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1 Polinomi
Zadatak 1.1 Odrediti kolicnik K i ostatak r pri dijeljenju polinoma P sa poli-nomom Q ako je
1. P (x) = x3 − 2x2 + x− 1 i Q(x) = x2 − x− 1,
2. P (x) = 2x4 + x3 − 2x2 − 3x + 2 i Q(x) = x2 + 3x + 2,
3. P (x) = x5 + 3x3 − x2 − 1 i Q(x) = x2 − x + 4,
4. P (x) = x4 + 5x3 − 3x2 + x− 6 i Q(x) = x2 + 1,
5. P (x) = x4 − 3x2 − 2 i Q(x) = x3 + x + 1,
6. P (x) = x5 − x4 + x3 − x2 + x− 1 i Q(x) = x2 + 2x− 1,
7. P (x) = x5 − x i Q(x) = x− 1.
Zadatak 1.2 Koristeci Hornerovu semu odrediti kolicnik K i ostatak r pri di-jeljenju polinoma P sa monomom Q ako je
1. P (x) = 2x3 + x2 + 4x− 5 i Q(x) = x− 1,
2. P (x) = x3 − 3x2 − 6x + 1 i Q(x) = x + 2,
3. P (x) = 4x4 − 2x3 + x2 + 2x− 6 i Q(x) = x− 8,
4. P (x) = x4 + 6x2 − 6x + 1 i Q(x) = x− 3,
5. P (x) = x4 − 3x2 + 2 i Q(x) = x + 1,
6. P (x) = x5 + x3 + x i Q(x) = x− 1,
7. P (x) = x5 − 1 i Q(x) = x− 1,
Zadatak 1.3 Odrediti ostatak pri dijeljenju polinoma P sa monomom Q akoje
1. P (x) = x3 − 2x2 + x− 2 i Q(x) = x− 3,
2. P (x) = x4 + x3 − 2x2 + x− 2 i Q(x) = x + 1,
3. P (x) = x5 + x3 − 2x i Q(x) = x + 2,
Zadatak 1.4 Polinom P pri dijeljenju sa x− 1 daje ostatak 3, a pri dijeljenjusa x− 2 ostatak 4. Koliki je ostatak pri dijeljenju polinoma P sa (x− 1)(x− 2)?
Zadatak 1.5 Polinom P pri dijeljenju sa x− 1 daje ostatak 3, a pri dijeljenjusa x + 1 ostatak 1. Koliki je ostatak pri dijeljenju polinoma P sa x2 − 1?
Zadatak 1.6 Sastaviti polinom (sa koeficijentom 1 uz najstariji clan) cije sunule
1
1. x1 = −1, x2 = 1 i x3 = 5,
2. x1 = 1, x2 = x3 = −2,
3. x1 = 3, x2 = 2 +√
3 i x3 = 2−√3,
4. x1 = −2, x2 = 1− 2i i x3 = 1 + 2i,
5. x1 = −1, x2 = 1, x3 = 2 i x4 = −3,
6. x1 = −1, x2 = 2, x3 =√
5 i x4 = −√5,
7. x1 = 1, x2 = 3, x3 = 2 + i i x4 = 2− i.
Zadatak 1.7 Rastaviti polinom P na faktore
1. P (x) = x3 + 2x2 − x− 2,
2. P (x) = x3 − 4x2 + x + 6,
3. P (x) = −x3 + 6x2 − 11x + 6,
4. P (x) = x3 + 4x2 − 11x + 6,
5. P (x) = x3 + x− 2,
6. P (x) = x4 + x3 − x2 + x− 2,
7. P (x) = x4 + 4x3 + 3x2 − 4x− 4,
8. P (x) = x3 − 2x2 − 7x + 2,
9. P (x) = x3 − 3x2 − 5x + 7,
10. P (x) = 2x3 + 3x2 − 11x− 6,
11. P (x) = 3x3 + x2 − 6x− 2,
12. P (x) = 2x3 − 3x2 + 6x + 4.
Zadatak 1.8 Odrediti vrijednost parametra α ∈ R tako da x1 bude jedna nulapolinoma P, pa za tako dobijenu vrijednost parametra α odrediti ostale nulepolinoma P ako je
1. P (x) = x3 + αx2 − 5x− 6 i x1 = 2,
2. P (x) = x3 + αx2 − 2x + 24 i x1 = −2.
Zadatak 1.9 Zadanu racionalnu funkciju rastaviti na parcijalne razlomke
1. R(x) = 5x+1x3+2x2−x−2 ,
2. R(x) = 2x2−8x−72x3+x2−13x+6 ,
2
3. R(x) = −2x2+5x−4x3−4x2+5x−2 ,
4. R(x) = 3x2−7x+8x3−3x2+4 ,
5. R(x) = x2−x−1x3−x2 ,
6. R(x) = 2x2+xx3−2x2+x−2 ,
7. R(x) = x2+2x−3x3+x2+3x+3 ,
8. R(x) = 4x2+x+5x3+2x−3 ,
9. R(x) = x3+x2−2x+3x2+x−2 ,
10. R(x) = 2x4−3x3+x2+5x−32x2−3x+1 .
2 Matrice i determinante
Zadatak 2.1 Izracunati αA + βB ako je
1. A =(
1 −2 40 −1 −3
), B =
( −1 3 −52 −2 0
), α = 2 i β = 3,
2. A =
2 0−1 −4
7 −31 1
, B =
1 1−2 −4−3 7−2 0
, α = 3 i β = −2,
3. A =
−1 6 −3
2 −2 10 −1 4
, B =
2 0 −21 −1 36 2 −3
, α = −2 i β = −1.
Zadatak 2.2 Izracunati (ako postoje) proizvode AB i BA ako je
1. A =
−1 1
3 −84 −2
i B =
( −2 1 2 −3−4 2 −5 −1
),
2. A =( −1 0 2
1 1 −3
)i B =
−1 1 0 2
0 2 −3 −11 2 −2 −1
,
3. A =
1 −13 2
−2 4
i B =
(0 −2 0
−1 3 0
),
4. A =(
3 −3 5)
i B =
2 −2 1−1 5 2
3 −3 2
,
3
5. A =
2−1
5
i B =
( −2 1 −5),
6. A =(
5 4−1 1
)i B =
(1 −32 −2
),
7. A =
−2 2 5
1 0 −13 −2 3
i B =
3 −2 1−5 4 −1
2 0 2
,
8. A =
−1 2 −2
1 3 10 5 2
i B =
−1 2 0
1 1 1−4 6 −2
.
Zadatak 2.3 Izracunati sljedece determinante
1.∣∣∣∣
3 −21 4
∣∣∣∣ ,
2.∣∣∣∣
x 22 x
∣∣∣∣ ,
3.∣∣∣∣
sin x cos x− cosx sin x
∣∣∣∣ ,
4.
∣∣∣∣∣∣
2 −1 33 1 1
−1 2 5
∣∣∣∣∣∣,
5.
∣∣∣∣∣∣
0 2 −31 0 −12 −2 4
∣∣∣∣∣∣,
6.
∣∣∣∣∣∣
2 1 43 −1 5
−2 2 −6
∣∣∣∣∣∣,
7.
∣∣∣∣∣∣
−3 5 62 −1 −43 −3 5
∣∣∣∣∣∣.
Zadatak 2.4 Odrediti (ako postoji) inverznu matricu zadane matrice
1. A =(
1 4−2 −7
),
2. A =( −2 −3−5 −8
),
4
3. A =(
2 −3−5 6
),
4. A =
−1 1 4−2 0 5
1 4 3
,
5. A =
1 2 −22 6 −1
−1 −1 3
,
6. A =
−2 1 −2−2 2 −7−2 −1 9
,
7. A =( −1 4
2 −8
),
8. A =
−1 2 −3
2 1 −23 4 −7
.
Zadatak 2.5 Rijesiti sljedece matricne jednacine
1.(
2 −3−5 8
)X =
(1 0 −12 3 2
),
2. X
1 −1 30 2 32 1 10
=
(1 0 −12 3 2
).
Zadatak 2.6 Izracunati rang sljedecih matrica
1. A =
−1 3 2 −21 2 −1 −22 −1 −3 0
,
2. A =
−1 2 −21 3 2−1 7 −2
,
3. A =
−2 1 0 9 −9 24 −2 0 −18 18 −42 −1 0 −9 9 −2
,
4. A =
1 a −1 22 −1 a 51 10 −6 1
(u zavisnosti od parametra a ∈ R)
5
3 Sistemi linearnih jednacina
Zadatak 3.1 Rijesiti sljedece sistema linearnih jednacina
1.2x −y = 3−3x +2y = −6
2. 2x −y = 3−4x +2y = −6
3. x −y = 2−2x +2y = −3
4.x +4y +3z = −14x +15y +17z = −105x +18y +22z = −14
R: (−2, 1,−1)
5.x −2y +2z = 44x −12y +5z = 134x −16y +3z = 11
R: (2, 0, 1)
6.x +6y −2z = −17−3x −16y +9z = 535x +28y −12z = −85
R: (−1,−2, 2)
7.x −2y +z = −22x −3y −z = −6−3x +5y +z = 9
R:(−1, 1, 1)
8.x −y +2z = 83x −2y +5z = 20x +2z = 7
R: (1,−1, 3)
9.x −2y +4z = −5−2x +5y −13z = 193x −7y +18z = −26
R: (1,−1,−2)
10.x −2y +z = −63x +2y −3z = 16x −2y +4z = −15
11.2x −y −z = 43x +4y −2z = 113x −2y +4z = 11
6
12.x +y +z = 4x +2y +3z = 8x +3y +4z = 11
13.2x +2y +2z = 03x +2y +2z = 14x −3y +4z = −14
14.3x +2y +2z = 14x −3y +2z = −84x −3y +4z = −14
15.x +2y −z = 22x −y +z = 43x +y = 6
16.x −y = 1−x +y = −1−x +z = −2
17.12x −3y −2z = −1−x −y +z = 32x −4y +2z = 8
R: (t, 2t− 1, 3t + 2), t ∈ R
18.−x −3y −z = −2x −y −z = 02x −4y −3z = −1
R: (t,−t + 1, 2t− 1), t ∈ R
19.2x −3y −z = −1−x +3y −z = 2x +3y −5z = 4
R: (2t− 1, t, t− 1), t ∈ R
20.2x −2y +3z = −4−x +y −z = 23x −3y −5z = −6
R: (t− 1, t + 1, 0), t ∈ R
21.2x +3y −z = 44x −y −2z = 1−2x −5y +z = −6
R: (t, 1, 2t− 1), t ∈ R
22.x −y −2z = 4−x +y +z = −3−2x +2y +z = −5
R: (t, t− 2,−1), t ∈ R
7
23.2x +y +z = 23x +y +z = 07x +2y +2z = 4
24.x +3y −2z = 12x +6y −4z = 3−x −3y +2z = 0
Zadatak 3.2 U zavisnosti od parametra a ∈ R rijesiti sistem linearnih jednacina
1.ax +y +z = 1x +ay +z = 1x +y +az = 1
2.ax +y +z = 12x +2ay +2z = 3x +y +az = 1
3.(a + 3)x +y −z = 1x +(a + 3)y −z = a + 2−x −y +z = 1
4.(a + 3)x −y −z = 1−x +(a + 6)y −z = a + 2x +y −z = 1
5.−x +y +z = 1−x −y +(a + 3)z = a + 2−x +(a + 6)y −z = 1
6.ax +y −z = −1x +ay +z = −1x +y +(a− 4)z = −1
7.(a− 5)x +y −z = 2x +(a− 5)y +z = 2x +y +az = 2
8.(a− 1)x +y +z = −2x +(a + 1)y +z = −1x +y +(a− 1)z = −2
9.(a + 1)x −y +z = 1−x +(a− 1)y +z = a− 1x +y −z = 1
8
4 Granicna vrijednost funkcije
Zadatak 4.1 Izracunati
1. limx→∞
3x5 − x4 + 2x3 − x2 + x− 6x4 + x3 + x2 + x + 1
,
2. limx→∞
x4
x3 − x2 + 2x− 9,
3. limx→∞
3x4 + 2x3 − x2 − x− 12x4 − x2 + 6
,
4. limx→∞
4x3 − x2 + 2x− 32x3 + x
,
5. limx→∞
x3 − x2 + x− 1x4 − 1
,
6. limx→∞
2x
x2 + 1,
7. limx→∞
(x2 + 4x + 12)3(2x− 3)4
(x2 + 2x + 2)5,
8. limx→∞
√4x− 1√
x + 2√
x + 1,
9. limx→∞
2x + 53√
8x3 + 2√
x− 1.
Zadatak 4.2 Izracunati
1. limx→2
x2 + x− 6x2 − x− 2
,
2. limx→1
x3 + x2 + x− 3x2 − 1
,
3. limx→−3
x3 − 7x + 6x3 + 3x2 + x + 3
,
4. limx→ 1
2
2x3 − 3x2 + 5x− 22x2 + 5x− 3
,
5. limx→−1
2x3 + 3x2 − 1x2 + 2x + 1
.
Zadatak 4.3 Izracunati
1. limx→0
sin 2x
3x,
9
2. limx→0
sin 3x
sin 6x,
3. limx→0
tan 3x
x,
4. limx→0
sin 3x− sin x
sin 2x,
5. limx→0
1− cosx
x2,
6. limx→0
1− cos2x
x sin x.
Zadatak 4.4 Izracunati
1. limx→∞
(x− 2x + 3
)2x
,
2. limx→∞
(x− 1x + 4
)−x
,
3. limx→∞
(x + 5x− 1
)x
,
4. limx→∞
(x + 3x− 4
) 3x4
,
5. limx→∞
(x + 4x + 2
) x−1x+1
.
5 Izvod funkcije
Zadatak 5.1 Koristeci se tablicom izvoda i pravilima diferenciranja izracunatiizvod zadane funkcije (u bilo kojoj tacki)
1. y = x3 + 2x2 − 6x + 5− 4√
x + x−2 + 3x−4,
2. y =x5
5− 2x3
3+
(1− x2
2
)2
+1x
+1
2x2+
16x3
,
3. y = 6 3√
x2 − 4 4√
x +(
1− 13√
x
)2
+84√
x− 6
3√
x,
4. y = 5x + 4,
5. y =2x
3− 4
5,
6. y =3x
2,
10
7. y =2x− 1
3,
8. y =x4 − 2x2 + 4
√x− 3
x
3x2 − 2x + 1,
9. y =x2 + 3x + 4√
x− 2x
,
10. y = (2x2 + 3x− 1)3x,
11. y = (x4 + 2x3) exp (x),
12. y = (√
x− 4√
x) log4 x,
13. y =2x− 3
5ln x,
14. y =3x− 1
3x,
15. y =log2 x
x− 1,
16. y =1 + ln x
x,
17. y =exp (x)(2x + 3)
(x + 1) ln x,
18. y =x4x
4x− 3,
19. y = 2x +(
12
)x
20. y = 2−x + 3−x
21. y = exp (−x) + exp (−2x) +√
exp (x)
22. y = x2 cos x,
23. y = x2 exp (x),
24. y = x2 cot x− x + sin x +tan x√
x,
25. y = tan x,
26. y = cot x,
27. y =cosx
1 + 2 sin x,
11
28. y =cosx
1− sin x,
29. y =√
1− x2 arcsin x,
30. y = exp (−x) arcsin x,
31. y = x arctan x,
32. y =arctanx
1 + x2,
Zadatak 5.2 Koristeci se pravilom za izvod slozene funkicje izracunati izvodzadane funkcije (u bilo kojoj tacki)
1. y = (2x− 3)12,
2. y =
√4x + 3
5,
3. y = exp (2x− 1),
4. y = exp (6x),
5. y = exp (−x),
6. y = exp(x
4
),
7. y = exp(
4x− 33
),
8. y = ln(
2x + 34
),
9. y = lnx
2,
10. y = sin (ax + b),
11. y = tanx
4,
12. y = arcsin (x + 3),
13. y = arctan ax,
14. y = (x2 + 2x− 1)14,
15. y = 4√
1− x2,
16. y =1
arctanx,
17. y = ln2 x,
12
18. y =√
cos x,
19. y = exp (− tanx),
20. y = log4 (arctan x),
21. y = sin 2x,
22. y = sinx2
2x + 1,
23. y = cos√
x,
24. y = tan x2,
25. y = arctanx
x + 1,
26. y = 4
√1 + cos2
2x
3,
27. y =√
1− sin 2x +√
1 + sin 2x,
28. y =1
(2 + cos3 2x)4,
29. y =x2 − 1√
ln 2x,
30. y = x√
x3 + 2,
31. y = lnx2
1− x2,
32. y = ln
√x + 1x− 1
,
33. y = ln(tan
(π
4+
x
2
)),
34. y = lnx2
√ax2 + bx + c
,
35. y = ln(√
x +√
x + 1 +√
x + 2),
36. y = ln(sinx +
√1 + cos 3x
),
37. y = ln 4
√sin 2x
1 + sin2 x2
,
38. y = exp (−x2),
13
39. y =√
x
x− 1exp
(√x
2
),
40. y = exp (sin 2x) + cos (2 exp (−x)),
41. y = ln[exp (ax) + x exp (−x)
],
42. y =exp (x) + exp (−x)exp (x)− exp (−x)
,
43. y = ln
√exp (ax)
exp (ax) + 1.
Zadatak 5.3 Detaljno ispitati zadanu funkciju, a zatim skicirati njen grafik
1. f(x) =x2 − 5x + 4
x− 5,
2. f(x) =(x− 3)2
x− 5,
3. f(x) =x2 − 7x + 10
x− 6,
4. f(x) =x2 − 3x
x− 4,
5. f(x) =x2 − 2x− 4
x− 4,
6. f(x) =x2 + x− 6
x2 + x,
7. f(x) =x2 + 5x
x2 + 5x + 4,
8. f(x) =x2 − x− 6x2 − x + 1
,
9. f(x) =x2 + 3x− 4x2 + 3x + 1
,
10. f(x) =x2 − 5x
x2 − 5x + 9,
11. f(x) =x2 + x
x2 + x− 6,
12. f(x) = (x− 2) exp (3x),
13. f(x) = (3− x) exp (x),
14
14. f(x) = (2x + 1) exp (2x),
15. f(x) = (3x− 2) exp (−x),
16. f(x) = (5− 2x) exp(−x
2
),
17. f(x) = x ln x,
18. f(x) = (x + 2) ln (x + 2),
19. f(x) = (3− 2x) ln (3− 2x),
20. f(x) =x
2ln x,
21. f(x) = x lnx
3.
6 Neodredeni integral
Zadatak 6.1 Izracunati sljedece neodredene integrale
1.∫
5x4dx,
2.∫
8.3x−0.17dx,
3.∫
(3x + 1) dx,
4.∫ (
x5 − 4x3 + 2x− 1)dx,
5.∫
dx
x2,
6.∫ (
3x2 − 2x +3x−√x + 2 4
√x3
)dx,
7.∫
dx7√
x5,
8.∫ (
2x
+x
2
)dx,
9.∫
3√
x 3√
xdx,
10.∫
(x− 2) (√
x + 1)x2
dx,
15
11.∫
4√
x− 8√
x5
9√
x4dx,
12.∫
3x exp (x)dx,
13.∫
5 · 4x + 4 · 5x
4xdx,
14.∫
ax
(1 +
a−x
3√
x2
)dx,
15.∫
exp (−x)(
1 +exp (x)√
x5
)dx,
16.∫
4x − 6x
12xdx,
17.∫
cos 2x
cos2 x sin2 xdx,
18.∫
dx
cos2 x sin2 x,
19.∫
tan2 xdx,
20.∫
1 + sin3 x
sin2 x,
21.∫ (
sinx
2− cos
x
2
)2
dx,
22.∫
x2
x2 + 1dx,
23.∫
x2 + 2x2 + 1
dx,
24.∫
x3 + x− 2x2 + 1
dx,
25.∫
x4
x2 + 1dx.
Zadatak 6.2 Medodom zamjene izracunati sljedece integrale
1.∫
(3x− 2)14 dx,
2.∫
4√
2x + 8,
16
3.∫
dx
7√
(3x + 5)4,
4.∫
exp (2x− 9)dx,
5.∫
exp (−x)dx,
6.∫
exp(
3x− 54
)dx,
7.∫
exp(−x
3
)dx,
8.∫
sin 2xdx,
9.∫
dx
cos2 (−x + 1),
10.∫
dx
x− a,
11.∫
dx
5x + 6,
12.∫
dx
(5x− 4)2 + 1,
13.∫
dx√−x2 + 4x− 3,
14.∫
dx√2x− x2
,
15.∫
dx
exp (−x) + exp (x),
16.∫
exp (x)dx
1 + exp (2x),
17.∫
(ax + b)n,
18.∫
dx
x2 + 4,
19.∫
dx
x2 + 6,
17
20.∫
dx√3− x2
,
Zadatak 6.3 Medodom zamjene izracunati sljedece integrale
1.∫
2xdx
x2 + 1,
2.∫
(2x− 3)dx
x2 − 3x + 6,
3.∫
tan xdx,
4.∫
cot xdx,
5.∫
sin xdx
cosx− 6,
6.∫
dx
x ln x,
7.∫
xdx
x2 − 2,
8.∫
(x2 + 1)dx
x3 + 3x− 9,
9.∫
exp (2x)dx
exp (2x)− 4,
10.∫
dx
(x2 + 1) arctan x,
11.∫
xdx
(x2 + 6)4,
12.∫
(x2 + 2)dx5√
x3 + 6x− 1,
13.∫
cosxdx
sin4 x,
14.∫
5√
x4 + 2x3 − 6(2x3 + 3x2
)dx,
15.∫ √
ln xdx
x,
16.∫
(arctanx)3 dx
1 + x2,
18
17.∫
cos4 x sinxdx,
18.∫
sin (x2 + 3)xdx,
19.∫
x2dx
cos2 x3,
20.∫
2x + arctan x
1 + x2dx,
21.∫
2x + 3√
arcsin x√1− x2
dx,
22.∫
dx
x ln4 x.
Zadatak 6.4 Metodom parcijalne integracije izracunati sljedece integrale
1.∫
x exp (x)dx,
2.∫
x sin xdx,
3.∫
(2x− 3) exp(−x
2
)dx,
4.∫ (x
2+ 4
)cos 3xdx,
5.∫
x2 sin xdx,
6.∫
(x2 + 3x− 1) cos xdx,
7.∫
(x2 − x + 4) sin 4xdx,
8.∫
x4 ln xdx,
9.∫
ln xdx4√
x3,
10.∫
ln xdx,
11.∫
arctanxdx,
19
12.∫
x arctan xdx,
13.∫
arcsin xdx,
14.∫
x · 2xdx,
15.∫
exp (2x) cos xdx,
16.∫
exp (x) sinx
2dx,
17.∫
exp(x
2
)sin 3xdx,
Zadatak 6.5 Izracunati sljedece integrale racionalnih funkcija
1.∫
dx
x2 − 1,
2.∫
4x2 + x− 6x3 − x2 − 2x
dx,
3.∫
4x2 − 9x− 1x3 − 2x2 − x + 2
dx,
4.∫
5x + 42x3 + 3x2 − 3x− 2
dx,
5.∫
x2 − 23x + 123x3 − 4x2 − 5x + 2
dx,
6.∫
3x2 − 2x− 2x3 + x2
dx,
7.∫
3x2 − 8x− 1x3 − 3x + 2
dx,
8.∫
3x2 − 2x + 8x3 − x2 + 2x− 2
dx,
9.∫
4x3 + 3x2 + 7x + 5
dx,
10.∫
x2 + 2x + 1x3 − x2 + x− 1
dx.
20
7 Odredeni integral
Zadatak 7.1 Primjenom Njutn-Lajbicove formule izracunati sljedece odredeneintegrale
1.∫ 2
0
x3dx,
2.∫ 2
1
(x2 − 2
x
)2
dx,
3.∫ π
3
0
sin xdx,
4.∫ π
4
0
cos2 xdx,
5.∫ π
3
π6
dx
cos2x,
6.∫ 9
1
√xdx,
7.∫ e2
1
dx
x,
Zadatak 7.2 Metodom zamjene ili metodom parcijalne integracije izracunatisljedece odredene integrale
1.∫ 2
0
dx
4 + x2,
2.∫ 1
0
xdx
(x2 + 1)2,
3.∫ 1
12
dx√2x− x2
,
4.∫ 1
0
exp (x)dx
1 + exp (2x),
5.∫ √
e
1
dx
x√
1− ln2 x,
6.∫ e
1
ln xdx,
7.∫ e
1
ln3 xdx,
21
8.∫ e−1
0
ln (x + 1)dx,
9.∫ 1
0
x exp (−x)dx,
10.∫ 1
0
x3 exp (2x)dx,
11.∫ π
2
0
x cosxdx,
12.∫ π
0
x2 sin xdx,
13.∫ π
4
π6
xdx
sin2 x,
14.∫ π
0
exp (x) sin xdx,
15.∫ π
2
0
exp (2x) cos xdx.
Zadatak 7.3 Izracunati povrsinu koju ogranicavaju
1. kriva y = cos x, prave x = −π
2, x =
π
2i 0x osa,
2. kriva y = x3, prave x = 0, x = 2 i 0x osa,
3. kriva y = tan x, prave x = 0, x =π
3i 0x osa.
Zadatak 7.4 Izracunati povrsinu koju ogranicavaju krive
1. y1(x) = −x2 − 2x + 1 i y2(x) = x + 3,
2. y1(x) = −x2 − 6x− 7 i y2(x) = −3x− 5,
3. y1(x) = −x2 + 4x− 5 i y2(x) = x− 3,
4. y1(x) = −x2 − 4x− 5 i y2(x) = −x− 3,
5. y1(x) = −x2 + 4 i y2(x) = x2 − 4,
6. y1(x) = x2 + 2x− 3 i y2(x) = −x2 − 2x + 3,
7. y1(x) = x2 + x− 7 i y2(x) = −2x2 − 2x + 29.
Zadatak 7.5 Odrediti parcijalne izvode prvog i drugog reda sljedecih funkcija
1. f(x, y) = x5 + 8 3√
y − 2x2y4 +√
xy + 6,
22
2. f(x, y) = 8x− 3yx ,
3. f(x, y) = xx2+y2 ,
4. f(x, y) =√
x2 + y2,
5. f(x, y) = xy + xy ,
6. f(x, y) = xyx+y ,
7. f(x, y) = arcsin xy ,
8. f(x, y) = ln exp (x) + exp (y).
Zadatak 7.6 Odrediti ekstremne vrijednosti zadane funkcije
1. f(x, y) = 14x3 + 27xy2 − 69x− 54y,
2. f(x, y) = x2 − 5xy − y2,
3. f(x, y) = 2x2 − xy − 3y2 − 3x + 7y,
4. f(x, y) = x3 − 3xy2 + y3.
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