Mathe 2 Uebungsaufgaben PDF

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Mathematik für Wirtschaftswissenschaftler II - Uebungen...

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Dr. Christian Aßmann Lehrstuhl für Statistik und Ökonometrie Otto-Friedrich-Universität Bamberg

y = f (L) = 10 L0.7, y

L > 0

g(L)

f

f ′(L) L0 = 100

f (111)

g(111)

δf = |f (111) − g(111)| x

p>0 m>0

x = f (p, m) = 10 e−0.1p ln(m + 1) fp′ (p, m) = −e−0.1p ln(m + 1),

′ fm (p, m) = 10 e−0.1p

1 m+1

g(p, m)

f

(p0, m0) = (2, 2) δf = |f (3, 3) − g(3, 3)| µ

y+1 x = f (y, p) = ln p

¶

p>1 y > p−1

x

fy′ (y, p) =

1 y+1

f p′ (y, p) = −

1 p

dx (y0, p0) = (2, 2) dx

∆y = ∆p = 0.5 δx = |∆x − dx|

√ p x, y > 0 5 x5 + 2 x y 2 ³ ´ p 00 g = 10·000

0

p

p> r = 0.9 w 3%

x = f (p, q) = q a p−b cq a b c > 0

p q > 0

b fp′ (p, q) = − f (p, q) p

fq′ (p, q) =

µ

¶ a + ln(c) f (p, q ) q

a b c > 0 f (p, q) f c=1 a b>0

x = f (p) = p−0.3

p>0

x

εx(p) = −0.3 ρx(p) p

p0 = 8

p0 = 8

ρx(p) = −0.1

ρx(p0) ∆x ∆p · x0

∆p = 1

εx(p0) ∆x · p0 ∆p · x0

∆p = 0.08 x = x(p) εx(p) =

a b > 0

p ∈ (0, b)

ap p−b p

Y (K, S) = 0.06 K 1.4 S 0.56 K>0 S>0 ρK (K, S) εS (K, S)

ρS (K, S) εK (K, S)

f (x) = 21−x f (x) f (n)(x)

n

n ∈ IN0

Tf2(x)

x0 = 0 Tf∞(x)

x0 =

0 f (x) = x5 − 5 x4 + 10 x3 − 10x2 + 5 x − 1 x0 = 1

f (x) Tf1(x)

Tf2(x)

x1 = 0 x0 x˜ 0 f (x) =

1 (2 − x)2

δf = |x0 − x˜ 0|

n

x 6= 2 n ∈ IN0

Tf∞(x) 0

f (x)

Tf1(x)

x0 =

x > 0 b>0 R 1x √ ( ) + x dx R 3−b b (x) dx x 2 R −4 dx exp(4x) R 10

2 dx √1 dx x R 10 −x e dx 0 R 10 x 2 dx 0 R π/2 cos(x) dx R 010 |x| dx 0

0 R 10 1

R

λ2xe−λx dx

R4 1

ln(x) x4 dx

λ>0

f (x) = cos(x)

A

x

Z

Γ(α) := α>0

R1

−1

∞

xα−1e−x dx

0

Γ(1) Z

0

a ∈ R \ {0} 1

xa−1 dx

[0, π ] |x| dx

 T 3 7 2

¡

1 −3 0

¢T

T  T 9  9  −5   µ ¶ 4 1 +7 5 7

  

    0 3 8 9 +  6 8· 6 8 4       −7 9 6  8− 5  9 · 0  4 0 −3 µ ¶ µ ¶ ¶ µ −4 −2 3 −5 · + 0 −2 2         3 3 0 0 7 ·   1  +  −4   7 ·  1  +  −4  9 6 9 6 µ

¶

T 2 0 µ 6 9¶  −2 −6  8· 6 7 0 0 ¶ µ ¶ µ µ 0 6 2 −4 6 + −2 −4 3 1 −4     −2 3 0 2 4 −6 5 ·  3 −1 2  +  5 8 −1  0 5 7 4 0 −2  T  5 1 2 5 −4  −8 ·  −4 9 −7  +  4 −2 3 2 8 −7 2 



 60 x1 =  24  , 64 x1



 53 x2 =  28  54 x2 p

x1 + x2

 −4 9  ¶ 8 + 6 3  −5 2 5

 6  7  7



 3 p =  2 .5  . 2 .5

x1 + p x1 · x2

xT1 · x2 xT1 · p √ p

A=

µ

1 2 2 4

¶

¶

µ

µ

4 −6 2 1 , C= −2 3 3 2 µ ¶ ¡ ¢ −2 7 D= , E= 1 1 . 5 −1 ,

B=

A·A

A·B

A·C

A·D

ET · A · E

E · A · ET

Ap := |A · .{z . . · A} p−mal

A ∈ Mn,n A=

A2 A3 Ap

µ

1 1 1 1

¶

p ∈ IN

¶

,



Ap

A=

µ 

0 1 0 0

0 A=0 0  0  0 A= 0 0  0 0   A=   0

¶

1 1 ··· 1     1 1 ··· 1   ∈ Mn,n    A = 1 1 ··· 1 A2

 1 1 0 1 0 0 1 0 0 0

1 1 0 0

A2 

1  1  1 0

A2 

1 ··· ··· 1 0 1 ··· 1     ∈ Mn,n  1 0 ··· ··· 0

A2

x, y, z ∈ IRn

< −x, −x > ≤ 0

< x + y + z, z > = < x, z > + < y, z > + < z, z > A ∈ Mm,n B ∈ Mp,q C ∈ Mr,s m, n, p, q, r, s ∈ IN (A + B) · C

A ∈ Mn,n

A·A = A

I ∈ Mn,n

⇒ AT

A

⇒I−A

A

6⇒ A − I

A





 9 a = 0  3

 −7 b =  7 7

d(a, b) |a|

a∗ ˆa

a 

3  −5  x =   9 −1 a ∈ IR < x, y > = 1

b     

a

b α 



1  −2    y =    4 a

y x µ ¶ 1 x= 1

d(x, y) = y

x=

y=

√ 47 α = 45o µ

1 −2

ˆy

y

x

ˆx

x

y

µ

4 a

¶

¶

y=

µ

b 8

¶

a, b ∈ IR d(x, y) = 10

x⊥y

.

IR4                                    

 2     3  −1    3     0 4       0  0 ,  0   −3     0 1           2  4 6 −1 −9            5   9   8   0   −7   ,  , ,  , 2   −9   0   −9   8      −2 −4 −9 0 9

     6  3    9          0   −4   −1         4 8 −4           , ,  −9 5  7 IR2

• • •

IR2 ¶ µ ¶¾ ½µ 4 2 , −5 −5 ½µ ¶ µ ¶¾ 0 1 , 0 0 ¶ µ ¶¾ ½µ ¶ µ 2 −3 2 , , 4 6 4

IR3

• • •        

IR3    4  4 4 , 4   3 2      0 9 2  0 , 2 , 9   0 5 7

      8  8  8  5 5 3 , ,   7 3 0





0 0 0 0    −2 2 0 0  A=   1 0 1 −1  0 −1 −1 1



 1 3 2 A=3 5 0 0 5 7  1 2 4 A=8 6 3 8 6 2 

a ∈ IR 

 1 −1 a A= 2 2 a 1 0 a A−1

a





0 −1 0 0 0 0 1 0 0 0 0 0      0 0 −2 −3 0 0  A=  0 0 5 7 0 0   0 0 0 0 1 0 0 0 0 0 0 1 A−1 =

µ

10 2 −2 1

¶

B−1 =

µ

0 −1 −1 1

¶

(B · A)−1 (106 · A)−1 (AT · B)−1 A, B ∈ Mn,n AT A+B A·B A−1

½

7x1 + 2x2 = −2 2x 1 − 6x 2 = 2

 

2x1 −7x2  6x2 ½ x1 −4x1

+ 4x2 − 8 = 0 + x3 − 9 = 0 − 3x3 − 7 = 0 − x2 + 7x3 = 0 + 7x2 + 5x3 = 0

x − y = 1 x + y = 1 x − 2y = a a > 0

2x1 + 3x2 − 2x4 2x1 + 3x2 + x3 − 2x4 − 3x5 x1 + 3x2 + x3 − 3x5 3x2 + x3 + 2x4 − 3x5

= = = =

0 0 0 0

4x1 + 2x2 + 18x4 x1 + 2x2 + 3x3 + 12x4 −x1 − x2 − x3 − 7x4 2x1 + x2 + 9x4

= = = =

20 8 −6 10 A·x = b

n

n |L| |L| 0 1 ∞ a1 , . . . , an Ax = 0 ⇒ x = 0 A rg(A) = rg(A|b) < n A−1 A rg(A) ≤ n rg(A|b) > rg(A) = n rg(A) = n b ∈ / Span(a1, . . . , an ) Span(a1, . . . , an ) = Rn rg(A) ≥ n a1′ , . . . , a′n



 0 4 0 det  9 3 −1  −1 0 8





5 4 7 det  5 0 −9  0 0 9



3  0   A= 2   0 −3

0 0 5 0 9

 3 0 5 0 7 0   0 0 −7   1 −7 0  2 −3 9

A V



A · e1, . . . , A · e5 {e1, . . . , e5}

IR5 

a11 a12 a13 B ∈ M3,3 A =  a21 a22 a23  a31 a32 a33 det(A) = −10 det(B) = −2

det(−2 · AT )   a11 a12 a13 det  a21 a22 a23  −2a31 −2a32 −2a33   a13 a11 a12 det  a23 a21 a22  a33 a31 a32 det(A · B · A−1)   a11 a12 a13 A =  0 a22 a23  B ∈ M3,3 0 0 a33 det(A) = −8 det(B) = −3

det(−10 · A)   0 0 a33 det  a11 a12 a13  0 a22 a23

det((A−1)T ) det(A · B · A−1) a22 = a33 = −2

a11 P ∈ Mn,n PT

⇒ det(P) = 0

P

P=

P P·P=P det(P) = 1 XT = X−1

X ∈ Mn,n ⇒ det(X) = ±1

X

q(x) = x21 + 4x1x2 − 2x1x3 + 3x22 + 2x2x3 + x32 x ∈ IR3

xT · A · x

q(x) A A

A

q(x) = 7x21 + 4x1x2 − 3x1x3 + 2x22 + 4x23 xT · A · x

q(x)

x ∈ IR3

A A A

xt yt

t µ

A=

µ

0.6 0.1 0.4 0.9

t−1 ¶ ¶ µ xt xt−1 =A· yt yt−1

¶ A

t=1 t = 2, 4, 8 t xt yt xt yt xt yt

1 2 4 8 10 16 21 22 100 94 90 88 25 100 100 70 48 40 100 130 153 160

xi,t

i t

t−1 µ

x0,t x1,t

¶

=A·

µ

A=

µ

0 2 1 0 2

x0,t−1 x1,t−1

¶

¶

x0,t A x0, x1 ∈ [−40, 40]

t x0,t x1,t x0,t x1,t

1 20 20 20 10

2 40 10 20 10

xt

µ

xt yt

¶

=A·

3 20 20 20 10

4 40 10 20 10

yt

µ

xt−1 yt−1

t t−1 ¶

A=

µ

1 1 0 0

¶ A·A=A

A A µ µ

x0 y0

¶

=

µ

1 0

x1 y1

¶

¶ µ ¶ µ ¶ −1 1 , , 1 1 ¶ µ x10 y10

µ

x ∈ [−2, 2]

A=

µ

0 2 − 12 0

¶ A

A

IR2

−2 1

¶

y ∈ [0, 2]...