Facit liste

Siden indeholder svarene på nogle eksamensopgaver (med forbehold for tryk- og andre fejl)

C - niveu opgaver

maj 94 1a: 6.2 1b: 28 1c: 552600kr 2: f(x) = x + 3, L = {–1, 3} 3: 31.8m, 2.86° 4: 0.0038, 0.5893 5: 27.72% 6: 78.3shore, f(x) = 7.82·1.033x, 3.3%, 17.6% 7a: (5år, 8år, 12.4år), 66.8% 7b: [4; 10]

august 94 1a: 1755.00kr 1b: 1.7 1c: 0.136 2: 222.2, 187.5 3: 62.68%, 55.52% 4: 2.56cm3, 6.45cm3 5: 32ppb, 25, 45, 26, 4 6: 7.6, f(x) = 5.48·1.09544x, 15.74 7a: P(0)=16/36 P(1)=8/36 P(2)=9/36 P(3)=2/36 P(4)=1/36, 1/3 7b: f(x)=–0.4x+1.2, 1.2, 3. 3.23, 21.8°

maj 95 1a: (12, 18, 30) 1b: 23.46° 1c: 20982.25kr 2: 14.97, 4.42 3: 28, 0.70 4: ]–2; 8], [–1.5; 6], 5.5, {–1; 5} 5: 33.56°, 3.41m, 21.80° 6: f(x)=0.096x+1.0, 987atm, 0.96atm, 302m 7a: 113.65·0.8629x, 7.89° 7b: 0.9914, 0.0815, 7

august 95 1a: 0.7625 1b: 30015kr 1c: 11.0 2: 5.61, 35.4°, 54.6° 3: 1863.45kr, 2117.50kr, 4.40% 4: 40.7d, 0.0401x+3.371, modellens anvendelses område overskredet 5: 6.92t, 9.5%, 24.51t 6: 64, 76%, 0.14 0.24 0.30 0.22 0.10 7a: 64.4%, 78.18% 7b: 63.6°, 2.22m

dec 95 1a: 3017.73kr 1b: 4.05 1c: ]–2; 4], [–1; 2] 2: 1/2, 8.5, 11, 13.5, 13 3: 6, 0.238, 0.1916 4: 2.23·1.219x, 3.5, 12.0 5: 19.42°, 20.1cm 6: 450kr, 19477.85kr, 45 7a: 7.17·1.357x, 35.7%, 15486 7b: 3.96m, 5.5m, 54000

feb 96 1b: 19.0% 1c: 0.7358 2: 3326.15kr, 4.0% 3: –0.5x+2, [–3; 5.5], 6 4: 4.4, 0.4, 0.9 5: AC=8.65, AB=4.60, CD=9.8, A=123.7°, C=84.3° 6: 3.65, 9.054·0.957x, x < 4.70 7a: 2002g, 2.6% 7b: 38034km, 5.96kr/l

maj 96 1a: 3.70 1b: [–1; ¥[ 1c: 5.38 2: 0.75x+1.5, (–2, 0) 3: 5.79, AC=11.59 AB=13.05 A=27.37° 4: 0.0214, 0.6578, 0.6722, 26.7 5: 0.47%, 5.16% 31.24kr 6: 31.4år, 1297mio, 297mio, 2.30mio 7a: 84.51%, 19.32%, 39.98% 7b: (71.4, 81.25, 87.5), 35%, 93.3

august 96 1a: 48/11 1b: 9500kr 1c: 0.087 1d: 37.78° 2: [–1; 3] 3: 1.5, 2.7, ]0: 9], 2, 1.5·1.2917x 4: 6.53, 3.01, 2.79, 5.80, 6.70 5: 252.5sek, 1677sek, modellen overskredet 6: (21.8år, 26.9år, 32.7år), 12% 7a: 34201kr, 137605kr 7b: 12.04m

august 96 1a: 29 1b: 3.22 1c: 6396.49kr 2: 12, 53.13°, 20, 16, 150 3: 2, –2, –x – 2 for x£2 2x – 8 for x > 2, {–4, 8} 4: 0.02087, 0.68721 5: 7635.50kr, nej, 4500kr 6: 392·0.9806x, 3.92, 35.36m, 81m, 69.2% 7a: 37.9%, 122.5, 135.47b: 0.3, 5.4, 0.6, 0.5

dec 96 1a: 5.05 1b: 926200kr 1c: 132215.24kr 2: 2.81 3: 53.25, (46.5m, 53.0m, 58m), 4: Y=0.27x+83.99, 83.99g, 89.4g 5: 37.8°, 62.2°, 4.8, 6.66. 6.33 6: 7.09%, 6200·1.0709x, 10,12år, 2003.14 7a: 7/16, 0.0013, 0.01, 0.17 7b: 35.87°, 25, 11.5, 6.25

feb 97 1b: 12.74 1c: x <= 3 2: 3071.12kr, 4.25% 3: 0.23x+8, (44, 19) 4: 40, 21.2°, 110.1m 5: AB=8, AC=DC=6.93, BAC=30°, CAD=ADC=68.85° 6: [3.09; 27.51], 7.55, 16.167·0.8475x 7a: 0.9087, 0.0032, 0.8609 7b: 1299.46kr, nej, 221md

maj 97 1a: 0.25 1b: 1837.56kr 1c: 1.49 1d: 1.25 2: 1/3 x – 1, 6 3: 7.21, 33.69°, AB=6.22, A=40°, AC=10.77 4: (12.53, 25.83, 49.89)h, 25% største > 49.89h, 40.24h 5: 5180·0.966x, 1835, 0.034 6: 227kJ, 32°C, 1355 7a: 0.64%, 1666.56kr 7b: 0.0544, 0.80

august 97 1a: 1/2 1b: 52% 1c: 15.9° 2: [–2; 6.5[, ]–2.5; 3.5], 2.5, {–0.5, 1}, {–1, 6} 3: 10.98·1.131x, 5.63, 36.654: 0.1, 0.4, 0.3, 0.2, 2.6 5: 151g/m3, 60% 6: 200cm, 215.4cm 7: –0.01, 25, –0.005, 12.5, 3000m 8a: 9348.23kr, 1.5% 8b: 24.8%, 94.6%

dec 97 1a: (1.3, 3.3, 4.3) 1b: 69282.28kr 1c: 132215.24kr 2: 0.5x+1.5, 1.25 3: 4.92, 15.1 4: 67.91x+213.1, 824, modellen overskredet 5: 12.0m, 36.87°, 7.05m, 2.57m, 1.09m 6: 0.7179, 0.2707, 0.77 7a: 3798.71kr, nej, 67md 7b: 21.6%, 9915, 29.15%

jan 98 1a: 351kr 1b: 2.18 1c: 2: 3x+110, x<=17 3: 6081.87kr, 6132.74kr, 3.72% 4: 17.9mio, 22.6mio, 2004, 2.34% 5: 46,4 105, 40, 77.4, 32.78°, 100.73° 6: 70%, 16 7a: 0.4032, 0.1268, 0.9002, 6 7b: {–1, 7}, ]–¥; 9]

maj 00 1a: –7 1b: x >= –2.5 1c: 1/6 2: 75.9, 22.08, 5 3: 35775.56kr, 53351.07kr, 5.5% 4: 2.8 4.15, 49.45°, 12.1 5: y = 0.3x + 10.23, 47.57g, 2.4cm 6: 0.1215, 0.5925, 14 7a: 0.268, 0.38 7b:

jan 01 1a: 143 1c: 58.9 2: 6.34, 5.74, a=4.04, C=138.0° 3: a=–0.75 b=1.5, {2, 7}, ]–2; 8[ 4: 10389.30kr, 262.05kr, 12 5: 124km/t, 17%, >144km/t 6: 114.3 0.9564x, 15.55, –48.6, 4.36% 7a: 0.2938, 0.6062, 12 7b:

maj 01 1a: 3.8 1b: 5.42 1c: x>=25/6 1d: 1208.80 kr 2: 105740.50kr, 3.57% 3: 3.00, 2.00, [2; ¥[, ]1; ¥[ 4: 4.20, 63.46°, 6.13, 3.18 5: 0.10, 0.30, 0.20, 0.15, 0.25 1.5 6: 171575·1.59x, 59%, 2770782>>1568414 anvendelsesområde overskredet 7a: 0.584, 0.061, 0.624 7b: y=0.236x+3.09, 61, 131

nov 01 1a: –12 1b: 2784.40kr 1c: 0.1239 2: ]–7; 8] og [–3; 1], [–2; 5] 3: 2.6, 0.47 1.226x, 12.73, 50% 4: 9.42 og 7.42, 19.22 og 17.57° og 20.43° 5: 3.45kg, 13%, >4.15kg 6: –0.02x+151.2, 61, 10 7a: 3.84%, 78.3, 125.5 7b: 2.50m og 2.0m og 2.66m, 17

jan 02 1a: 9.5 1b: 8.3% 1c: 1.58 2: 34000kr, 30%, 29840kr 3: 1586.13kr, 1541.57kr ca.= 1542kr, 30 4: f(5) = 10 og f(–3) = –2, f(x) = 1.5x + 2.5, 3 1/3 5 5: 23.02, 66.5°, 12.0, 6.56, 82.3 6: 16.1m, 4.99 0.958x, 4.2% 7a: 7b:

maj 02 1a: 7200 kr 1b: 73.18% 1c: 19.06% 2: 25.52°, 16.76, 52.06°, 13.42° 3: 3.25, 75% 4: {-3, 11}, ]1; 8[, f(x) = 0.5x+1.5 x <= 5, f(x) = -2/3 x + 7 1/3 x > 5 5: f(x) = 1274.5 · 0.854x, 13.56 km, 7.59%, 4.39 km 6: 19.28%, 2.72% 7a: 25%, > 1.5 ppm 7b: 13000.66 ca.= 12999, 23%, 441.50 kr

dec 02 1a: [–4; 6] og [–3; 2] 1b: 1.3 1c: 1.5 2: 31.0°, 5.83, 7.68, 7.07, 6.22 3: f(4) = 2 og f(–2) = 5, x >= 6 4: (12.3, 13.4, 14.6)km/l, 60% 5: 25445.40 kr, 38, 1006.08 kr 6: 69.9428 1.0959x, 145.5 enhed, 5.89°, 7.57%deg;,150% 7a: 2.85%, 25.3% 7b: 4.2%, 33.80 kr, 40.58 kr

maj 03 1a: [3; 5[ og [–1; 5] 1b: 0.431 1c: 15000 kr 2: f(x) = –0.25x + 65 , 252 3: 14, 9.51, 4.45, 6.32, 44.77° 4: f(x) = 2.778 1.011543x, 7.80 l, 107.3 km/t , 60.45 km/t 5: (2.6 cm, 3.1 cm, 7.7 cm), 37% 6: 10864.98 kr, A = 12155.77, G = 12144.00 kr, 14.03% 7a: 5074 kr, 3903x + 3895, 7.9 år, x > 2.33 år, forbrug = 3116 kr 7b: 5.96% årligt, 5.35 l/pers, år 2014.7, 11.27 år

dec 03 1a: 1.2 1b: 3.45 1c: 660000 kr 2: 40.7°, ED = 2.15, BC = 1.31, HC = 1.10, AB = 1.4 3: a = 1.101, b= 145, f(10) = 380 mio, 2007, 61.8% 4: 477157.30 kr, 15 år, 5.99%, 648000 kr 5: a = 10.17, b = –11.7, 34 g, 2.1 md, 2.5 g 6: 0.952, 0.037, 0.553 7a: 0.0696, 0.109 7b: 3687, 177, 154

maj 04 1a: x>= –6 1b: 46.4 mia 1c: 2.3% 2: 12m, 29.74°, 24.19m, 26.34m, 45.48m 3: 2908 kr/md afdrager 249993.56 kr, 114 mdr svarer til 3014 kr/md og 115 mdr svarer til 2995.4 kr/md 4: f(x) = 0.218 x + 56.02, 41.25°, 1.01 ohm 5: f(x) = 729 (1/3)x, 0.631, –2 6: 51.3 cm, 28.6 % 7a: 657.41 kr, 7.67 % årligt, 575 kr 7b: 0.35, 0.20, 5.4, 0.45

B - niveu opgaver

august 93 1a: 53.13° 1b: {0.4905, 2.6511} 1c: 3x/Ö(3x2+1)2: –1, –2+Ö2, –2–Ö2 3: y=4x+4 4: 2Ö5, (x–7)2+(y–3)2=20, (7±Ö11, 0) 5: 26.11°, 47.65°, 197mm 6: 7a: 0.254m 7b: 52·x0.621, 32.32·x–0.379), 7.6

maj 94 1a: 2(3 ln5+1) 1b: Ö(13+x2) 1c: e2 2: (–2, 3), 5, y=–4/3 x – 2 3: y=9x+22, (3, 49) 4: 2.336, 1.188, 3, 1/3 5: 44.38°, 67.52°, 76.52°, 30.8, 36.86°, B=67.52°, C=30.66°,P=101.82°, 6.7 6a: 94,71s, 3.00mmol/l, 0.0348e–0.0116t 6b: (p/2, p/2), (p/6, p/6 + Ö3), (5p/6, 5p/6 – Ö3)

august 94 1a: –ln4 1b: 14 sinx cosx 1c: ]–1; 3[ 2: 69.32, 67.34, 1454 3: y=x+2, x=–1 4: 0.5 T=(2, –2) 5: (x–1)2+(y–2)2=25, (5, 5), (6, 2) 6: 2.0449, 0.0001156, 56% 7a: y=–0.5x+0.5+ln2, (1/3, 5/3–ln3) 7b: Ö48

maj 95 1b: 0.5 cosx/Ö(7+sinx) 1c: 33.69° 2: 16.84, 26.81, 11,36 3: y=16x+48, –3, 1 4: 71.37db, 0.2995, ja 5: –ln12 6a: 1.783, 11.56, 39.79, 4.95, 18.57% 6b: (–2, 3), 5, (–2, 8), (2, 0), y = 0.5x – 3, 14/Ö5

august 95 1a: 7.67 1b: 1c: 14/Ö10 2: a=Ö40, b=Ö52, c=6, A=56.31°, B=71.56°, C=52.13°, 19.0 3: f(x)=2x+3+4/(x–3) 4: (–4, –1), (0.8, –3.4), 63.43°, y=2x+1 5: 5.1+9.2/x, 4 6: 16200/x, x=155.88, b=103.92 7a: 1.7967, 1.4, [0.2, 1.4[ 7b: 1, 4, y=6x–6, [–9, ¥[

dec 95 1a: (2x+x2ln5)5x 1b: 2x2+5x+19+69/(x–4) 1c: 2: ]–¥; 2[ È ]5; ¥[ 3: 0.63, –0.6316, –10.9% 4: 4.77, 37.1°, 5.65, 15.31 5: {–2, 2}, y=3/2, [–3; 2.4] 6a: (x+1)2+(y+2)2=25, y=–4/3 x+5, 73.7° 6b: Ö(x2+2x+8), 3/4 x +2.5

maj 02 1a: 1b: ex(Öx + 1/2Öx) 1c: 2: (2, 2), y = 2x – 1 3: 27.56, 133.36°, 40.89°, 508,08 4: 5: (7, 0, 5Ö3), y = 2x – 14, (4, –10), (10, 6) 6 f '(x) = (2x2–20x+332)/(x–5)2, voks: x < 2 og x > 8, f(x) = 2x + 3 + 18/(), y = 2x + 3, x = 5 7a: 1.19 m/s, 1.60 m, 0.146 7b: 40%, 3748.7

maj 03 1a: 60.95° 1b: (2+cos(x)+x sin(x)) / (2+cos(x))2 1c: x = 1, y = 2 2: {–5, –1}, (–3, 8), ]–5; –1[ 3: 14514, 12445, 54.46° 4: C = (–9, 3) og r = 5, y = 4/9 x + 20 / 9, |CQ| ca. = 18 > 5 + 11: intet fælles punkt 5: aft i ]–¥; –2[ og ]0; 2[ voks i ]–2; 0[ og ]2;¥[ 6 f(x) = 3.579 x1.281, 14.58 W 7a: 41.45°, 28.66 min, f '(t) = –0.667e–0.023t, f '(15) = –0.47° hast, hvormed temp ændres 7b: R, {2}, y = ln(2), (l.b. = 0 r.b. = 1 g = 0.5) {0.75552809}

maj 04 1a: T = (2, –6) 1b: 5x4sin(x) + x5cos(x) 1c: y = 5x – 18 2: 34.7°, (x – 1)2 + (y + 1)2 = 25, (1, 4), (–2, –5) 3: 71, 21.89°, 59.94°, 29.33 4: f(x) = 0.9466 x0.5401 , 64.77, 1548, 45.4% 5: 2x – 2/x2, aft i x < 0 og 0 < x < 1 voks for x > 1, y = 3.5x – 2, (–0.5, –3.75) 6 V(x) = px2/2 ·l, p(19 – 1.5x2), 141.6 7a: 65, 3.16, 6.02 7b: 6438, 35.12, 9560

aug 04 1a: 4/√10 ≈ 1.26 1b: –7/(5x–7)2 1c: C: (4, –1) r = 6 2: 6.04, 44.27°, 19.73°, 18.64, 45.53 3: (3, 6), y = 2x + 2, (7, 2) 4: f '(x) = 0.5x3 + 1.5x2 – 3x – 4, f aft. i ]–∞; 4] og [–1; 2], [–8; ∞[ 5: 14.96, 17,87, –1208·0.207e–0.207x < 0, minimal klækningstemp 6 f(x) = 4295x–1.20, 24, 49, 38.6%, 123 7a: 5, 96, 3.63 7b: y = 5, x = –4 og x = 1, b2 + 16 > 0

maj 05 1a: 54.5° 1b: (1 – ln(x))/x2 1c: y = 2.5 og x = –3 2: y = 2x + 1, 2√5 ≈ 4.47 3: 34.92, 16.60°, 15.10, 148.0 4: f(x) = 54.86x–4.223, 247, 0.43 5: f '(x) = 3x2 – 5x – 2, aft for [–1/3; 2], –1.578 6 (4, –2) √50, y = 7x – 80, (–3, –3) og (–1, 3) 7a: 1095 og 1466, 24.22, 17e0.085x, 12.7 7b: 22.24, 26.4, 7.026 ≈ 2 min over 7, 2.444cos(0.26t + 0.99), –2.029

A - niveau opgaver

1.001: 5 1/3 1.009: 1 1/5 1.011: e – 1 og 1 1.015: 5/6 og 2 5/6

3.038: 2 og 2p(ln(2) + 3)

maj 00 1a: (2, 2, 1) 1b: (5, 6) 1c: 3·1 + (–6)·½ = 0 1d: 5/6 1e: (x+1)2 + (y–7)2 + (z+2)2 = 25 1f: check 2: 4, 42.27° 3: 2 2/3, 2.4p 4: (3, –1, 8), –(x–3) + (y+1) + 3(z–8) = 0 eller –x + y + 3z = 19 5: (0, 2Ö2), y = 3x + 6, (1, 1) 6: y = 2/3 x, y = –2 / (ln(x2–1)+2), (x < –1) u (x >1) 7a: 90.91 – 30.91 e–0.22x, 4.73 7b: 15/ln3 – 6/(ln3)2 = 8.68, 1/4

august 00 1a: 2 1b: 4, (3, –1, 2) 1c: (1, 0) og (e2, 0), (2, –1) 1d: 1, 2 1e: 0.3 1f: 4, x – 2y + 2z – 6 = 0 2: –1, 41, 29, 1/Ö29 = 0.19 3: 2 ln(5/3), 8p/15 = 1.676 4: 5: –½cos(x2+1), (x+1)4x/ln4 – 4x/(ln4)2 6: 303, 7a: y = x – e + 1, f(x) = exlnx–x, R+ 7b: y = Ö2 sin(2x)

X nov 00 1a: 2 1b: (–1, 4, –5), 6 1c: 40 1d: 4/3 1e: y = 2x+2 1f: 3y + 2x = 7 1g: f(x) = –x2+3x+6, g(x) = –x2+3x+8 2: (8, 6, 0) + t(–4, –3, 8), 2x + z = 16, 63.43° 3: 76.11°, 60.67, 6.25 4: 3.5p + 2 – e½p 5: (15 – 0.084t)2, 59.5 s 6a: (8, 0), (0, 8), (0, 16), (&3150;1, 9), (0, 8), 34 2/3 6b: y = etan(x) – 1, 87.47°

nov 00 1a: –1.5 1b: 3x + 7y + z = 49 1c: (x, y, z) = (–4, 5, 7) + t(3, –2, 1) 1d: (–3, 5), Ö34 1e: –17, –6 1f: check 1g: 6 2: Ö68, (0.6, 1.8), (–1 +-Ö5)/2 3: 3/4 p2 + 2 = 9.402, 91.37 4: (–3, 6, 4), 22.2°, 5: –2000, 11128 e–0.4t, 30 – 23.87e–0.2t, (3.4386, 656100) 6a: ½e(e + 1) = 2.335, (11e12 + 1)/144 6b:y = sin(½px) – 2/p cos(½px)

jan 01 1a: 1 1b: (x – 3)2 + (y – 1)2 + (z + 4)2 = 1/4 1c: y = 2x – 8 1d: (–4, 0), (0, 0), (0, 8), ortogonale 1e: test 1f: 36 2: (–10, 24), 12x + 5y – 109 = 0, –4, 4 3: s = –1, t = –2, 71.57° 2x + y + 2z – 3 = 0 3.375 4: 11/6, 116, ½ 5: y = –(sqrt(e3x2–10x+6 – 1)) 6a: v(t) = 0.1/a (1–e–at), 575 sek, 0.1/a a>0.002 6b: -50e–4+20e–1 = 6.44

maj 01 1a: 16 1b: 2e + 1 1c: {–1, –5} 1d: y = –1.5x + 2 1e: 1f: 2 – 5 = –3 2: 2/3, 206, Ö321 3: 152.8°, 8.94, (10,20,5) + (3,0,2)t, (310/13,20,185/13) 4: 12.2, 361/(1+46.53e–0.20577x), 22.6 5: 4/ln(2) = 5.77, 8(1 – ½t)/ln(2), 8/ln(2) = 11.54, 6p/ln(2) = 27.194 6a: f(x) = ln(x3 + e), x > –e1/3 6b: ln(2), p/8 – 1/4 = 0.1427

aug 01 1a: 5 1b: (3, –4, 2), 6 1c: 6.5 1d: (–1, 1, 2) 1e: 9/4, 2 1f: –x2/4 +5x/2 + 10 2: (–4, –2), (1, 6), (–3/13 (–1, 5), AD = BC, 22 3: 38.68°, 4: y = 50000/(1 + 4 exp(–1.96t)), 882.7 5: –2.5x + 10.5, y = Ö(2x2–10x+4) 6: 6p, 8p/3(Ö27–Ö8) 7a: p2/2 + 2, (e2 + 1)/4 7b: (0, 1), (0, 2.5), (0, 1), y = 2 + Ö3p/6(x + 3)

nov 01 1a: s=–1/3 1b: 2/3 1c: Ö45 1d: C(4, –2, 3), r = 6 1e: ln4 + 14/3 1f: a = 1, b = –2 2: 82.87°, (0.4, –0.2), t = 5 3: (x, y, z) = (0, 0, 220) + t(16, –16, –20), S(176, –176, 0), 20x – z – 120 = 0, 87.12°, 1201.5 4: y = –4x + 2, f(x) = 1/(ln(x+1)+0.5) 5: V = p(1/6 - e–4/2 + e–6/3), 2/3 (1 – e–2)3/2 6a: y = –0.019x2 + 0.0017 x3, –0.0625 6b: (0, 0), (–15, –3), (15, 3), t = ±Ö3

maj 02 1a: 16 1b: –2.5, 5 1c: (xex)' = y + ex 1d: (3.4, 1.4, 0)+t(–4, 1, 5) 1e: 0.52, 0.78 1f: 2x – y + 2z = 15, (–1, 3, 1) 2: 69.83°, (2.16, 2.88) 3: e2 – 3, p(e4/2 – 2e2 – 1/6) 4: (6.72, 11.76, 20.16), 12.14, 42/(5Ö13) 0 2.33, 147.58° 5: y = –0.5x – 1.5, f(x) = –Ö(2 ln(x) + 4) 6: ln(11/7), 7463/6 – 14197/42 7a: (0, 0) og (3, 0), t = ±Ö(3/5), 16 7b: 20.78, (8, 2, 13), (16, -22, -3)

aug 02 1a: x2 -3x + k 1b: 22 1c: (x, y) = (1, –:3) + t(–3, 5) 1d: 1e: 1f: 2: 3: 4: t = 0.25, t = –0.25, t = –1, t = 0.75 5: N(t) = 200/(1 – 0.5e–0.02t) -> 200, R(t) = 300 + 100e–0.05t -> 300 6a: (2p, 4), 2, 12p

dec 02 1a: ln 2 + 7/3 1b: {2, 3} 1c: x – 5y – 2z + 15 = 0 1d: (2, 0, 1) og (8, 3, –4) passer i lign 1e: 6 1f: {0, 3} 1g: y = ex – e – 1/3 2: (2.8, 1.4), 16, s = 0.5 og t = –1 3: 112.95°, (8, 0, 16), 161.10°, 15.36 4: 2, +–Ö5 5: 2.5, l(t) = 30 – 25.99 e–t/6, V(t) = (10 – 9.99 e–t/6)3, 1000 6a: 7/3 + 7 ln 2, ½ ln 1.125 6b: (0, 0), (p2, 0)

jan 03 1a: –1 1b: –10 1c: (3, 1) 1d: venstre side = 3x3 + 12x2 og højre side = 3x3 + 18x2 – 6x2 1e: (1, 4, 3) og (3, 2, 3) 2: 22.38°, (8.5, 8.5), –3.4 3: Ö7 / 4, 135.58° 4: 86.07, f(x) = 665 / (1 + 65.5e–0.54x) 5: y = 4/3 x + ln 1.5, f(x) = ln(ex – e–x + 1.5) Dm(f) : x > ln 0.5 6: 2 ln 4 – ln 2, ½(ln 4 – ln 2) 7b: 0.3, 3/28 p

maj 03 1a: F(c) = x2 – x – 5 1b: y = 2x + 8 1c: (x – 2)2 + (y + 1)2 + (z – 3)2 = 41, 3(x – 5) + 4(y – 3) – 4(z + 1) = 0 1d: venstre side = 2x lnx + x – 1 og højre side = 2x lnx – 2 + x + 1 1e: (x, y, z) = (–1, –1/3, 0) + (2, 1, 0)t 1f: h(3) – h(1) = 82, a = g(1) = 12 så y = 12(x –1) + 5 2: 5.91, (x, y, z) = (8, 12, 0) + (–4, 0, 32)t, (6, 12, 16), 232 3: 7.5, 1.5, 7, 21.79°, –0.3 4: 2/9 e3 + 1/9, 2/3 + 2Ö2 – 2, –5/64 5: y = 1/(cos x + 2), 0.08 Ö3, (p, 1) 6a: (0, ±Ö2) og (–2, 0) og (–1, 0), retningsvektorer: (2, –2) og (–2, –2), (–1, 0) og (–17/9, 8/27) 6b: 4, "symmetri", 1024/105 p

aug 03 1a:e2 + 1 1b: y = ½x 1c: y = 3x + 7 1d: check, 4 1e: (–1, 0, 1), Ö2, (–1, –1, 0), (–1, 1, 2) 1f: 22,15 2: (–0.2, –0.6), 82.87°, (2.8, 3.4), 16 3: Ö90, 5y + 8z = 45, 32.01°, 4: 1.6 l/m, V = (13–0.1t)3/27, 130 m 5: 2 – Öe, p/12 6a: (–6, 0), (–5, 0.5), (Ö6)3/3 = 4.9 6b: N = 2500/(1+249 e–1.2 t), 927.25, 4.16 uger

dec 03 1a: 2/3, 32/5 + e2 – 1 1b: 1, 6, –9 1c: (x – 3)2 + (y + 1)2 + z2 = 36 1d: check 1e: fig1 er F, –2 2: (6, 4), 82.23, 2.75 3: y = 1/e x + 3, f(x) = (1/4 lnx + 7/4)2, R\{0} 4: 48.6, (15, –75, 60)·(1, 1, 1) = 0, 5: 17 1/15p = 53.62, 2/3 a3/2, 25/3 6a: (1, 0), (e, 5/e), 5/2 6b:

maj 04 1a: 42 1b: (7, –4) 1c: (0, –11), (0, 11), (3, –16), (3, 16) 1d: 6, (1, –2, 3), dist = 30/5 = 6 1e: f '(x) = 2 ln(2x+3) + 2 1f: 56 – 8 = 48, –1 – 0 = –1 2: 98.13°, 3.26, {7, 3} 3: (x,y,z) = (10,0,0) + t(–10,10,7), 3(x–10) + 2y + 10(z–6) = 0, FG® = ED®, 106.301, 3 4: 24, 4 5: y = –8x + 20, f(x) = –Ö(2x4–16), x> ÖÖ(8) 6a: (2, ½), (2et, e2t–1), 4e2t + e4t – 2e2t + 1 = (e2t + 1)2, ½(e3–e–2) + 2.5 = 12.475 6b: g(x) = 3 e0.4x, f(x) = 5 e0.4x, y = 2x + 5, 5(–3 e0.4x 0.4 – 2(–3 e0.4x) = 0 ok, 5(3 e0.4x 0.4 – 2(3 e0.4x – 4) = 8

dec 04 1a: 4 x4 – 3 x3 + c 1b: (4, –2, –5), 6 1c: check 1d: (2, 0), (–2, 0), (0, –4), (0, –4)·(1, 0) = 0 1e: s = 3, t = –5 1f: k = –0.75 2: 3: 4: 5: 6a: 6b:

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