Eksamenssett.no
Ressurser
Skolenytt
Hoderegning
ECON1200
Formelark
Matematikk for samfunnsvitenskap
eksamenssett.no
Derivasjonsregler
•
(
x
n
)
′
=
n
x
n
−
1
(x^n)' = nx^{n-1}
(
x
n
)
′
=
n
x
n
−
1
•
(
e
x
)
′
=
e
x
(e^x)' = e^x
(
e
x
)
′
=
e
x
•
(
ln
x
)
′
=
1
x
\displaystyle (\ln x)' = \frac{1}{x}
(
ln
x
)
′
=
x
1
•
(
f
g
)
′
=
f
′
g
+
f
g
′
(fg)' = f'g + fg'
(
f
g
)
′
=
f
′
g
+
f
g
′
(produktregel)
•
(
f
/
g
)
′
=
f
′
g
−
f
g
′
g
2
\displaystyle (f/g)' = \frac{f'g - fg'}{g^2}
(
f
/
g
)
′
=
g
2
f
′
g
−
f
g
′
(kvotientregel)
•
(
f
(
g
(
x
)
)
)
′
=
f
′
(
g
(
x
)
)
⋅
g
′
(
x
)
(f(g(x)))' = f'(g(x)) \cdot g'(x)
(
f
(
g
(
x
))
)
′
=
f
′
(
g
(
x
))
⋅
g
′
(
x
)
(kjerneregel)
Integrasjonsregler
•
∫
x
n
d
x
=
x
n
+
1
n
+
1
+
C
\displaystyle \int x^n dx = \frac{x^{n+1}}{n+1} + C
∫
x
n
d
x
=
n
+
1
x
n
+
1
+
C
(
n
≠
−
1
n \neq -1
n
=
−
1
)
•
∫
1
x
d
x
=
ln
∣
x
∣
+
C
\displaystyle \int \frac{1}{x} dx = \ln|x| + C
∫
x
1
d
x
=
ln
∣
x
∣
+
C
•
∫
e
a
x
d
x
=
1
a
e
a
x
+
C
\displaystyle \int e^{ax} dx = \frac{1}{a}e^{ax} + C
∫
e
a
x
d
x
=
a
1
e
a
x
+
C
•
∫
a
b
f
(
x
)
d
x
=
F
(
b
)
−
F
(
a
)
\displaystyle \int_a^b f(x) dx = F(b) - F(a)
∫
a
b
f
(
x
)
d
x
=
F
(
b
)
−
F
(
a
)
Optimering
•
FOB:
f
′
(
x
)
=
0
f'(x) = 0
f
′
(
x
)
=
0
eller
f
x
=
f
y
=
0
f_x = f_y = 0
f
x
=
f
y
=
0
•
AOB (én var.):
f
′
′
(
x
0
)
<
0
⇒
f''(x_0) < 0 \Rightarrow
f
′′
(
x
0
)
<
0
⇒
maks,
>
0
⇒
> 0 \Rightarrow
>
0
⇒
min
•
AOB (to var.):
D
=
f
x
x
f
y
y
−
(
f
x
y
)
2
D = f_{xx}f_{yy} - (f_{xy})^2
D
=
f
xx
f
yy
−
(
f
x
y
)
2
•
Lagrange:
L
=
f
(
x
,
y
)
−
λ
(
g
(
x
,
y
)
−
c
)
\mathcal{L} = f(x,y) - \lambda(g(x,y) - c)
L
=
f
(
x
,
y
)
−
λ
(
g
(
x
,
y
)
−
c
)
•
Elastisitet:
ε
=
p
D
(
p
)
⋅
D
′
(
p
)
\displaystyle \varepsilon = \frac{p}{D(p)} \cdot D'(p)
ε
=
D
(
p
)
p
⋅
D
′
(
p
)
Lineær algebra
•
det
(
a
b
c
d
)
=
a
d
−
b
c
\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc
det
(
a
c
b
d
)
=
a
d
−
b
c
•
A
−
1
=
1
det
(
A
)
(
d
−
b
−
c
a
)
\displaystyle A^{-1} = \frac{1}{\det(A)}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}
A
−
1
=
det
(
A
)
1
(
d
−
c
−
b
a
)
•
Cramers regel:
x
i
=
det
(
A
i
)
det
(
A
)
\displaystyle x_i = \frac{\det(A_i)}{\det(A)}
x
i
=
det
(
A
)
det
(
A
i
)
Differenslikninger
•
x
t
+
1
=
a
x
t
+
b
x_{t+1} = ax_t + b
x
t
+
1
=
a
x
t
+
b
:
x
∗
=
b
1
−
a
\displaystyle x^* = \frac{b}{1-a}
x
∗
=
1
−
a
b
•
Generell løsning:
x
t
=
(
x
0
−
x
∗
)
a
t
+
x
∗
x_t = (x_0 - x^*)a^t + x^*
x
t
=
(
x
0
−
x
∗
)
a
t
+
x
∗
•
Stabil hvis
∣
a
∣
<
1
|a| < 1
∣
a
∣
<
1
•
Karakteristisk likning:
r
2
+
a
1
r
+
a
2
=
0
r^2 + a_1r + a_2 = 0
r
2
+
a
1
r
+
a
2
=
0
Konsument-/produsentoverskudd
•
K
O
=
∫
0
x
∗
D
(
x
)
d
x
−
p
∗
x
∗
\displaystyle KO = \int_0^{x^*} D(x) dx - p^* x^*
K
O
=
∫
0
x
∗
D
(
x
)
d
x
−
p
∗
x
∗
•
P
O
=
p
∗
x
∗
−
∫
0
x
∗
S
(
x
)
d
x
\displaystyle PO = p^* x^* - \int_0^{x^*} S(x) dx
PO
=
p
∗
x
∗
−
∫
0
x
∗
S
(
x
)
d
x