Eksamenssett.no
Ressurser
Skolenytt
Hoderegning
TMA4100
Formelark
Matematikk 1
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
|
(
a
x
)
′
=
a
x
ln
a
(a^x)' = a^x \ln a
(
a
x
)
′
=
a
x
ln
a
•
(
ln
x
)
′
=
1
/
x
(\ln x)' = 1/x
(
ln
x
)
′
=
1/
x
•
(
sin
x
)
′
=
cos
x
(\sin x)' = \cos x
(
sin
x
)
′
=
cos
x
|
(
cos
x
)
′
=
−
sin
x
(\cos x)' = -\sin x
(
cos
x
)
′
=
−
sin
x
|
(
tan
x
)
′
=
1
/
cos
2
x
(\tan x)' = 1/\cos^2 x
(
tan
x
)
′
=
1/
cos
2
x
•
(
arcsin
x
)
′
=
1
/
1
−
x
2
(\arcsin x)' = 1/\sqrt{1-x^2}
(
arcsin
x
)
′
=
1/
1
−
x
2
|
(
arctan
x
)
′
=
1
/
(
1
+
x
2
)
(\arctan x)' = 1/(1+x^2)
(
arctan
x
)
′
=
1/
(
1
+
x
2
)
•
Produktregel:
(
f
g
)
′
=
f
′
g
+
f
g
′
(fg)' = f'g + fg'
(
f
g
)
′
=
f
′
g
+
f
g
′
•
Kvotientregel:
(
f
/
g
)
′
=
(
f
′
g
−
f
g
′
)
/
g
2
(f/g)' = (f'g - fg')/g^2
(
f
/
g
)
′
=
(
f
′
g
−
f
g
′
)
/
g
2
•
Kjerneregel:
(
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
)
Integrasjonsregler
•
∫
x
n
d
x
=
x
n
+
1
/
(
n
+
1
)
+
C
\displaystyle \int x^n\,dx = x^{n+1}/(n+1) + C
∫
x
n
d
x
=
x
n
+
1
/
(
n
+
1
)
+
C
(for
n
≠
−
1
n \neq -1
n
=
−
1
)
•
∫
1
/
x
d
x
=
ln
∣
x
∣
+
C
\displaystyle \int 1/x\,dx = \ln|x| + C
∫
1/
x
d
x
=
ln
∣
x
∣
+
C
•
∫
e
a
x
d
x
=
e
a
x
/
a
+
C
\displaystyle \int e^{ax}\,dx = e^{ax}/a + C
∫
e
a
x
d
x
=
e
a
x
/
a
+
C
•
∫
sin
x
d
x
=
−
cos
x
+
C
\displaystyle \int \sin x\,dx = -\cos x + C
∫
sin
x
d
x
=
−
cos
x
+
C
|
∫
cos
x
d
x
=
sin
x
+
C
\displaystyle \int \cos x\,dx = \sin x + C
∫
cos
x
d
x
=
sin
x
+
C
•
∫
1
/
(
1
+
x
2
)
d
x
=
arctan
x
+
C
\displaystyle \int 1/(1+x^2)\,dx = \arctan x + C
∫
1/
(
1
+
x
2
)
d
x
=
arctan
x
+
C
•
∫
1
/
(
x
2
+
a
2
)
d
x
=
(
1
/
a
)
arctan
(
x
/
a
)
+
C
\displaystyle \int 1/(x^2+a^2)\,dx = (1/a)\arctan(x/a) + C
∫
1/
(
x
2
+
a
2
)
d
x
=
(
1/
a
)
arctan
(
x
/
a
)
+
C
•
Delvis integrasjon:
∫
u
d
v
=
u
v
−
∫
v
d
u
\displaystyle \int u\,dv = uv - \int v\,du
∫
u
d
v
=
uv
−
∫
v
d
u
Taylorrekker
•
e
x
=
∑
x
n
/
n
!
\displaystyle e^x = \sum x^n/n!
e
x
=
∑
x
n
/
n
!
•
sin
x
=
∑
(
−
1
)
n
x
2
n
+
1
/
(
2
n
+
1
)
!
\displaystyle \sin x = \sum (-1)^n x^{2n+1}/(2n+1)!
sin
x
=
∑
(
−
1
)
n
x
2
n
+
1
/
(
2
n
+
1
)!
•
cos
x
=
∑
(
−
1
)
n
x
2
n
/
(
2
n
)
!
\displaystyle \cos x = \sum (-1)^n x^{2n}/(2n)!
cos
x
=
∑
(
−
1
)
n
x
2
n
/
(
2
n
)!
•
1
/
(
1
−
x
)
=
∑
x
n
\displaystyle 1/(1-x) = \sum x^n
1/
(
1
−
x
)
=
∑
x
n
for
∣
x
∣
<
1
|x| < 1
∣
x
∣
<
1
•
ln
(
1
+
x
)
=
∑
(
−
1
)
n
+
1
x
n
/
n
\displaystyle \ln(1+x) = \sum (-1)^{n+1} x^n/n
ln
(
1
+
x
)
=
∑
(
−
1
)
n
+
1
x
n
/
n
for
−
1
<
x
≤
1
-1 < x \leq 1
−
1
<
x
≤
1
•
arctan
x
=
∑
(
−
1
)
n
x
2
n
+
1
/
(
2
n
+
1
)
\displaystyle \arctan x = \sum (-1)^n x^{2n+1}/(2n+1)
arctan
x
=
∑
(
−
1
)
n
x
2
n
+
1
/
(
2
n
+
1
)
for
∣
x
∣
≤
1
|x| \leq 1
∣
x
∣
≤
1
Differensiallikninger
•
Separabel:
d
y
/
g
(
y
)
=
f
(
x
)
d
x
dy/g(y) = f(x)\,dx
d
y
/
g
(
y
)
=
f
(
x
)
d
x
•
Integrerende faktor:
μ
=
e
∫
P
(
x
)
d
x
\displaystyle \mu = e^{\int P(x)\,dx}
μ
=
e
∫
P
(
x
)
d
x
for
y
′
+
P
y
=
Q
y' + Py = Q
y
′
+
P
y
=
Q
•
Karakteristisk likning:
r
2
+
a
r
+
b
=
0
r^2 + ar + b = 0
r
2
+
a
r
+
b
=
0
for
y
′
′
+
a
y
′
+
b
y
=
0
y'' + ay' + by = 0
y
′′
+
a
y
′
+
b
y
=
0
•
Eksponentiell vekst:
y
′
=
k
y
⇒
y
=
C
e
k
x
y' = ky \Rightarrow y = Ce^{kx}
y
′
=
k
y
⇒
y
=
C
e
k
x
Vektorer
•
a
⃗
⋅
b
⃗
=
a
1
b
1
+
a
2
b
2
+
a
3
b
3
=
∣
a
⃗
∣
∣
b
⃗
∣
cos
θ
\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3 = |\vec{a}||\vec{b}|\cos\theta
a
⋅
b
=
a
1
b
1
+
a
2
b
2
+
a
3
b
3
=
∣
a
∣∣
b
∣
cos
θ
•
a
⃗
×
b
⃗
=
(
a
2
b
3
−
a
3
b
2
,
a
3
b
1
−
a
1
b
3
,
a
1
b
2
−
a
2
b
1
)
\vec{a} \times \vec{b} = (a_2b_3-a_3b_2, a_3b_1-a_1b_3, a_1b_2-a_2b_1)
a
×
b
=
(
a
2
b
3
−
a
3
b
2
,
a
3
b
1
−
a
1
b
3
,
a
1
b
2
−
a
2
b
1
)
•
Planlikning:
a
x
+
b
y
+
c
z
=
d
ax + by + cz = d
a
x
+
b
y
+
cz
=
d
, normalvektor
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
•
Avstand punkt-plan:
d
=
∣
a
x
0
+
b
y
0
+
c
z
0
−
d
∣
/
a
2
+
b
2
+
c
2
d = |ax_0+by_0+cz_0-d|/\sqrt{a^2+b^2+c^2}
d
=
∣
a
x
0
+
b
y
0
+
c
z
0
−
d
∣/
a
2
+
b
2
+
c
2
Komplekse tall
•
e
i
θ
=
cos
θ
+
i
sin
θ
e^{i\theta} = \cos\theta + i\sin\theta
e
i
θ
=
cos
θ
+
i
sin
θ
(Eulers formel)
•
z
z
ˉ
=
∣
z
∣
2
z\bar{z} = |z|^2
z
z
ˉ
=
∣
z
∣
2
,
∣
z
∣
=
a
2
+
b
2
|z| = \sqrt{a^2+b^2}
∣
z
∣
=
a
2
+
b
2
•
De Moivre:
(
e
i
θ
)
n
=
e
i
n
θ
(e^{i\theta})^n = e^{in\theta}
(
e
i
θ
)
n
=
e
in
θ
•
n
n
n
-te rotter:
w
k
=
∣
z
∣
1
/
n
e
i
(
θ
+
2
k
π
)
/
n
w_k = |z|^{1/n}e^{i(\theta+2k\pi)/n}
w
k
=
∣
z
∣
1/
n
e
i
(
θ
+
2
kπ
)
/
n
,
k
=
0
,
…
,
n
−
1
k = 0, \ldots, n-1
k
=
0
,
…
,
n
−
1
Konvergenstester
•
Geometrisk rekke:
∑
r
n
=
1
/
(
1
−
r
)
\displaystyle \sum r^n = 1/(1-r)
∑
r
n
=
1/
(
1
−
r
)
for
∣
r
∣
<
1
|r| < 1
∣
r
∣
<
1
•
p-rekke:
∑
1
/
n
p
\displaystyle \sum 1/n^p
∑
1/
n
p
konvergerer for
p
>
1
p > 1
p
>
1
•
Forholdstesten:
L
=
lim
∣
a
n
+
1
/
a
n
∣
L = \lim |a_{n+1}/a_n|
L
=
lim
∣
a
n
+
1
/
a
n
∣
; konv. hvis
L
<
1
L < 1
L
<
1
•
Leibniz: alt. rekke konv. hvis leddene er pos., avtagende,
→
0
\to 0
→
0