1196 Appendices
APPENDIX G: ANSWERS TO SELECTED PROBLEMS
Chapter 1
1.1 What is Calculus2, page 11
Le fimi, derivative, and incgral 3, The assumptions are comin refined
eh mt 1s, ;
", 19, 1G inches 2, 625 pounds
. 23.20 pounds 25. 15°C
27,240 29, 83.655. 31, S40S.045
320 M1
eb B20 Sh.
© 41.3 square units 4. G or T square units 43, 6 square units 47, a, 0.40
20h TLS years $3. 2,000 Ihvin?
1.2 Preliminaries, page 25
tea [-3.4) bjexes eoerel ae7
Ba ——: b -
= 0.4): a
ba vores
6
eof) Ue (fo) 23.
+0 15? = 006
OL Mx
we-2s 18
GE
3] 2813 27. 7.999,8.001] 28. (r+ PE HO
wed 1197
w.2- V3 0.2679
48. 2 b.
20.
period 2 period 2 petiod 2
aa b. 3 ”.
9.8 = 30,0 = 2D =
1.3 Lines in the Plane: Parametric Equations, page 33
Bovty-Se0
1 ae pay —
S.2y-
154y ty +3=0
0 98-756
arty
ca a.
(5.0.00, ~3) 1198
23. » 25.
Appendices
2
0), 0,-0.3)
ue
3 »
a
no slopes(—3.03
0.2.4.0) 0.-0.(-§.0)
41, DG,6), £2.16), or F(0,—19);thtee parallelograms can be Found. 43.
45,a,$40 b, 254 miles 47. 4.9, Appendices 1199
1a Fun
1s and Graphs, pnge 48
a2 f=: f
1D = (00,20); f(-=—1: [W=S; $10) =3 (=00.0) U (0,00): f
5.D = (4,00): fl = 1: (4) is undefined; £13) 2. = (00,00); f(-1) = sind OIE: £ (4)
JO) = Si) © O8MS 9D = (00,00); (3) =4 f()=2 fO)=4 LG 13. 10c + 5h a
19, a. not equal b. cual 21. no equal b. equal 23.a.even Deven 28.a,even b. oven 27. (fegytr) =v? + |
(eof) = 22 $2. 29. (FowyIx) = Sine +3) (eof) = Psion +3 He ueey =A? = lgGysu" Wales Se ~ 1
atu) = E35. u(x) =e) stan 3. uls) = sins: ge) = JB 3. PFS): QceosfCt0)) Ale ~.2
4B. VIO, 22V9,49VE 48, 24,45 47.03 9.0. 64 unies be Ad units SL a, S(O) = 23.344 ems
D. $6 x 10-7) 19.008 emis 5. a MA be = 3+ 392% 7.2 seconds
ao. 63 seconds, 288 1
55. a D = (06,0) (©,00) bem is. positive imeger e 7 minutes n= 12. e.sime approaches 3 minutes but never ges there
"maximum distance Wan @ = 48°" 38, §
1:5 Inverse Fu
ions; Inverse Trigonometric Functions, page 59
B.Yes &.NO_7.NO INO M.(6.4.0.0.0.D.49) 1y
ax ae x
Wy FAS aa § we Bang we wae VA; a
no inverse exiss inverse exists
BO7284 60.3968 4.25657 53.9 = tan 13 —tan~! 2 $5.0, 3.141592654: conece:
(9.38) = —0.1068, 1200 Appendices
Chapter | Proficiency Examination, page 62
18.a6¢48y 3720 bee + 1Oy— 41 0 ede—ay+5=0
1. , 20, a a. ?
& 3
+ Y 2
4 AE = 4 Tee ya
28,
28:
ts
ie 28, The 1wo functions are not the same.
—t STF 28. (fon) = sinV THIF (@=/100)= [cos]
10 |
ais WV = §xl2—24) Appendices 1201
Viste 1
2-112 38, does not exist 38. y
47 a3.sin“tx?
a5 7
vos
48.221 47.0 49,2 $1. =f and =} 53,0, function; not one-to-one; Ds (~00, 00% Rs 10.00)
b. not a futon e faretion: onetoone; D: (20,28); Ri (=e.98) A. not 8 neon, eft: Ao one-to-one;
Ds (96.2) (2.2) U QrowIeR: (8,1) UtTs09) $8. The altwdes meat a (3.34). The medians meet at (2
57.400; $208; in 3 years 99.0 bE eo B62 N se, yu E20) 64 False: =F
69. m, (96, 300) U (300.00) B.D Sx = 100 since x represents & percentage ¢. 120 warker-hours a. 300
va Bas =6ix dV =309=62 eV = NOS 2x dV = 15ei $8 =3evH
but 11 0.64209, ete deve e,trve 67. A, about 3.3562 m
eos
», about $9.3
60% Tha
mos tan! § 0.3906 81.
PS
at
Fe eT
Pieswon
a 4.6
0. Root location theorem gives result “
Chapter 2 Supplementary Problems, page 123
Ko R31 $.26v% 7.150 9.978025 11.281 13.4 1842 17.16 TE 2x 3235 25.4
2.2 WY Bl WE Bet HF WF Asin! 4B1 AKI ATO MD SL} SR —sinx 55.1 Appendices 1203
5] 65. continuous on [-5.5]
$7. secty 59. chy 6 pole atx =8 is not removable 63, continuous on [—
67. a, continuous on [0,5] b. discontinuous at x = —2- ¢, continuous on [5.5]
a
be. 1 6. The limit exists for any number that is not an even inieger
WAs oss 64 Tn
Wie
sang’ Yaa
seist 382
be, te eross the y-axis at (0,1), AS —+ 00,9 > 00. Asx + —90,y + 00. The smallest value of y = is #
77, ak =0.251n2, approximately 29.7% b. approximately 82.3% c. approximately 7.96% RS, Answers vary.
98. a. The wind chill for 20 mifh is 3.75” and for SO mifh is TF. bw 282 ea vad, TIL ar v= 45,7 ROR
99, This is problem Al in the morning session of 1956.
Chapter 3
M1 Am Introduction to the Derivative: Tangents, puge MMT
1. Some describe it asm five-step process:
1, Find fox
2. Find fr + xy
3. Find fer + ax) fee)
4 Find Ga + av) =Fanrar
5. Find im, o'er + Ax) = fos /a8
3. Continuity does not imply diferen
noe
lity but differeaabiity implies coniaiy,
fi
1a0 BO 13.8.2 b.2_BS.a,—dx 0 17.0: differentiable forall x 19.3; uillremiable fr all x
21.64 treble forall 23, —1/(2x2p diforonable forall 20 28.2¢~y-}3—0 W.3e—4+ l=
Deny 1=0 Bx43y—15=0 33,125" —5y 2190 35.2 bad
441 The derivative is 0 when x = J: the graph has a horizontal tangent at (4 y
43, The imi ofthe dternce quotient rom the lefts —t an rom the igh is +1 50 the fmit doesnot exis. 48. yes Ted
49} Sled Sia, 20; same praph, but shied yenially"b.2r $5. For the tangent lings hve negative slope and for 1204 Appendices
x > 0 the tangent ines have postive slope. Tere is coma at ad hares 20 angen atx =O, 30 href: drvative at
that point. $7. a.) = +E y= yO EE YAY by Bag + SE — BP2 — 29) + 67g +P — 2) 4 + Hy O=0
cole BERL 4, Soe Figure 3.14 in enbook y= 09= 0.7 ~05) 99. (0,
4.2 Techniques of Differentiation, page 182
110 11 03)2 Gd) de (1S) =2e (16) =2x 3,25)? BH)Gx N32 K)—% HT Sa F
Bead Dead? BAL 92-2 2S A eg PB tg Med 18, 3203 = $2
17 Bey Wea 4 ADEE Be MFG) = Set = 1S 1: $0) = BT = He: [ERI = De? — 30; FL4) = 12
2B, f" (0) = 4% PM) = ADAG FC) = BUH FM) = BAO 25.1 TTT HY +IRO
W.6e4y—620 Sex G+5—0 3 COand $B} 382. §) 37. 19,6) Meade —S Al 2r—y-2=0
Ba av-ty— 150 doy=0 45, (0,0) and (4,64) 47. The equation is not satis. 49, The equation isnot satisfied.
S51, IF isthe degree of the polynomial, PU cc) ~ 0 for every value of x
3.3 Derivatives of Trigonometrie, Exponential, and Logarithmic Funetions, page 160,
4
Feosx —xosctx toatx 9 —x¥ sins + 2x cos
sect — Qesee?x + 2tans
Lease sins 3.2¢—sing S.in20 2. Fain + $5
“eons ~ sing
n efeseu( cots) IS.x+2vlny 1 2etsiny 19% e-* (cose —sinx} 2 zt
gy, [eoeeb Roos =2—sint yg Lg 2e0sr—siny=I gy =?
ae oor 2 — cos Gin = eos =
3S, 2sec Brand 37. sec + secOun? 9 39, —sinx —eosx AL. —2e sins 4,
a7 Viv -2v + (1
fae £0,f00) 0.
9.2% -y=0 Shxwytl=0 ShAS
of Rates of Change: Modeling Rectilinear Motion, page 170
31 34 148 HF MLO 1X6 18.m2—2 b2 G2 dealways accelerating 17.9032 18-415 b, 6 ~
©.46 d, decorating on (0,3); accelerating on (3,6) 19a. 21-2 —2-3 B.A HOM! 6. de aoeeeratng on [1,3]
Bia. -Jsint b,—Scosr 6.12 a decelerating on [0,$) ad on (2..2] aeelerating on (3 2) 28. quadratic model
25, exponential model 27. quadrase model 29. cubic model 31 logarithmic model 38., ~6 b. The decline will be the same
cach year. 38, 136.nits 37.0, 9945 mimin b.9.960m 3.64 fvs b. 336 fe —32r +64 TMS 160 US 41, 30 1
SRT a5. ny = 24 U8, 126 47-8, 2001 + SDInr + 450 newspapers pot year. 1.530 newspapers Der year. 1,635,
vewapapers 4.8.02 ppm BOIS ppm «028 ppm SIs JI our 3.4420 pero pox me 0.395 per mo
ae s2N
aoe
4s hal th surface aca
BEN Jeost b2z G14 $9. V"(x) =?
Ising ale)
0.55(2), $0 the rate of change
35 The
in Rule, page 178,
3.6402) 5 BES 7. fafa + $)-+ Gr + SYsec? Gr HIS $] Base bare Grete OF Mea Tue
b, 8 — 24x c. ~56(3N +15 — Bx — 1267) 13, 25¢5x — 2)! 15, 83x — 1x7 - Ix +1? 17. desde +2)
19, (-2e + 3e-P 2, eM seex tame 23, (2 + Lexpl? +145) 28, Sx cos Sx +sinSx 27, 6(1 = 2x)-#
L¢stt3y7[_=tee 19 +5 yy cosv—siny 45 2
29. 4 , 3, SAME 35.2 37.1,7 OF Mal be ets
Be 3) le —sF] arse Snereose ASE ARNT OS A :
43.0.3 bE cf 48-2 a7. decreasing by 6 lbw 49, a ineresing by 0035 lus b. 71S m S172) 5.23
ls 2
1 sm plpicos sin B)cos? S8.6rty-+1S=0 59.
iss going houee 88.10) = OT
siolesrvor'e
"CO and
3.46 Implicit Differentiation, page 190
ysiny a 2" a 2
1 SRE 9. et — aby tage Soap ae
0. * 9, Ss, *
2
a Bae
ae DED Appendices 1205
28.2e ay 4
0 Mat benyte=0 Byao ML
28
wa
O normal line $3. 2.0),
3.7 Related Rates and Applications, page 198
1-242 S$ Dob Ob MIS HS IE I-20 HVE 210 28-3. 25-30% 27 Increasing at ate
of 48x in?/s 29. 126 untsyr 34. $4 poople per year 38. qinJs 37.2.5 fUs 38,S0 ih 41. 1.2 m/min 43. 30 fs
48. = pin $ in2/min 47.0.31 fms 49, a, 000139 atmospheres (approximately) b. aller 66 seconds
(approximately): 0006125 atmospheres $1.8 [us 83. seas $8. a, Vip) = 7.592 4 20. where ys the dep of the water
b.3.3 ialnin $7. 1.28 fumin 59, b, 8x kmnin
3.48 Linear Approsimation and Differentials, page 211
LGede Rx Fay $. (cose —xsinayde Te
Scords Me Se talnvids
SR ee EE EEE 5. the 19, 0.995; by calculator 0934884371
21,217.64 by caleulaor CY YESEEE 23.0106 oF 6% 25.0.03 oF 3% 27. 0.05 pas per million 29, reduced by 12,000 wits
31, $6.05 the aetsa norese I8 $6.15. 3.0.00245 mee! 38.2% 37. increases by 2% and ¥ increases by 35%
4, ~6.95 (or about 7 panilefunit ares 4B. u, 472.7 D. 468.70- 48.1.2 units 7, 14142 49. 05671
53. 5% ZS56GH4 57. Ar = —3, (97) = 9S; by caleulator SEMBLE: i As = 16, (97) 9189
‘59, No, as any positive choice for vo quickly leads to a negative xy for f(x) =x", and negative numbers are not in the domain
off.
Chapter 3 Prtceney Examination page 218
ints
* xfln3aF
a] 28.0 26. 20r= 354m? a +9
17, et 4 ful 2sin2e IB seo =H v9, 24
2,
Ss A ooo 2s -
maa ae Woe HT
21 Ge 2B. 1dr —y—G= 0 2% tangent tine y — $= FO — De normal ine y —
40-0 M29 fss
Chapter 3 Supplementary Problems, page 216
Seossx
Lav e6r—7 3
Hi, Ge + Shexpe? +e —3)
DIG
UEC) Ay sane am
rr wae ~
ese? VT oot JF 23)
ws tinsnotcas 28.5 ey 2) SELES amt caeeate a RDS
4x2 siny — 20s?
bey Ae 0 6 GB) 47.211 —2shvsine?) + 10s Geoen?) A, =
v4 sy? = 20y ~502
ya oy
59, 7.90 per year in 2014 61.0.3! pp
Mayes
39. 2ax tay = 0 45a, 4s
Le ty=2
shy’
‘53, Many examples exist, One such is y = by — Sh, $5. (3-99 and (—3,-9)
2667. Reads 11. 187217129. 75,72.0625. 77. -00012 rain
Co
19. dae does 0: va ABs 2A AE ASL Mmmm AI nC) = EY
dy Hae sind
dt” ed
fadians’s 95. For the velociy for acceleration,
7000
99, This is Panam Problem 6. ming esion in 1946 1206 Appendices
Chapter 4
4.1 Extreme Values of a Continuous Funetion, page 235
L
ria noes pons
is not mas)
the domain, £8
napa ileal manors Te
M =f3=9 £0= wafcne.2 |
M 9 | m=s(-2)=-16
9. endpoints ML
m= h@) =0
hQ) = 20?
SL AF Step 1: Find the value of the funtion atthe endpoints of an interval
13, SEs ese IN EY Peres ates ‘Step 2: Find the critical points; that is, points at which the derivative:
70) FO=t of the funetion is zero or undefined.
m=foysoaioe7 | M =s(3)=§ ‘Step 3: Find the value of the function at each ertial point,
Step 4: Sute the absolue extrema,
17. The meximorn value is 1 andthe ernimum value Is ©, 19. The maximum value fs 48 ad the minimum volue i677. 24. The
maximum value is 3 and the misiwum value i ~8§8. "23, The anton is not continuous and tere is no maximar or minizaum,
25. The mani vale is “e™/ andthe min value is Ge", 27."The maxim va 69 and te minimum vale
is B9, The wp Smale vale foeave Is eguve apd when > O° the vals get sale wou ll. 3de'The sles
vale eI. 30, The sae va /S— (32), 3, THe lags salu Onde sales ae ik—S. 37. The ge
value is 20 and the smallest value is ~20. 39. The lagest value is 0 and the smallest value is ~ 9/708, 41, The laegest value is
appzoximately 0.501 and tbe smallest value is 0. Answers 10 43 —48 nay vary
saa sor={
efor -ler Oand concave down for x <0
17a, cttieal number:
bs increasing on (€-",00), decreasing on (0. €-* eo
cc ered point (e~!--e~)rlave minimum ts
dno sovond onder erica numbers: concave up on (0,00)
as 1208 Appendices
25,
38. Att = $.rolative maximum: ate = 1,
relative minimum
37. As = 4, relative minimum
39, relative minimum at ty
‘lative maximum at -9
41. relative misisurn at 3;
relative maximum at ©
Answers for Problems 42-45 may vary
By a
49.8, —2) £60 + 1312) = 42; —}Oy' + JO}*+ 252) = 49h
b.NG) = 2 62+ Ly fd ot Ld $25) I=
fr bebeckis 1030s. Shanon sk ats = 2 shy" 4
57. f(x) = =x + 9x7 = 1
‘when x = 2.5, so the optimum time Appendices 1209
44.Ci
-ve Sketching with Asymptotes: Limits Involving Infinity, page 271
5.0. 7.3 99 ILL 130 15-} Moo IL 2-0 23.0 25, asympuotes:
(00,7) U7.00); concave up am (08.7); eoneave down on (7.00% no ertcal points; no points
2. 2%. i
raph rising on
a. »
a
49. populion wil be largest JSR Fase 55. False
after 26 mines: palo
poaches 5:00 fn the Loner
inecion pont at 9 minutes
Mtr which the ele a which the
Fepuimion deerenss pr mini
inst decease
Typ tai aiionisito
as 1210 Appendices.
45 VHopital’s Rule, poge 280)
1.8 The imi set an indeterminate Fors The cortet limit fs 2. b. The tim isnot an indeterminate form, The core iit is
134 154 17.4 19.0 24.0 230 25.0 27.2% 29.0 3Loo Boo
exist 4 hp Ay =O My se? f(a) = eta) =0
bl S3a bel
44 Optimization in the Physical Sciences and Engineering, page 291
1, y(—8)= 12 isthe maximum, 3. y(@5) =8 ig the asim, 5, y(-$)=8 isthe masimom. 7.
15. 10/2 x5v2_ 17. Height of cylinder is 2h usisr= BV6 19.r-= 203, 2h
2B, SOOR 28.33 fx 49 fe 27,200 niles after 1S he. BM, Is117 feet aller 2 seconds 33. a origin is atthe bouom center with
is (w) poting rit and y-axis(A) poining up. be 18k GSH de IGM @ 10M 38. 11,664 in. = 67580
2 mites owing al ihe way by 84 minules cowing 1 a pois 4.5 miles from B and running the rest of the way
39, 960,00 41. approsimately 403.1 £0? when @ = 1.153 43. b. When p = 12, the largest value of fis 6. 48. x
Mad
e relnp + ve*dnpy — ake ing
aaroe-t[ cto tne] ZTE ,
49. a. The maximum bei SE
9. 0. The he 32(m? + 1)
Ashe BR) oe
53.r= 3.8dem and = 7.67 em, $5, 15v3 emis a minimum, $7. The minimum value occurs when T = 4°C,
5.4, D. Boas = 018207, Bin = a COS Y Appendices 4211
hes ; 4.008 11.21—-s0-200- 8s
rea Res (SE); coy 50+: Pte) Dre eb The mim ol esse wie i Ss
20
item, The maxim pro's $700, 17, Profit is maximized when x = $60. 19,
rie Is 2%, 21. 208 years from now. 23.400 cases 25.a,7=8 es
people per year bh. The percentage
27, Since the optimum solution is over 100 years, you should will the book to your heirs so they can sell it in 117.19 years, 29, Sell
the boards ata price of $42, 31, Lower the fare $250, 33, Plant 80 lola ees. 38, 62 vines 37. a, The most profable time to
conclude the project is 10 days from now. b. Assume R is gortinaous on [0,10} and thas the glass is coming in ata constant zate
fivouuhout the time period. 39. a, és) = aptx) =
min ma (04)
%
decreasing on ($0) m3 Revere P= FT
fineton
aa risa maximum by
a
) Bye Grae "The largest survival percemage
4s 60.6% ancl tho smaliost
survival percentage is 22%
‘The largest hatching occurs when «
T 23,58 and the smallest wlien T= 30
0 oF
cate
this is abou 55°, 1212 Appendices
Chapter 4 Proficiency Examination, page 313
18.2 19.5 200 2e® 22,
2
29,19 in. x
% 10in, 30,7
Chapter 4 Supplementary Problems, page 34
1M 2X 7.
=p te Lan ae
%. y Appendices 1213
vw. y . minimum £2
27. avin ((/3/2) 0.6% minimum f(0)=0 29.0 3} 3.0 3S. doesnot exis 31 39 40 M2 45.0
47.55 ABulnS $1.0) $3, does no exit cosines) 85. fis Keton and gis dervacve
8. 2b=0
The derivative does nol exist ate
63. The maximum profit of $108,900 is rehed when 165 units ate rented at $740 cach, 68, The maximum yield is 6.125 th for 35
trees per are, 67, 28.072 ft of pipe laid on the shore gives the miniowm cos, 69, "The dimensions are 20 fe by 30 f 74. The
price is $90 and the maximum profit is $1,100, 8. 0p = V2. 79.x = Va leads wa relative minimum; x = Va leads to
areatve maximum. 83.06") > Oont—J.—-H)UC.2) BL < Von t—H1} es") > Don (4.2) de f"Ex) Don
J) < 0 0M (-0.876,0) U(0,0.876) € {"C8) > 0 0N00,2) da f"Ix) <0 ONG
x= RO8T6 L/"(s)docs nor exist atx =O Bf") 20 Base b.oo cm doo eco fe —oo 89, Disance
ved 10 be approximately 1.7812 when x ©0:460355, 91. eelative minima at (9.0) and approximately (4.8. 107.9: relative
(1.0.95) and approximazely (0.8.0.7). 93.4 = Kim “—* = fim.o © 14 ina 98. This is Putnam Problem 1
oF ehe morsing session of 1941.97, This i Purnam Probl
the moraing session of 1961.
‘of the morning session of 1985, 99, This is Putnam Problem | of
Chapter 5
1 Antiifferentiation, page 334
LUE RAEI HS SEES TAMkltC, %t0—adnwsC tank +e 18 taser
15.Ssinty- $C 17 de aPC i -tree
ie Beare 2 § VEC 2x sine eC Appendices
1214
38. Fer) = Ina} a
+9 = 40
3
+
?
4
i
3
z
AT.att) = 43 tus
20 us
45k
38. 1¢In2 97.
g
E
i
S88 23. 1.183
tg
ae
25
a3
Se
_&
a4
24
ge
a3
“5
6
25,0415
‘S11 square unit
49. 3.4 square unit
42.1 square unit
23 Riemann Sums and the Definite Integral, poge 364
28875
19, 1.512
17.0791
13,675 15, ~0.125
1 175
3.0714 33,
31079 50875 7.1183 9, 1.942
228
53.2.3; P|
a
lus, page 362
S44 The Fundamental Theorems of Cale
1 maT
tat
Te 1K Mh
35
5.
3.16480
Lo
TT 9,
»
me mA
2
=o
bee
53. a, relative minimum, 1215
Appendices.
47, 10 V2
Ore 3° 4C
+340
iH
THI EC BM Mla +E WinVEFNC Sine + HE
1a +e) —In2 a.a, We take 1 Frdor asthe variable, so the note from the
The Dirichlet fonction is defined as
Funetion so that f(s) equals 2 determined constant ¢ (usually |) when the variable x takes a rational value, and another constant
11, —foos3bx74C 19. Fine
VIVA +C wfar-sec 9,
Hint 3ezi48 87.20%: §
g
i
Ege
225 a8 sea
Bae fu ’
g 21 a
aL Ties yt 22
Tei gecde ta
“sae Ee Big
wou eee i
Serer) i
ee gE ES oa z
Zee Shey 5
ote B22 Er ug «OF
falegEe a
neog BEE SR “3 8
it eh goess 72 4
ig oes'soi 2
2idgbisas 4:
sbazegs?s fB E
fiadeligas eg
=C Shy=tr
© Bay
Jane
© m-ju-y
25, cos. + si: 1216 Appendices
s :
Orthogonal
vrajectores
‘@ andl d are onhogonal irajecoris; b and e are orthogonal trajectories: ¢ and fate onthogonal trajectories
approximately 10,523 years $3. $ min $8.0, 985 fsb. «© 3.986.067 mi; h 382 fe 57, 2 hr and $3 2
‘The Mean Value theorem for Integrals; Average Value, page 389
1, 1.85 is in ie interval. 3, 1.055 is in che imerval. 8, V3 is in the interval. 7. The mean value theorem does nae apply because
1be function is discontinuous at 0. 9. 0.0807 isin the interval, 1. 0.4427 is in the interval
13. The mean value theorem does nat apply hecause the funetion is discontinuous at 0,
1A =35 Ask 19. A 1.839
y
2g 2-2 28.0 an $—Fing 29.4 (4-2v2) ang 9B aso 37,
41 43. $16.10 45,5318 47.60F 49,
=k, C=O gti) = he
erm) 9. § (Vd~
E(t + tob-b ty SL. 45.2°F $3, 18 months $5, 14 minutes $7, sect Appendices 1217
58 Numerical Integration: The Trapezoid Role and Simpson's Rote, page 398
Ra 10 125 61267 Sa LAL WINGS 61267 10.016 B02 0125 90.0135 ¥.0.125 «0.125
1. Tripeoidal ule, 2.975 Simprons rae, 2.39333; exact vale, 13, Taporidal rule, 5.146; Simpson's rule, $.252 exact valve
HE 150.0.7828 b. 0785. I7.m. 2057871 b. 2.018596 19.
0.580" b.0594 21, A'* 0.75; the ens asver is between
0175~ 005 and 0.775 +005 23.0455; the exet answer is beween 0455 —O.OHIS and DSS +OONIS. 25,A~ 3.25; the
ct answer is between 3.25 O00 and 325-4001 27. = Ott the exact answer is between O44 ~ 0.01 and 0.44 +001
Wan = 164 bw =I Mean = 184 ben Ban =82 bow 8 353.1 3% = SIR 39 100 yd 41. 50.38
Ini 43.79.17 48, The onder of consergnge is 12.47, The oer of convergence is. 49. Simpson's rule yieks 2006095
tpproxinutely because the ero is 0 singe f"%x) ~ 0 for cues, Sh. Answers vary, but the calelated value should be
appoxinntely 314 em’, 3. Left endpoint, 20841; Trapezoid 25525: Newton-Cots, 25975; Exact, 2.5958 frm computer $8.
S72
