad the remaider R = s s = k=+. Taylor s Iequality: If T (x = ( k a k satisfies the iequality k=0 R <+ the remaider R (x =f(x T (x satisfies the iequality o the iterval where f (+ (x <M. f (k (c k! (x c k is the th Taylor polyomial of f(x cetered at c, the R (x M x c + ( +! We use this iformatio, whe applicable, to fid maximum errors whe approximatig a fuctio by a Taylor polyomial as well. Parametric Curves We leared how to defie curves parametrically. That is, we leared how to describe a curve give by a equatio H(x, y =0 i terms of a pair of fuctios You will eed to be able to do the followig: x = f(t, y = g(t. (a Graph a curve from it s parametric equatios. (b Recogize the curve of a set of parametric equatios. (c Elimiate the parameter of the parametric equatios to fid a equatio i x ad y describig the curve. (d Costruct a set of parametric equatios for a curve writte i cartesia coordiates. 0. Calculus of Parametric Equatios I the discussio below, we will assume that a curve ca be described parametrically by Slopes dy dy dx = dt x = f(t, y = g(t. This gives a formula for the slope as a fuctio of parameter. Arclegth s = t dx dt d y dx = d dt = g (t f (t. ( dy dx dx dt (f (x +(g (x dx
Surface Area Rotated about the x axis: Rotated about the y axis: Area uder the curve Area betwee the curve ad the x axis: Area betwee the curve ad the y axis: S = S = t t A = A = πf(x (f (x +(g (x dx πg(x (f (x +(g (x dx t y dx = t x dy = t t g(tf (t dt f(tg (t dt Questios Try to study the review otes ad memorize ay relevat equatios before tryig to work these equatios. If you caot solve a problem without the book or otes, you will ot be able to solve that problem o the exam. For problems to 4, determie whether the series coverges or diverges. State the test you used.... 4. = =0 = = + 6 4 +7 +8 5 +7 4 + + 9 + l( cos + 5. Show that 5 is a upper boud o the error of = 4 if the sum is approximated by the first +7 two terms. 6. Approximate the sum of = + by summig the firs terms. Fid a boud o the error of your approximatio. For problems 7 through, determie whether the series is absolutely coverget, coditioally coverget, or diverget. 7. 8. 9. = ( l (! = = cos(π + 0... = ( ( + ( (l = ( 5 + =. Show that 4 is a upper boud o the error of = 4 if the sum is approximated by the first +7 two terms. 4. Suppose the power series (x+ has a radius = of covergece R = 5. List all possible itervals of covergece. 5. Fid the radius ad iterval of covergece of (x = 6. Fid the radius ad iterval of covergece of ( 4 (x + = 7. Fid the radius ad iterval of covergece of x = 8. Fid the radius ad iterval of covergece of x =
9. Fid the radius ad iterval of covergece of! x 00 = 0. Fid a power series represetatio i powers of x for the fuctio f(x = +x with iterval of covergece.. Fid a power series represetatio i powers of (x for the fuctio f(x = +x ad give the iterval of covergece.. Fid a power series represetatio i powers of (x for l( + x.. What is the power series represetatio of x ( x? 4. Fid the Maclauri series for f(x = l( x from the defiitio of a Maclauri series. Fid the radius of covergece. 5. Fid a Taylor series for f(x = cos(πx cetered at x =. Prove that the series you fid represets cos(πx for all x. 6. Use multiplicatio to fid the first 4 terms of the Maclauri series for f(x =e x cosh(x. 7. Use divisio to fid the first terms of the Maclauri series for g(x = x cos x. 8. Use the power series of +x to estimate. correct to the eares.000. Justify that the error is less tha 0.000 usig the Alteratig Series Estimatio Theory or Taylor s Iequality. 9. Fid the sum: (a (b (c ( ( =0 + (+! = =0 ( x (+! (d 4! + 8! + 6 4! + 5! +... 0. Fid the Taylor polyomial T (x for the fuctio f(x = arcsi x, at a = 0.. Approximate f by a Taylor polyomial with degree at the umber a. Ad use Taylor s Iequality to estimate the accuracy of the approximatio f(x T (x whe x lies i the give iterval. (a f(x = x, a =8, =, 7 x 9 (b f(x =x si x, a =0, =4, x. Fid the Taylor polyomial T (x for the fuctio f(x = cos x at the umber a = π/. Ad use it to estimate cos 80 0 correct to five decimal places.. A car is movig with speed 0m/s ad acceleratio m/s at a give istat. Usig a secod-degree Taylor polyomial, estimate how far the car moves i the ext secod. Would it be reasoable to use this polyomial to estimate distace traveled durig the ext miute? 4. Show that T ad f have the same derivatives at a up to order. I problems 5 to 7, graph the parametric curve. 5. x(t = cos t, y(t = si(t. 6. x(t =e t, y(t = l(t +. 7. x(t = t, y(t =t / t. I problems 8 to 40, elimiate the parameter to fid a Cartesia equatio of the curve. 8. x(t = cos t, y(t = si(t. 9. x(t =e t, y(t = l(t +. 40. x(t = t, y(t =t / t. I problems 4 to 4, fid parametric equatios for the curve 4. x + y 4 = 4. y = x +x 4. Fid a equatio of the taget to the curve at the give poit. x = cos(θ+si(θ, y = si(θ+cos(θ; θ = 0 44. For which values of t is the taget to curve horizotal or vertical? Determie the cocavity of the curve. x = t t, y =t 6t 45. Fid the area eclosed by the curve x = t t, y = t ad the y-axis. 46. Fid the area of oe quarter of the ellipse described by x = 5 si(t,y = cos(t. 47. Fid the exact legth of the curve: x = t +t,y = l( + t; 0 t. 48. Fid the exact legth of the curve: x = e t +e t,y = 5 t; 0 t. 49. Fid the exact surface area by rotatig the curve about the x-axis: x = t,y = t ; 0 t.
Aswers +. 6 < + 6 = ( ad ( coverges (geometric r =. Thus + 6 coverges by Com- pariso Test.. lim Hece = 4 +7+8 5 +7 4 + +9+ =0 Compariso Test.. l >. Thus Test. 4 +7+8 5 +7 4 + +9+ = l( = ad diverges. diverges by Limit diverges by Compariso cos 4. + < + < ad coverges (p-series. Thus coverges by Compariso Test. = cos + 5. Sice 4 +7 < it is sufficiet to show that 4 5 is a boud o the sum. The R 4 x dx = 4 4. = 6. 0 = + =.7479977. Note that 0 x + x dx =.05, x + x dx =0.09504. 5. cos(πx = Thus, the sum lies i the iterval (.84409,.859977. 7. Coverges by the Alteratig Series test. By the Itegral Test, it does ot coverge absolutely. So it coverges coditioally. 8. Diverges by the Test for Divergece. 9. Coverges by the Alteratig Series test. By the Limit Compariso test (with b =, it does ot coverge absolutely. So it coverges coditioally. 0. Coverges absolutely by the Limit Compariso test (with b =.. Coverges by the Alteratig Series test (Use L Hôpital s rule. By the Itegral Test, it does ot coverge absolutely. So it coverges coditioally.. Coverges absolutely by the Limit Compariso test (with b =. /. R = 4 +7 = 4 = x dx = 4 4 4. (-6,4, (-6,4], [-6,4, [-6,4] 5. a+ 0 for all x: R =, I =(, 6. a+ 4 x < : R = 4, I =(7 4, 9 4 ] 7. a+ x < : R =, I =(, 8. 9. a+ x < : R =, I =[, ] a+ for all x: R = 0, I = {0} ( ( ( x = x =0 = ( x for x (, + 0. +x = =0. +x = +(x = ( = =0 (x ( (x for x (,. Itegrate the previous solutio to get l ( + x =C + ( (+ (x + : (C = l x. ( x = x d dx = x = ( =0 x x = 4. l( x = =0 =0 R (x π+ x + (+! ( = x d dx x =0 x + = f ( (0! x = l + f ( ((x! = =0 0 for all x x = : (R = ( + π (x (! 6. e x cosh x = ( + x + x (x! +...( +! + = +x + 5 x + 6 x + 7. x cos x = x x! + x4 4! x = 6 x4 0 + 8. +x = x + x 9 4x 8 + Thus,. 0 + 450. Sice the series is alteratig the error for this sum is less tha the size 7 of the ext term, which is 40500, which is less tha 0.00. 9. Fid the sum: (a ( ( = =0 (+ ta ( (b! = e (c =0 =0 ( x (+! = si x x = π 6 (d 4! + 8! + 6 4! + 5! +... = e 0. T (x =x + 6 x. (a + x 8 (x 8 88 : R f ( (7! 0.0004 (b x x4 6 : R 4 f (5 ( 5 5! 0.096. (x π + 6 (x π : cos 80 = cos 4π 9 0.74. T (x =s(0 + s (0x + s (0 x = 0x + x T ( = m: No
4. Prove by mathematical iductio or directly cosider the k th derivative of the polyomial T. 8. y =x x 9. y = l(l(x l( + 40. y = x x 5. 4. x = cos(t, y = si(t 4. x = t, y = t +t 4. y = x 44. vertical at t = /, horizotal at ±, cocave up whe t>/, cocave dow whe t</. 6. 45. 8 5 46. 5π 47. 0/ + l( + 0 + l( + 7. 48. e e 49. 5 π(47 + 64.