examen final

1. Interpolacion por el método de Lagrange.
Hallar el polinomio interpolador para
{{7, -2}, {4, -7}, {6, -3}, {12, -3}}
Los nodos son :
{x0 = 7, x1 = 4, x2 = 6, x3 = 12}
Las imagenes son :
{f0 = -2, f1 = -7, f2 = -3, f3 = -3}
In[1]:= lo[x_] ==
-2 {x - (4)} {x - (6)} {x - (12)}
{7 - (4)} {7 - (6)} {7 - (12)}
Out[1]= lo[x_] ⩵ 
2
15
(-12 + x) (-6 + x) (-4 + x)
l1[x_] ==
-7 {x - (7)} {x - (6)} {x - (12)}
{4 - (7)} {4 - (6)} {4 - (12)}
Out[2]= l1[x_] ⩵ 
7
48
(-12 + x) (-7 + x) (-6 + x)
In[4]:= l2[x_] ==
-3 {x - (7)} {x - (4)} {x - (12)}
{6 - (7)} {6 - (4)} {6 - (12)}
Out[4]= l2[x_] ⩵ -
1
4
(-12 + x) (-7 + x) (-4 + x)
In[5]:= l3[x_] ==
-3 {x - (7)} {x - (4)} {x - (6)}
{12 - (7)} {12 - (4)} {12 - (6)}
Out[5]= l3[x_] ⩵ -
1
80
(-7 + x) (-6 + x) (-4 + x)
el polinomio interpolante de la grage es :
In[6]:= Simplify
2
15
(-12 + x) (-6 + x) (-4 + x) +
7
48
(-12 + x) (-7 + x) (-6 + x) -
1
4
(-12 + x) (-7 + x) (-4 + x) -
1
80
(-7 + x) (-6 + x) (-4 + x)
Out[6]=
1
60
-1548 + 414 x - 37 x2
+ x3

polisolu =
1
60
-1548 + 414 x - 37 x2
+ x3


2. In[37]:= Plot
1
60
-1548 + 414 x - 37 x2
+ x3
, {x, -1, 12},
Epilog → {Point[{{7, -2}, {4, -7}, {6, -3}, {12, -3}}]}
Out[37]=
2 4 6 8 10 12
-35
-30
-25
-20
-15
-10
-5
Interpolacion por sistema de ecuaciones.
Hallar el polinomio interpolador para
{{7, -2}, {4, -7}, {6, -3}, {12, -3}}
Los nodos son :
{x0 = 7, x1 = 4, x2 = 6, x3 = 12}
las imagenes son :
{f0 = -2, f1 = -7, f2 = -3, f3 = -3}
In[8]:= -2 ⩵ 73
j + 72
i + 71
g + 70
l
Out[8]= -2 ⩵ 7 g + 49 i + 343 j + l
In[9]:= -7 ⩵ 43
j + 42
i + 41
g + 40
l
Out[9]= -7 ⩵ 4 g + 16 i + 64 j + l
In[10]:= -3 ⩵ 63
j + 62
i + 61
g + 60
l
Out[10]= -3 ⩵ 6 g + 36 i + 216 j + l
In[11]:= -3 ⩵ 123
j + 122
i + 121
g + 120
l
Out[11]= -3 ⩵ 12 g + 144 i + 1728 j + l
El sistema de ecuaciones lineales es :
-2 ⩵ 7 g + 49 i + 343 j + l
-7 ⩵ 4 g + 16 i + 64 j + l
-3 ⩵ 6 g + 36 i + 216 j + l
-3 ⩵ 12 g + 144 i + 1728 j + l
El conjunto solución del sistema de ecuaciones es :
2 Israel final.nb

3. In[13]:= Solve[{-2 ⩵ 7 g + 49 i + 343 j + l, -7 ⩵ 4 g + 16 i + 64 j + l,
-3 ⩵ 6 g + 36 i + 216 j + l, -3 ⩵ 12 g + 144 i + 1728 j + l}, {l, j, i, g}]
Out[13]= l → -
129
5
, j →
1
60
, i → -
37
60
, g →
69
10

El polinomio interpolante es :
Polisolu = -
129
5
+
1 x3
60
+ -
37 x2
60
+
69 x
10

In[36]:= Plot-
129
5
+
1 x3
60
+ -
37 x2
60
+
69 x
10
, {x, -1, 12},
Epilog → {Point[{{7, -2}, {4, -7}, {6, -3}, {12, -3}}]}
Out[36]=
2 4 6 8 10 12
-35
-30
-25
-20
-15
-10
-5
Interpolacion por diferencias divididas :
Hallar el polinomio interpolador para la colección de puntos :
{{7, -2}, {4, -7}, {6, -3}, {12, -3}}
Los nodos son :
{x0 = 7, x1 = 4, x2 = 6, x3 = 12}
las imagenes son :
{f0 = -2, f1 = -7, f2 = -3, f3 = -3}
In[16]:= LV = {{7, -2}, {4, -7}, {6, -3}, {12, -3}}
Out[16]= {{7, -2}, {4, -7}, {6, -3}, {12, -3}}
In[17]:= LV // MatrixForm
Out[17]//MatrixForm=
7 -2
4 -7
6 -3
12 -3
7 -2
4 -7
6 -3
12 -3
5
3
2
0
-1
3
-1
4
1
60
Israel final.nb 3

4. In[18]:= Simplify-2 +
5
3
(-7 + x) -
1
3
(-7 + x) (-4 + x) +
1
60
(-7 + x) (-6 + x) (-4 + x)
Out[18]=
1
60
-1548 + 414 x - 37 x2
+ x3

In[35]:= Plot
1
60
-1548 + 414 x - 37 x2
+ x3
, {x, -1, 12},
Epilog → {Point[{{7, -2}, {4, -7}, {6, -3}, {12, -3}}]}
Out[35]=
2 4 6 8 10 12
-35
-30
-25
-20
-15
-10
-5
4 Israel final.nb