Matematik

Question 1

110 Integral –> e^5x / e^5x/2

Answer 1

e^5x - 5x/2 = e^5x/2 = e^2.5x / 2.5

Question 2

112 Integral –> (5-3x)^-1

Answer 2

Där u = 5-3x => u’ = -3
du=u’ · dx = -3 · dx
=> dx= -du/(-3)

u^-1 / -3 du = - ln (u) / 3 + C

Question 3

117 Integral –> (x^2 -1) / x^3 - 3x + 1

Answer 3

Substitution
u = x^3 - 3x + 1
u’ = 3x^2 - 3 = 3(x^2 -1)

(x^2 -1) = u’ / 3

1/u * u’/3 dx = 1/3 * 1/u * u’ dx = 1/3 * 1/u du

= 1/3 * ln (u) + c

Question 4

119 Integral (1-x) ^0.5

Answer 4

-

Question 5

121 Integral (ln x)^2 / x

Answer 5

u=ln(x)
=> u’ = 1/x

Question 6

122 Integral sin x * cos x

Answer 6

-

Question 7

125 Integral ln(x+1)^0.5

Answer 7

= 0.5 * ln(x+1)
u= ln (x+1)
v’ = 0.5

Question 8

126 Integral x^3 * ln(x)

Answer 8

-

Question 9

127 Integral x sin(2x)

Answer 9

-

Question 10

128 Integral x cos(3x)

Answer 10

-

Question 11

133 Integral x e^-x^2 mellan 0 och 1

Answer 11

u = x^2
u’ = 2x

Question 12

136 Integral (2x + 5)^-1 mellan 0 och 5

Answer 12

u = 2x + 5
u’ = 2

(u)^-1 * u’/2 dx

Question 13

138 Integral (x +1) / x^0.5 mellan 0 och 1

Answer 13

x^0.5 + x^-0.5

Question 14

139 Integral e^x / 3e^x - 2 mellan 0 och 2 ln(2)

Answer 14

u=3ex -2
=> u’ =3ex
=> e^x =u’ / 3

Question 15

143 Integral x / (x + 3) ^0.5 mellan 0 och 1

Answer 15

u=x+3
=> u’ =1
x =u-3

Question 16

146 !!!!!!!!

Question 17

148 !!!!!!!

Question 18

Lim x—> Oändlighet e^-x = 0 (exempel tentamen 131030)

Answer 18

= 0

Question 19

Bestäm gränsvärdet
lim𝑥→∞ (𝑥3 ― 3𝑥) / (2𝑥3 + 3𝑥)

Answer 19

Lösning = 0.5

Question 20

Derivera 𝑦 = sin (𝑥) / (x2 +1)

Answer 20

cos (𝑥) ∙ (𝑥2 + 1) ― 2𝑥 ∙ sin (𝑥) / (𝑥2 + 1)^2

Question 21

Derivera 𝑦 = (𝑥2 ― 𝑥^0.5)2,5

Answer 21

= 2,5 (𝑥2 ― 𝑥^0.5) 1,5 ∙ (2𝑥 ― 0,5𝑥^-0.5)

Question 22

Derivera 𝑦 = 𝑒^ 𝑥3 ∙ cos (𝑥)

Answer 22

𝑒𝑥3 ∙ cos (𝑥) ∙ (3𝑥2 ∙ cos (𝑥) ― sin (𝑥) ∙ 𝑥3)

Question 23

Bestäm ∂𝑓∂𝑥 och ∂𝑓∂𝑦 för tvåvariabel funktionen
𝑓(𝑥,𝑦) = 15𝑥4𝑦2 ― 𝑥2𝑦3 +2𝑦 + 3

Answer 23

∂𝑓∂𝑥 = 15 ∙ 4𝑥3𝑦2 ― 2𝑥 ∙ 𝑦3 +2 ∙ 0 + 0
= 60 ∙ 𝑥3𝑦2 ― 2𝑥 ∙ 𝑦3

∂𝑓∂𝑦 = 15𝑥4 ∙ 2𝑦 ― 𝑥2 ∙ 3𝑦2 +2 + 0
= 30𝑥4 ∙ 2𝑦 ― 3𝑥2𝑦2 +2

Question 24

Bestäm ∂𝑓∂𝑥 och ∂𝑓∂𝑦 för tvåvariabel funktionen
𝑓(𝑥,𝑦) = (5𝑦3 + 𝑦 ∙ ln(𝑥))3

Answer 24

∂𝑓∂𝑥 = 3 ∙ (5𝑦3 + 𝑦 ∙ ln(𝑥))2 ∙ 𝑦 ∙ 𝑥 ―1

och

∂𝑓∂𝑦 = 3 ∙ (5𝑦3 + 𝑦 ∙ ln(𝑥))2 ∙ (15𝑦2 + ln(𝑥))

Question 25

Undersök om 𝑔(𝑥,𝑦) = (𝑥3 ― 𝑦3)2 är en lösning till den partiella differentialekvationen
𝑦2∂𝑔∂𝑥 + 𝑥2∂𝑔∂𝑦 = 0

Answer 25

JA

Question 26

∫(𝑥2 + 5𝑥) ∙ ln (𝑥)𝑑𝑥

Answer 26

𝑢 = ln (x) => 𝑢′ = 1/𝑥
𝑣′ = 𝑥2 + 5𝑥 => 𝑣 = x3/3 + 5x2/2

= ln (x) ∙ (x3/3 + 5x2/2) ― (x3/9 + 5x2/4) + 𝐶

Question 27

∫3𝑥2cos (𝑥3) 𝑑𝑥

Answer 27

= 𝑠𝑖𝑛 (𝑢) + 𝐶 = 𝑠𝑖𝑛 (𝑥3) + 𝐶

Question 28

∫𝑥(2𝑥 + 9)3,5 𝑑𝑥

Answer 28

𝑢 = 2𝑥 + 9 => 𝑥 = 𝑢 ― 9
2
𝑜𝑐ℎ 𝑢′ = 2 => 𝑑𝑥 = 𝑑𝑢/𝑢′ = 𝑑𝑢/2

((2𝑥 + 9)5,5 / 22 ― (2𝑥 + 9)4,5 / 2 ) + 𝐶

Question 29

𝑦′ = x3 + x3 ∙ 𝑦 𝑦(0) = 9

Answer 29

Sätt 𝑢 = 𝑥4/4 => 𝑢′ = 𝑥3

𝑦(𝑥) = ―1 + 10 ∙ 𝑒x4 / 4

Question 30

Fråga 3a) 131030

Answer 30

-

Question 31

Fråga 3b) 131030

Answer 31

gx = y2 ex + ex + x ex
gxx = y2 ex + 2ex + x ex

gy = 2y ex
gyy= 2 ex

SVAR ÄR JAA

Question 32

Fråga 4a) 131030

Answer 32

5/8 e2 - 3/8

Question 33

Fråga 4b) 131030

Answer 33

2 - 2(cos (Pi / 4))^0.5

Question 34

Fråga 5) 131030

Answer 34

y’ = cos x - y/x
y’ + 1/x * y = cos x

P(x) = 1/x
Q(x) = cos x

C = 1

y = sin x + (cos x) / x + 1/x

Question 35

Fråga 2c) 2014

Answer 35

-

Question 36

Fråga 4b) 2014

Answer 36

-

Question 37

Fråga 4c) Integral (5x - 2) / (x + 1) ^1/3
2014

Answer 37

u = 1 + x
u’ = 1

5x = 5u - 5

Question 38

Fråga 5 2014

Answer 38

f(x) = 2(x+1)
g(x) = y^-2