Soluciones mat4es

Soluciones Problemas.
Lista 1.
1) a) 1500; b) 50t − 2. c) 98 bacterias/hora.
3) 40Km/h.
4) Depreciaci´on m´axima con trozos iguales.
π
π 3
π 4
5) T4 (x) = 1 + x − x3 /3 − x4 /6. 1 + 16
− 31 ( 16
) − 16 ( 16
) . |R4 | ≤ π 5 /(120 · 164 ).
6) T3 (x) = e−1 (1 − (x + 1) + (x + 1)2 /2 − (x + 1)3 /6). e−0.99 ≈ 0.99005e−1 . |R3 (−0.99)| < 0.25 · 10−9 .
√
π
π 2
π 3
+ 16 60
.
7)cos(7π/30) ≈ T3,π/4 (7π/30) = 22 1 − 60
− 12 60
8) T2 (x) = 1 − x2 /2. cos(0.001) ≈ 0.9999995. |R2 (0.001)| < 10−9 /6. T3 (x) = 1 − x2 /2. cos(0.001) ≈
0.9999995. |R3 (0.001)| < 10−12 /24. Error real 4.16 · 10−14 .
9) √
9a) M´ınimo Local
√ (3/2, −27/16). Inflexiones (0, 0), (1, −1). 9b) M´aximo Local (0, 1). Inflexiones
(− 2/2,
√ 2/3), ( 2/2, √2/3). 9c) M´aximos Locales (0, 0), (4, 0). M´aximoLocal (2, 16). Inflexiones
(2 − 2/ 3, 64/9), (2 + 2/ 3, 64/9).
10) 10a) fx = 3 + 2xy 2 , fy = −1 + 2x2 y. 10b) fx = 3 cos(3x − y), fy = − cos(3x − y) − 1/y 2 . 10c)
fx = 3x2 y 2 − 2x, fy = 2x3 y − ey . 10d) fx = xy−1 y x (x log(y) + y, fy = xy y x−1 (y log(x) + x).
11) 11a) UP = 2V /(V − P )2 , UV = −2P/(V − P )2 . 11b) UP = 1/P , UV = −1/V . 11c) UP =
V sec2 (P V ), UV = P sec2 (P V ), UN = −eN. 11d) UP = (V /P )(log(P ))V −1 , UV = (log(P ))V log(log(P )).
12) a) L0 = 1, a = 1/2, α = 2. b) y = 2T + 1. c) L (0) = 2. 2L0 .

Lista 2.
1) a) 27; b) m´ınimo -3, m´aximo 897. c) P2 (t) = 112 + 27(t − 1) + 15(t − 1)2 . R = 0.
2) T3 (x) = x − x2 /2 + x3 /3. |R3 (0.4)| < 1/640.
3) a) b = 1. b) T3,1 (x) = 1 + 3(x − 1) + (x − 1)2 /2 + x3 /3. |R3 (1.05)| < 2 · 10−6 .
4a) A = LU



1
3 1
1
 , U = 0 8/3 2/3
L = 1/3 1
1/3 1/4 1
0
5/2
4b) A = LU



1
L = 1 1  ,
1 1 1




1 1 1
U = 0 3 3 
0
4

5a) Sin pivotaje A = LU , soluci´on x = (7/3, 5/3, −1/3)T .




1
1 1 0
 , U = 0 −2 2
1
L = 2
0 −3/2 1
0
6
5b) Con pivotaje parcial, soluci´on x = (7/3, 5/3, −1/3)T . P A = LU .






1
2 0 2
0 1 0
 , U = 0 3 3  P= 0 0 1 
1
L= 0
1/2 1/3 1
0
−2
1 0 0

1

6a) Sin pivotaje A = LU ; x = (1, 0, 0, −1)T .


4 −2 8 4

9 0 3

U =

1 1
9




1
−1/2

1

L=
 2
0
1 
1
−1/3 1 1

6b) Con pivotaje P A = LU ; x = (1, 0, 0, −1)T .




8 −4 17
9
1



9 1/4 −11/4 
1
 L = −1/4

U =

 1/2 −1/3

7/12 115/12
1
54/7
1/2
0
−6/7 1

0
0

P =
0
1

0
1
0
0

6c) Cholesky, sin permutaci´on A = GGT ; x = (1, 0, 0, −1)T .


2
−1 3


G=
4
0 1 
2 −1 1 3

Lista 3.
1) A2 = A, A3 = A, An = A, ∀n. B 2 = I, B 3 = B. C 2 = 0, C 3 = 0,C n = 0, ∀n ≥ 2.
Bn =

1
0
0 (−1)n

3) Sin pivotaje A = LU , soluci´on u = 3, v = 1, w = 0.




1
2 −3 0
1
1
L = 2 1  , U = 
1 2 1
−5
4) Sinpivotaje A = LU , soluci´on u = 3, v = −2, w = 1.




1
1 1 1
L = 1 1  , U =  2 2
1 1 1
2
5) Sin pivotaje A = LU ; (u, v, w, z) = (1, 2, 3, 4).




2 −1 0
0
1
 3/2 −1 0 


1
 L = −1/2

U =

 0

4/3 −1 
−2/3
1
5/4
0
0
−3/4 1
6a) Con pivotaje P A = LU .



1
L =  0 1 ,
−2 0 1




1 4 2
U =  1 1
7
2




1 0 0
P = 00 1 
0 1 0

1
0
0
0


0
0

1
0

Soluci´on de Lz = P b,z = (−2, 1, 28)T . Soluci´on de U x = z, x = (u, v, w) = (2, −3, 4)T .
6b) Con pivotaje P A = LU .






1
1 1 0
0 1 0
L = 0 1  , U =  1 1  P = 1 0 0 
1 0 1
1
0 0 1
Soluci´on de Lz = P b,z = (0, 0, 1)T . Soluci´on de U x = z, x = (u, v, w) = (1, −1, 1)T .
7a) Incompatible.






0 10
1 −1 0
1
1 −1 P = 1 0 0
L = 0 1  , U = 
0 0 1
0
1 1 1
7b) Compatible 1-indeterminado, (u, v, w) = λ(1, 1, 1).




1 −1 0
1
1 −1
L = 0 1  , U = 
0
1 1 1
7c) Compatible determinado.


1
L = 0 1  ,
1 1 1




1 1 0
U =  1 1
2



0 1 0
P = 1 0 0
0 0 1



0 1 0
P = 1 0 0 
0 0 1

Soluci´on (u, v, w) = (1/2, 1/2, 1/2)T .
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