{"id":52324,"date":"2025-01-19T13:28:43","date_gmt":"2025-01-19T12:28:43","guid":{"rendered":"https:\/\/www.valcon.it\/pp\/?p=52324"},"modified":"2026-09-10T19:32:03","modified_gmt":"2026-09-10T17:32:03","slug":"polinomio-di-newton-100","status":"publish","type":"post","link":"https:\/\/www.valcon.it\/pp\/polinomio-di-newton-100\/","title":{"rendered":"Polinomio di Newton – 100"},"content":{"rendered":"\n
\n
Interpolazione e regressione<\/a><\/div>\n<\/div>\n\n\n\n

<\/p>\n\n\n\n

Esercizio del libro di testo a pagina 100.
Calcola il valore approssimato per x=3 utilizzando il polinomio interpolatore di Newton passante per i punti dati.
Svolgi l\u2019esercizio per 2 \/ 3 \/ 4 \/ 5 \/ 6 punti.<\/em><\/p>\n\n\n\n

Otterrai gli stessi risultati del polinomio di Lagrange con gli stessi punti<\/a>.<\/em><\/p>\n\n\n\n

\n\n\n\n

Punti = 2, Grado = 1<\/h2>\n\n\n\n
<\/th>x<\/th>y<\/th>\"f_1\"<\/th><\/tr><\/thead>
0<\/td>2<\/strong><\/td>-3<\/strong><\/td><\/td><\/tr>
<\/td><\/td><\/td>\"\displaystyle \frac{-3-15}{2-1}\" = -18<\/td><\/tr>
1<\/td>1<\/strong><\/td>15<\/strong><\/td><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
\"p_1(x)\"<\/td>= \"b_0 +b_1 (x-x_0)\"<\/td><\/td><\/tr>
<\/td>= \"-3 -18 (x-2)\"<\/td><\/td><\/tr>
<\/td><\/td><\/td><\/tr>
\"p_1(3)\"<\/td>= \"-3 -18 (3-2)\"<\/td><\/td><\/tr>
<\/td>= -21<\/strong><\/td><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
\n\n\n\n

Punti = 3, Grado = 2<\/h2>\n\n\n\n
<\/th>x<\/th>y<\/th>\"f_1\"<\/th>\"f_2\"<\/th><\/tr><\/thead>
0<\/td>2<\/strong><\/td>-3<\/strong><\/td><\/td><\/td><\/tr>
<\/td><\/td><\/td>\"\displaystyle \frac{-3-15}{2-1}\" = -18<\/strong><\/td><\/td><\/tr>
1<\/td>1<\/strong><\/td>15<\/strong><\/td><\/td>\"\displaystyle \frac{-18-\left(-\displaystyle\frac{13}{4}\right)}{2-5}\" = \"\displaystyle \frac{59}{12}\"<\/td><\/tr>
<\/td><\/td><\/td>\"\displaystyle \frac{15-2}{1-5}\" = \"\displaystyle -\frac{13}{4}\"<\/td><\/td><\/tr>
2<\/td>5<\/strong><\/td>2<\/strong><\/td><\/td><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
\"p_2(x)\"<\/td>= \"b_0 +b_1 (x-x_0) +b_2 (x-x_0)(x-x_1)\"<\/td><\/tr>
<\/td>= \"\displaystyle -3 -18(x-2) + \frac{59}{12}(x-2)(x-1)\"<\/td><\/tr>
<\/td><\/td><\/tr>
\"p_2(3)\"<\/td>= \"\displaystyle -3 -18(3-2) + \frac{59}{12}(3-2)(3-1)\"<\/td><\/tr>
<\/td>= \"\displaystyle -\frac{67}{6}\" = -11,166…<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
\n\n\n\n

Punti = 4, Grado = 3<\/h2>\n\n\n\n
<\/th>x<\/th>y<\/th>\"f_1\"<\/th>\"f_2\"<\/th>\"f_3\"<\/th><\/tr><\/thead>
0<\/td>2<\/strong><\/td>-3<\/strong><\/td><\/td><\/td><\/td><\/tr>
<\/td><\/td><\/td>\"\displaystyle \frac{-3-15}{2-1}\" = -18<\/strong><\/td><\/td><\/td><\/tr>
1<\/td>1<\/strong><\/td>15<\/strong><\/td><\/td>\"\displaystyle \frac{-18-\left(-\displaystyle\frac{13}{4}\right)}{2-5}\" = \"\displaystyle \frac{59}{12}\"<\/td><\/td><\/tr>
<\/td><\/td><\/td>\"\displaystyle \frac{15-2}{1-5}\" = \"\displaystyle -\frac{13}{4}\"<\/td><\/td>\"\displaystyle \frac{\displaystyle\frac{59}{12}-\displaystyle\frac{9}{20}}{2-6}\" = \"\displaystyle -\frac{67}{60}\"<\/td><\/tr>
2<\/td>5<\/strong><\/td>2<\/strong><\/td><\/td>\"\displaystyle \frac{-\displaystyle\frac{13}{4}-(-1)}{1-6}\" = \"\displaystyle \frac{9}{20}\"<\/td><\/td><\/tr>
<\/td><\/td><\/td>\"\displaystyle \frac{2-1}{5-6}\" = -1<\/td><\/td><\/td><\/tr>
3<\/td>6<\/strong><\/td>1<\/strong><\/td><\/td><\/td><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
\"p_3(x)\"<\/td>= \"b_0 + b_1 (x-x_0) + b_2 (x-x_0)(x-x_1) + b_3 (x-x_0)(x-x_1)(x-x_2)\"<\/td><\/tr>
<\/td>= \"\displaystyle -3 -18 (x-2) + \frac{59}{12}(x-2)(x-1)-\frac{67}{60}(x-2)(x-1)(x-5)\"<\/td><\/tr>
<\/td><\/td><\/tr>
\"p_3(3)\"<\/td>= \"\displaystyle -3 -18 (3-2) + \frac{59}{12}(3-2)(3-1)-\frac{67}{60}(3-2)(3-1)(3-5)\"<\/td><\/tr>
<\/td>= \"\displaystyle -\frac{67}{10}\" = -6,7<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
\n\n\n\n

Punti = 5, Grado = 4<\/h2>\n\n\n\n
<\/th>x<\/th>y<\/th>\"f_1\"<\/th>\"f_2\"<\/th>\"f_3\"<\/th>\"f_4\"<\/th><\/tr><\/thead>
0<\/td>2<\/strong><\/td>-3<\/strong><\/td><\/td><\/td><\/td><\/td><\/tr>
<\/td><\/td><\/td>\"\displaystyle \frac{-3-15}{2-1}\" = -18<\/strong><\/td><\/td><\/td><\/td><\/tr>
1<\/td>1<\/strong><\/td>15<\/strong><\/td><\/td>\"\displaystyle \frac{-18-\left(\displaystyle-\frac{13}{4}\right)}{2-5}\" = \"\displaystyle \frac{59}{12}\"<\/td><\/td><\/td><\/tr>
<\/td><\/td><\/td>\"\displaystyle \frac{15-2}{1-5}\" = \"\displaystyle -\frac{13}{4}\"<\/td><\/td>\"\displaystyle \frac{\displaystyle\frac{59}{12}-\displaystyle\frac{9}{20}}{2-6}\" = \"\displaystyle -\frac{67}{60}\"<\/td><\/td><\/tr>
2<\/td>5<\/strong><\/td>2<\/strong><\/td><\/td>\"\displaystyle \frac{\displaystyle-\frac{13}{4}-(-1)}{1-6}\" = \"\displaystyle \frac{9}{20}\"<\/td><\/td>\"\displaystyle \frac{-\displaystyle\frac{67}{60}-\left(-\displaystyle\frac{13}{20}\right)}{2-4}\" = \"\displaystyle \frac{7}{30}\"<\/td><\/tr>
<\/td><\/td><\/td>\"\displaystyle \frac{2-1}{5-6}\" = -1<\/td><\/td>\"\displaystyle \frac{\displaystyle\frac{9}{20}-\left(-\displaystyle\frac{3}{2}\right)}{1-4}\" = \"\displaystyle -\frac{13}{20}\"<\/td><\/td><\/tr>
3<\/td>6<\/strong><\/td>1<\/strong><\/td><\/td>\"\displaystyle \frac{-1-\displaystyle\frac{1}{2}}{5-4}\" = \"\displaystyle -\frac{3}{2}\"<\/td><\/td><\/td><\/tr>
<\/td><\/td><\/td>\"\displaystyle \frac{1-0}{6-4}\" = \"\displaystyle \frac{1}{2}\"<\/td><\/td><\/td><\/td><\/tr>
4<\/td>4<\/strong><\/td>0<\/strong><\/td><\/td><\/td><\/td><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
\"p_4(x)\"<\/td>= \"b_0 + b_1(x-x_0) + b_2(x-x_0)(x-x_1) + b_3(x-x_0)(x-x_1)(x-x_2)\" + \"b_4(x-x_0)(x-x_1)(x-x_2)(x-x_3)\"<\/td><\/tr>
<\/td>= \"\displaystyle -3 -18 (x-2) + \frac{59}{12}(x-2)(x-1)-\frac{67}{60}(x-2)(x-1)(x-5)\" + \"\displaystyle\frac{7}{30}(x-2)(x-1)(x-5)(x-6)\"<\/td><\/tr>
<\/td><\/td><\/tr>
\"p_4(3)\"<\/td>= \"\displaystyle -3 -18 (3-2) + \frac{59}{12}(3-2)(3-1)-\frac{67}{60}(3-2)(3-1)(3-5)\" + \"\displaystyle\frac{7}{30}(3-2)(3-1)(3-5)(3-6)\"<\/td><\/tr>
<\/td> = \"\displaystyle -\frac{39}{10}\" = -3,9<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
\n\n\n\n

Punti = 6, Grado = 5<\/h2>\n\n\n\n
<\/th>x<\/th>y<\/th>\"f_1\"<\/th>\"f_2\"<\/th>\"f_3\"<\/th>\"f_4\"<\/th>\"f_5\"<\/th><\/tr><\/thead>
0<\/td>2<\/strong><\/td>-3<\/strong><\/td><\/td><\/td><\/td><\/td><\/td><\/tr>
<\/td><\/td><\/td>\"\displaystyle \frac{-3-15}{2-1}\" = -18<\/strong><\/td><\/td><\/td><\/td><\/td><\/tr>
1<\/td>1<\/strong><\/td>15<\/strong><\/td><\/td>\"\displaystyle \frac{-18-\left(\displaystyle-\frac{13}{4}\right)}{2-5}\" = \"\displaystyle \frac{59}{12}\"<\/td><\/td><\/td><\/td><\/tr>
<\/td><\/td><\/td>\"\displaystyle \frac{15-2}{1-5}\" = \"\displaystyle -\frac{13}{4}\"<\/td><\/td>\"\displaystyle \frac{\displaystyle\frac{59}{12}-\displaystyle\frac{9}{20}}{2-6}\" = \"\displaystyle -\frac{67}{60}\"<\/td><\/td><\/td><\/tr>
2<\/td>5<\/strong><\/td>2<\/strong><\/td><\/td>\"\displaystyle \frac{\displaystyle-\frac{13}{4}-(-1)}{1-6}\" = \"\displaystyle \frac{9}{20}\"<\/td><\/td>\"\displaystyle \frac{-\displaystyle\frac{67}{60}-\left(-\displaystyle\frac{13}{20}\right)}{2-4}\" = \"\displaystyle \frac{7}{30}\"<\/td><\/td><\/tr>
<\/td><\/td><\/td>\"\displaystyle \frac{2-1}{5-6}\" = -1<\/td><\/td>\"\displaystyle \frac{\displaystyle\frac{9}{20}-\left(-\displaystyle\frac{3}{2}\right)}{1-4}\" = \"\displaystyle -\frac{13}{20}\"<\/td><\/td>\"\displaystyle \frac{\displaystyle \frac{7}{30}-\left(\displaystyle-\frac{17}{60}\right)}{2-0}\" = \"\displaystyle \frac{31}{120}\"<\/td><\/tr>
3<\/td>6<\/strong><\/td>1<\/strong><\/td><\/td>\"\displaystyle \frac{-1-\displaystyle\frac{1}{2}}{5-4}\" = \"\displaystyle -\frac{3}{2}\"<\/td><\/td>\"\displaystyle \frac{\displaystyle -\frac{13}{20}-\left(\displaystyle-\frac{11}{30}\right)}{1-0}\" = \"\displaystyle -\frac{17}{60}\"<\/td><\/td><\/tr>
<\/td><\/td><\/td>\"\displaystyle \frac{1-0}{6-4}\" = \"\displaystyle \frac{1}{2}\"<\/td><\/td>\"\displaystyle \frac{\displaystyle -\frac{3}{2}-\displaystyle\frac{1}{3}}{5-0}\" = \"\displaystyle -\frac{11}{30}\"<\/td><\/td><\/td><\/tr>
4<\/td>4<\/strong><\/td>0<\/strong><\/td><\/td>\"\displaystyle \frac{\displaystyle\frac{1}{2}-\left(-\displaystyle\frac{3}{2}\right)}{6-0}\" = \"\displaystyle \frac{1}{3}\"<\/td><\/td><\/td><\/td><\/tr>
<\/td><\/td><\/td>\"\displaystyle \frac{0-6}{4-0}\" = \"\displaystyle -\frac{3}{2}\"<\/td><\/td><\/td><\/td><\/td><\/tr>
5<\/td>0<\/strong><\/td>6<\/strong><\/td><\/td><\/td><\/td><\/td><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
\"p_5(x)\"<\/td>= \"b_0 + b_1(x-x_0) + b_2(x-x_0)(x-x_1) + b_3(x-x_0)(x-x_1)(x-x_2)\" + \"b_4(x-x_0)(x-x_1)(x-x_2)(x-x_3)\"
+ \"b_5(x-x_0)(x-x_1)(x-x_2)(x-x_3)(x-x_4)\"<\/td><\/tr>
<\/td>= \"\displaystyle -3 -18 (x-2) + \frac{59}{12}(x-2)(x-1)-\frac{67}{60}(x-2)(x-1)(x-5)\" + \"\displaystyle\frac{7}{30}(x-2)(x-1)(x-5)(x-6)\"
+ \"\displaystyle\frac{31}{120}(x-2)(x-1)(x-5)(x-6)(x-4)\"<\/td><\/tr>
<\/td><\/td><\/tr>
\"p_5(3)\"<\/td>= \"\displaystyle -3 -18 (3-2) + \frac{59}{12}(3-2)(3-1)-\frac{67}{60}(3-2)(3-1)(3-5)\" + \"\displaystyle\frac{7}{30}(3-2)(3-1)(3-5)(3-6)\"
+ \"\displaystyle\frac{31}{120}(3-2)(3-1)(3-5)(3-6)(3-4)\"<\/td><\/tr>
<\/td>= -7<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
\n\n\n\n
\n\n\n\n

Ripeti i passi precedenti calcolando il polinomio interpolatore<\/strong> di Newton passante per 2 \/ 3 \/ 4 \/ 5 \/ 6 punti<\/p>\n\n\n\n

\n\n\n\n

Punti = 2, Grado = 1<\/h2>\n\n\n\n
<\/th>x<\/th>y<\/th>\"f_1\"<\/th><\/tr><\/thead>
0<\/td>2<\/strong><\/td>-3<\/strong><\/td><\/td><\/tr>
<\/td><\/td><\/td>\"\displaystyle \frac{-3-15}{2-1}\" = -18<\/td><\/tr>
1<\/td>1<\/strong><\/td>15<\/strong><\/td><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
\"p_1(x)\"<\/td>= \"b_0 +b_1 (x-x_0)\"<\/td><\/td><\/tr>
<\/td>= \"-3 -18 (x-2)\"<\/td><\/td><\/tr>
<\/td>= \"-18 x +33\"<\/td><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n
\n\n\n\n

Punti = 3, Grado = 2<\/h2>\n\n\n\n
<\/th>x<\/th>y<\/th>\"f_1\"<\/th>\"f_2\"<\/th><\/tr><\/thead>
0<\/td>2<\/strong><\/td>-3<\/strong><\/td><\/td><\/td><\/tr>
<\/td><\/td><\/td>\"\displaystyle \frac{-3-15}{2-1}\" = -18<\/strong><\/td><\/td><\/tr>
1<\/td>1<\/strong><\/td>15<\/strong><\/td><\/td>\"\displaystyle \frac{-18-\left(-\displaystyle\frac{13}{4}\right)}{2-5}\" = \"\displaystyle \frac{59}{12}\"<\/td><\/tr>
<\/td><\/td><\/td>\"\displaystyle \frac{15-2}{1-5}\" = \"\displaystyle -\frac{13}{4}\"<\/td><\/td><\/tr>
2<\/td>5<\/strong><\/td>2<\/strong><\/td><\/td><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
\"p_2(x)\"<\/td>= \"b_0 +b_1 (x-x_0) +b_2 (x-x_0)(x-x_1)\"<\/td><\/tr>
<\/td>= \"\displaystyle -3 -18(x-2) + \frac{59}{12}(x-2)(x-1)\"<\/td><\/tr>
<\/td>= \"\displaystyle \frac{59}{12} x^2- \frac{131}{4}x +\frac{257}{6}\"<\/td><\/tr>
<\/td><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n
\n\n\n\n

Punti = 4, Grado = 3<\/h2>\n\n\n\n
<\/th>x<\/th>y<\/th>\"f_1\"<\/th>\"f_2\"<\/th>\"f_3\"<\/th><\/tr><\/thead>
0<\/td>2<\/strong><\/td>-3<\/strong><\/td><\/td><\/td><\/td><\/tr>
<\/td><\/td><\/td>\"\displaystyle \frac{-3-15}{2-1}\" = -18<\/strong><\/td><\/td><\/td><\/tr>
1<\/td>1<\/strong><\/td>15<\/strong><\/td><\/td>\"\displaystyle \frac{-18-\left(-\displaystyle\frac{13}{4}\right)}{2-5}\" = \"\displaystyle \frac{59}{12}\"<\/td><\/td><\/tr>
<\/td><\/td><\/td>\"\displaystyle \frac{15-2}{1-5}\" = \"\displaystyle -\frac{13}{4}\"<\/td><\/td>\"\displaystyle \frac{\displaystyle\frac{59}{12}-\displaystyle\frac{9}{20}}{2-6}\" = \"\displaystyle -\frac{67}{60}\"<\/td><\/tr>
2<\/td>5<\/strong><\/td>2<\/strong><\/td><\/td>\"\displaystyle \frac{-\displaystyle\frac{13}{4}-(-1)}{1-6}\" = \"\displaystyle \frac{9}{20}\"<\/td><\/td><\/tr>
<\/td><\/td><\/td>\"\displaystyle \frac{2-1}{5-6}\" = -1<\/td><\/td><\/td><\/tr>
3<\/td>6<\/strong><\/td>1<\/strong><\/td><\/td><\/td><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
\"p_3(x)\"<\/td>= \"b_0 + b_1 (x-x_0) + b_2 (x-x_0)(x-x_1) + b_3 (x-x_0)(x-x_1)(x-x_2)\"<\/td><\/tr>
<\/td>= \"\displaystyle -3 -18 (x-2) + \frac{59}{12}(x-2)(x-1)-\frac{67}{60}(x-2)(x-1)(x-5)\"<\/td><\/tr>
<\/td>= \"\displaystyle -\frac{67}{60} x^3+ \frac{277}{20}x^2- \frac{776}{15}x +54\"<\/td><\/tr>
<\/td><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n
\n\n\n\n

Punti = 5, Grado = 4<\/h2>\n\n\n\n
<\/th>x<\/th>y<\/th>\"f_1\"<\/th>\"f_2\"<\/th>\"f_3\"<\/th>\"f_4\"<\/th><\/tr><\/thead>
0<\/td>2<\/strong><\/td>-3<\/strong><\/td><\/td><\/td><\/td><\/td><\/tr>
<\/td><\/td><\/td>\"\displaystyle \frac{-3-15}{2-1}\" = -18<\/strong><\/td><\/td><\/td><\/td><\/tr>
1<\/td>1<\/strong><\/td>15<\/strong><\/td><\/td>\"\displaystyle \frac{-18-\left(\displaystyle-\frac{13}{4}\right)}{2-5}\" = \"\displaystyle \frac{59}{12}\"<\/td><\/td><\/td><\/tr>
<\/td><\/td><\/td>\"\displaystyle \frac{15-2}{1-5}\" = \"\displaystyle -\frac{13}{4}\"<\/td><\/td>\"\displaystyle \frac{\displaystyle\frac{59}{12}-\displaystyle\frac{9}{20}}{2-6}\" = \"\displaystyle -\frac{67}{60}\"<\/td><\/td><\/tr>
2<\/td>5<\/strong><\/td>2<\/strong><\/td><\/td>\"\displaystyle \frac{\displaystyle-\frac{13}{4}-(-1)}{1-6}\" = \"\displaystyle \frac{9}{20}\"<\/td><\/td>\"\displaystyle \frac{-\displaystyle\frac{67}{60}-\left(-\displaystyle\frac{13}{20}\right)}{2-4}\" = \"\displaystyle \frac{7}{30}\"<\/td><\/tr>
<\/td><\/td><\/td>\"\displaystyle \frac{2-1}{5-6}\" = -1<\/td><\/td>\"\displaystyle \frac{\displaystyle\frac{9}{20}-\left(-\displaystyle\frac{3}{2}\right)}{1-4}\" = \"\displaystyle -\frac{13}{20}\"<\/td><\/td><\/tr>
3<\/td>6<\/strong><\/td>1<\/strong><\/td><\/td>\"\displaystyle \frac{-1-\displaystyle\frac{1}{2}}{5-4}\" = \"\displaystyle -\frac{3}{2}\"<\/td><\/td><\/td><\/tr>
<\/td><\/td><\/td>\"\displaystyle \frac{1-0}{6-4}\" = \"\displaystyle \frac{1}{2}\"<\/td><\/td><\/td><\/td><\/tr>
4<\/td>4<\/strong><\/td>0<\/strong><\/td><\/td><\/td><\/td><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
\"p_4(x)\"<\/td>= \"b_0 + b_1(x-x_0) + b_2(x-x_0)(x-x_1) + b_3(x-x_0)(x-x_1)(x-x_2)\" + \"b_4(x-x_0)(x-x_1)(x-x_2)(x-x_3)\"<\/td><\/tr>
<\/td>= \"\displaystyle -3 -18 (x-2) + \frac{59}{12}(x-2)(x-1)-\frac{67}{60}(x-2)(x-1)(x-5)\" + \"\displaystyle\frac{7}{30}(x-2)(x-1)(x-5)(x-6)\"<\/td><\/tr>
<\/td>= \"\displaystyle \frac{7}{30} x^4 -\frac{263}{60} x^3 +\frac{1741}{60} x^2 -\frac{1168}{15} x + 68\"<\/td><\/tr>
<\/td><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n
\n\n\n\n

Punti = 6, Grado = 5<\/h2>\n\n\n\n
<\/th>x<\/th>y<\/th>\"f_1\"<\/th>\"f_2\"<\/th>\"f_3\"<\/th>\"f_4\"<\/th>\"f_5\"<\/th><\/tr><\/thead>
0<\/td>2<\/strong><\/td>-3<\/strong><\/td><\/td><\/td><\/td><\/td><\/td><\/tr>
<\/td><\/td><\/td>\"\displaystyle \frac{-3-15}{2-1}\" = -18<\/strong><\/td><\/td><\/td><\/td><\/td><\/tr>
1<\/td>1<\/strong><\/td>15<\/strong><\/td><\/td>\"\displaystyle \frac{-18-\left(\displaystyle-\frac{13}{4}\right)}{2-5}\" = \"\displaystyle \frac{59}{12}\"<\/td><\/td><\/td><\/td><\/tr>
<\/td><\/td><\/td>\"\displaystyle \frac{15-2}{1-5}\" = \"\displaystyle -\frac{13}{4}\"<\/td><\/td>\"\displaystyle \frac{\displaystyle\frac{59}{12}-\displaystyle\frac{9}{20}}{2-6}\" = \"\displaystyle -\frac{67}{60}\"<\/td><\/td><\/td><\/tr>
2<\/td>5<\/strong><\/td>2<\/strong><\/td><\/td>\"\displaystyle \frac{\displaystyle-\frac{13}{4}-(-1)}{1-6}\" = \"\displaystyle \frac{9}{20}\"<\/td><\/td>\"\displaystyle \frac{-\displaystyle\frac{67}{60}-\left(-\displaystyle\frac{13}{20}\right)}{2-4}\" = \"\displaystyle \frac{7}{30}\"<\/td><\/td><\/tr>
<\/td><\/td><\/td>\"\displaystyle \frac{2-1}{5-6}\" = -1<\/td><\/td>\"\displaystyle \frac{\displaystyle\frac{9}{20}-\left(-\displaystyle\frac{3}{2}\right)}{1-4}\" = \"\displaystyle -\frac{13}{20}\"<\/td><\/td>\"\displaystyle \frac{\displaystyle \frac{7}{30}-\left(\displaystyle-\frac{17}{60}\right)}{2-0}\" = \"\displaystyle \frac{31}{120}\"<\/td><\/tr>
3<\/td>6<\/strong><\/td>1<\/strong><\/td><\/td>\"\displaystyle \frac{-1-\displaystyle\frac{1}{2}}{5-4}\" = \"\displaystyle -\frac{3}{2}\"<\/td><\/td>\"\displaystyle \frac{\displaystyle -\frac{13}{20}-\left(\displaystyle-\frac{11}{30}\right)}{1-0}\" = \"\displaystyle -\frac{17}{60}\"<\/td><\/td><\/tr>
<\/td><\/td><\/td>\"\displaystyle \frac{1-0}{6-4}\" = \"\displaystyle \frac{1}{2}\"<\/td><\/td>\"\displaystyle \frac{\displaystyle -\frac{3}{2}-\displaystyle\frac{1}{3}}{5-0}\" = \"\displaystyle -\frac{11}{30}\"<\/td><\/td><\/td><\/tr>
4<\/td>4<\/strong><\/td>0<\/strong><\/td><\/td>\"\displaystyle \frac{\displaystyle\frac{1}{2}-\left(-\displaystyle\frac{3}{2}\right)}{6-0}\" = \"\displaystyle \frac{1}{3}\"<\/td><\/td><\/td><\/td><\/tr>
<\/td><\/td><\/td>\"\displaystyle \frac{0-6}{4-0}\" = \"\displaystyle -\frac{3}{2}\"<\/td><\/td><\/td><\/td><\/td><\/tr>
5<\/td>0<\/strong><\/td>6<\/strong><\/td><\/td><\/td><\/td><\/td><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
\"p_5(x)\"<\/td>= \"b_0 + b_1(x-x_0) + b_2(x-x_0)(x-x_1) + b_3(x-x_0)(x-x_1)(x-x_2)\" + \"b_4(x-x_0)(x-x_1)(x-x_2)(x-x_3)\"
+ \"b_5(x-x_0)(x-x_1)(x-x_2)(x-x_3)(x-x_4)\"<\/td><\/tr>
<\/td>= \"\displaystyle -3 -18 (x-2) + \frac{59}{12}(x-2)(x-1)-\frac{67}{60}(x-2)(x-1)(x-5)\" + \"\displaystyle\frac{7}{30}(x-2)(x-1)(x-5)(x-6)\"
+ \"\displaystyle\frac{31}{120}(x-2)(x-1)(x-5)(x-6)(x-4)\"<\/td><\/tr>
<\/td>= \"\displaystyle \frac{31}{120}\ x^5 - \frac{53}{12}\ x^4 + \frac{215}{8}\ x^3 -\frac{805}{12}\ x^2 + \frac{1601}{30}\ x + 6\"<\/td><\/tr>
<\/td><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n
\n
\"\"<\/figure>\n<\/div>\n\n\n
\n\n\n\n

Osserva i risultati successivi<\/p>\n\n\n\n

Punti<\/th>Polinomio<\/th>x=3<\/th><\/tr><\/thead>
2<\/td>\"p_1(x) = -18x+33\"<\/td>-21<\/strong><\/td><\/tr>
3<\/td>\"p_2(x) = \displaystyle \frac{59}{12} x^2 - \frac{131}{4}x + \frac{257}{6}\"<\/td>-11<\/strong>,166…<\/td><\/tr>
4<\/td>\"p_3(x) = \displaystyle -\frac{67}{60} x^3 + \frac{277}{20}x^2 - \frac{776}{15}x+54\"<\/td>-6,7<\/strong><\/td><\/tr>
5<\/td>\"p_4(x) = \displaystyle \frac{7}{30} x^4 -\frac{263}{60} x^3 +\frac{1741}{60} x^2 -\frac{1168}{15} x + 68\"<\/td>-3,9<\/strong><\/td><\/tr>
6<\/td>\"p_5(x) = \displaystyle \frac{31}{120}\ x^5 - \frac{53}{12}\ x^4 + \frac{215}{8}\ x^3 -\frac{805}{12}\ x^2 + \frac{1601}{30}\ x + 6\"<\/td>-7<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n","protected":false},"excerpt":{"rendered":"

Esercizio del libro di testo a pagina 100.Calcola il valore approssimato per x=3 utilizzando il polinomio interpolatore di Newton passante per i punti dati.Svolgi l\u2019esercizio per 2 \/ 3 \/ 4 \/ 5 \/ 6 punti. Otterrai gli stessi risultati del polinomio di Lagrange con gli stessi punti. Punti = 2, Grado = 1 x … Leggi tutto<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[114],"tags":[48],"class_list":["post-52324","post","type-post","status-publish","format-standard","hentry","category-calcolo","tag-newton"],"_links":{"self":[{"href":"https:\/\/www.valcon.it\/pp\/wp-json\/wp\/v2\/posts\/52324","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.valcon.it\/pp\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.valcon.it\/pp\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.valcon.it\/pp\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.valcon.it\/pp\/wp-json\/wp\/v2\/comments?post=52324"}],"version-history":[{"count":143,"href":"https:\/\/www.valcon.it\/pp\/wp-json\/wp\/v2\/posts\/52324\/revisions"}],"predecessor-version":[{"id":64506,"href":"https:\/\/www.valcon.it\/pp\/wp-json\/wp\/v2\/posts\/52324\/revisions\/64506"}],"wp:attachment":[{"href":"https:\/\/www.valcon.it\/pp\/wp-json\/wp\/v2\/media?parent=52324"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.valcon.it\/pp\/wp-json\/wp\/v2\/categories?post=52324"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.valcon.it\/pp\/wp-json\/wp\/v2\/tags?post=52324"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}