Интерполирование функций

Ʌɚɛɨɪɚɬɨɪɧɚɹɪɚɛɨɬɚɂɧɬɟɪɩɨɥɢɪɨɜɚɧɢɟɮɭɧɤɰɢɣ ɐɟɥɶ ɪɚɛɨɬɵ ɢɡɭɱɟɧɢɟ ɦɟɬɨɞɨɜ ɢɧɬɟɪɩɨɥɢɪɨɜɚɧɢɹ ɮɭɧɤɰɢɣ ɫɪɚɜɧɢɬɟɥɶɧɵɣ ɚɧɚɥɢɡ ɪɚɫɫɦɨɬɪɟɧɧɵɯ ɦɟɬɨɞɨɜ ɩɪɚɤɬɢɱɟɫɤɨɟ ɢɧɬɟɪɩɨɥɢɪɨɜɚɧɢɟɮɭɧɤɰɢɣɧɚɗȼɆ ɉɨɫɬɚɧɨɜɤɚɡɚɞɚɱɢ ɉɭɫɬɶɮɭɧɤɰɢɹ y  f (x) ɡɚɞɚɧɚɬɚɛɥɢɰɟɣ y0  f ( x0 ), y1  f ( x1 ), !, y n  f ( xn ) . Ɂɚɞɚɱɚɢɧɬɟɪɩɨɥɢɪɨɜɚɧɢɹɫɬɚɜɢɬɫɹɨɛɵɱɧɨɜɫɥɟɞɭɸɳɟɣɮɨɪɦɟ ɧɚɣɬɢɦɧɨɝɨɱɥɟɧ P( x)  Pn ( x) ɫɬɟɩɟɧɢɧɟɜɵɲɟnɡɧɚɱɟɧɢɹɤɨɬɨɪɨɝɨɜ ɬɨɱɤɚɯ xi (i = 0, 1, …, n  ɧɚɡɵɜɚɟɦɵɯ ɭɡɥɚɦɢ ɢɧɬɟɪɩɨɥɹɰɢɢ ɫɨɜɩɚɞɚɸɬɫɨɡɧɚɱɟɧɢɹɦɢɞɚɧɧɨɣɮɭɧɤɰɢɢɬɟ P( xi )  yi . Ƚɟɨɦɟɬɪɢɱɟɫɤɢ ɷɬɨ ɨɡɧɚɱɚɟɬ ɱɬɨ ɧɭɠɧɨ ɧɚɣɬɢ ɚɥɝɟɛɪɚɢɱɟɫɤɭɸ ɤɪɢɜɭɸɜɢɞɚ y  a0 x n a1 x n 1 ! an ɩɪɨɯɨɞɹɳɭɸɱɟɪɟɡɡɚɞɚɧɧɭɸ ɫɢɫɬɟɦɭɬɨɱɟɤ M i ( xi , yi ) (i = 0, 1, …, n  ɪɢɫ  ȼ ɬɚɤɨɣ ɩɨɫɬɚɧɨɜɤɟ ɡɚɞɚɱɚ ɢɧɬɟɪɩɨɥɢɪɨɜɚɧɢɹ ɧɚɡɵɜɚɟɬɫɹ ɩɚɪɚɛɨɥɢɱɟɫɤɨɣ Ɇɧɨɝɨɱɥɟɧ P(x) ɧɚɡɵɜɚɟɬɫɹ ɢɧɬɟɪɩɨɥɹɰɢɨɧɧɵɦ ɦɧɨɝɨɱɥɟɧɨɦ Ⱦɨɤɚɡɚɧɨɱɬɨɜɭɤɚɡɚɧɧɨɣɩɨɫɬɚɧɨɜɤɟɡɚɞɚɱɚɢɧɬɟɪɩɨɥɢɪɨɜɚɧɢɹ ɜɫɟɝɞɚ ɢɦɟɟɬ ɟɞɢɧɫɬɜɟɧɧɨɟ ɪɟɲɟɧɢɟ ɂɧɬɟɪɩɨɥɹɰɢɨɧɧɵɟ ɮɨɪɦɭɥɵ ɨɛɵɱɧɨ ɢɫɩɨɥɶɡɭɸɬɫɹ ɩɪɢ ɧɚɯɨɠɞɟɧɢɢ ɧɟɢɡɜɟɫɬɧɵɯ ɡɧɚɱɟɧɢɣ f (x) ɞɥɹ ɩɪɨɦɟɠɭɬɨɱɧɵɯ ɡɧɚɱɟɧɢɣ ɚɪɝɭɦɟɧɬɚ ɉɪɢ ɷɬɨɦ ɪɚɡɥɢɱɚɸɬ ɢɧɬɟɪɩɨɥɢɪɨɜɚɧɢɟɜɭɡɤɨɦɫɦɵɫɥɟɤɨɝɞɚɯ ɧɚɯɨɞɢɬɫɹɦɟɠɞɭ x0 ɢ xn , ɢɷɤɫɬɪɚɩɨɥɢɪɨɜɚɧɢɟɤɨɝɞɚɯ ɧɚɯɨɞɢɬɫɹɜɧɟɨɬɪɟɡɤɚ> x0 , xn ]. Ɋɢɫ1.1 ɉɪɢ ɨɰɟɧɤɟ ɩɨɝɪɟɲɧɨɫɬɢ ɪɟɡɭɥɶɬɚɬɨɜ ɞɨɥɠɧɵ ɭɱɢɬɵɜɚɬɶɫɹ ɤɚɤ ɩɨɝɪɟɲɧɨɫɬɶ ɦɟɬɨɞɚ ɢɧɬɟɪɩɨɥɹɰɢɢ ɨɫɬɚɬɨɱɧɵɣ ɱɥɟɧ  ɬɚɤ ɢ ɩɨɝɪɟɲɧɨɫɬɢɨɤɪɭɝɥɟɧɢɹɩɪɢɜɵɱɢɫɥɟɧɢɹɯ ɂɧɬɟɪɩɨɥɹɰɢɨɧɧɚɹɮɨɪɦɭɥɚɅɚɝɪɚɧɠɚ ɉɭɫɬɶ xi (i = 0, 1, …, n) – ɩɪɨɢɡɜɨɥɶɧɵɟ ɭɡɥɵ ɚ yi  f ( xi ) – ɡɧɚɱɟɧɢɹ ɮɭɧɤɰɢɢ f (x) . Ɇɧɨɝɨɱɥɟɧɨɦ ɫɬɟɩɟɧɢ n ɩɪɢɧɢɦɚɸɳɢɦ ɜ ɬɨɱɤɚɯ xi ɡɧɚɱɟɧɢɹ yi , ɹɜɥɹɟɬɫɹɢɧɬɟɪɩɨɥɹɰɢɨɧɧɵɣɦɧɨɝɨɱɥɟɧɅɚɝɪɚɧɠɚ Pn ( x)  ( x x1 ) ( x x2 )...( x xn ) y0 ( x0 x1 ) ( x0 x2 )...( x0 xn ) ( x x0 ) ( x x2 )...( x xn ) y1 ... ( x1 x0 ) ( x1 x2 )...( x1 xn ) ( x x0 ) ( x x1 )...( x xn 1 ) yn . ( xn x0 ) ( xn x1 )...( xn xn 1 ) Ɉɫɬɚɬɨɱɧɵɣɱɥɟɧɪɚɜɟɧ f ( n 1) (X) Rn ( x)  (x (n 1)! x0 )( x x1 )!( x xn ) , ɝɞɟȟɟɫɬɶ ɧɟɤɨɬɨɪɚɹ ɬɨɱɤɚ ɧɚɢɦɟɧɶɲɟɝɨ ɩɪɨɦɟɠɭɬɤɚ ɫɨɞɟɪɠɚɳɟɝɨ ɜɫɟɭɡɥɵ xi (i = 0, 1, …, n ɢɬɨɱɤɭɯ. ȼɵɪɚɠɟɧɢɹ Pi( n) ( x)  ( x x0 )( x x1 )!( x xi 1 )( x xi 1 )!( x xn ) ( xi x0 )( xi x1 )!( xi xi 1 )( xi xi 1 )!( xi xn ) (1.1) ɧɚɡɵɜɚɸɬɫɹɤɨɷɮɮɢɰɢɟɧɬɚɦɢɅɚɝɪɚɧɠɚ i xi yi -0,5 1 0,1 2 0,3 0,2 3 0,5 1 ɉɪɢɦɟɪ  ɇɚɩɢɫɚɬɶ ɢɧɬɟɪɩɨɥɹɰɢɨɧɧɵɣ ɦɧɨɝɨɱɥɟɧ Ʌɚɝɪɚɧɠɚ ɞɥɹ ɮɭɧɤɰɢɢ f (x) , ɡɧɚɱɟɧɢɹɤɨɬɨɪɨɣɞɚɧɵɬɚɛɥɢɰɟɣ Ɋɟɲɟɧɢɟ ɉɨ ɮɨɪɦɭɥɟ   ɩɪɢn ɩɨɥɭɱɢɦɜɵɪɚɠɟɧɢɹ Pi(3) ( x) ɩɪɢi=0, 2, 3: P0(3) ( x )  ( x 0,1)( x 0,3)( x 0,5) x3  ( 0,1)( 0,3)( 0,5) 0,9 x 2 0,23 x 0,015 , 0,015 P2(3) ( x)  x( x 0,1)( x 0,5) x3  0,3 – 0,2 – ( 0,2) P3(3) ( x)  0,6 x 2 0,05 x , 0,012 x( x 0,1)( x 0,3) x 3  0,5 – 0,4 – 0,2 0,4 x 2 0,03 x 0,04 ( P1(3) ( x) ɜɞɚɧɧɨɦɫɥɭɱɚɟɜɵɱɢɫɥɹɬɶɧɟɫɥɟɞɭɟɬɬɤ y1  0 ). Ɍɨɝɞɚɢɫɤɨɦɵɣɦɧɨɝɨɱɥɟɧɛɭɞɟɬɢɦɟɬɶ ɜɢɞ 3 P3 ( x)  ¤ yi Pi(3) ( x)  i 0 125 3 x 3 30 x 2 73 x 0,5 . 12 ɂɧɬɟɪɩɨɥɹɰɢɨɧɧɵɟ ɮɨɪɦɭɥɵ ɇɶɸɬɨɧɚ ɧɚɡɵɜɚɸɬɫɹɪɚɜɧɨɨɬɫɬɨɹɳɢɦɢɟɫɥɢ xi xi  $ xi  h , 1 ɍɡɥɵ ɢɧɬɟɪɩɨɥɹɰɢɢ (i  0,1,!, n 1) . Ʉɨɧɟɱɧɵɦɢɪɚɡɧɨɫɬɹɦɢɮɭɧɤɰɢɢ y  f (x) ɧɚɡɵɜɚɸɬɫɹɪɚɡɧɨɫɬɢ ɜɢɞɚ $ yi  yi yi – ɤɨɧɟɱɧɵɟɪɚɡɧɨɫɬɢɩɟɪɜɨɝɨɩɨɪɹɞɤɚ 1 $2 yi  $ yi 1 $ yi – ɤɨɧɟɱɧɵɟɪɚɡɧɨɫɬɢɜɬɨɪɨɝɨɩɨɪɹɞɤɚ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $k yi  $k 1 yi 1 $k 1 yi – ɤɨɧɟɱɧɵɟɪɚɡɧɨɫɬɢk-ɝɨɩɨɪɹɞɤɚ ɇɢɠɟɞɚɟɬɫɹɬɚɛɥɢɰɚɤɨɧɟɱɧɵɯɪɚɡɧɨɫɬɟɣɩɪɢn  ɬɚɛɥ  ɉɟɪɜɚɹɢɧɬɟɪɩɨɥɹɰɢɨɧɧɚɹɮɨɪɦɭɥɚɇɶɸɬɨɧɚɢɦɟɟɬɜɢɞ y ( x)  Pn ( x)  y0 q (q 1) 2 $ y0 2! q $ y0 ! q (q 1)!(q n! n 1) n $ y0 , (1.2) ɝɞɟ q  ( x x0 ) h . Ɂɚɦɟɬɢɦ ɱɬɨ ɜ ɮɨɪɦɭɥɟ ɢɫɩɨɥɶɡɭɟɬɫɹ ɜɟɪɯɧɹɹ ɝɨɪɢɡɨɧɬɚɥɶɧɚɹ ɫɬɪɨɤɚɬɚɛɥɈɫɬɚɬɨɱɧɵɣɱɥɟɧ Rn (x) ɮɨɪɦɭɥɵ  ɢɦɟɟɬɜɢɞ f ( n 1) (X) n 1 Rn ( x)  h q (q 1)!(q (n 1)! n) . (1.3) ɉɪɢ ɧɚɥɢɱɢɢ ɞɨɩɨɥɧɢɬɟɥɶɧɨɝɨ ɭɡɥɚ xn 1 ɩɨɥɶɡɭɸɬɫɹɛɨɥɟɟɭɞɨɛɧɨɣɩɪɢɛɥɢɠɟɧɧɨɣɮɨɪɦɭɥɨɣ $n 1 y0 Rn ( x) z q (q 1)!(q (n 1)! ɧɚ ɩɪɚɤɬɢɤɟ n) . ɉɨɫɥɟɞɧɹɹ ɮɨɪɦɭɥɚ Ɍɚɛɥɢɰɚ ɩɨɥɟɡɧɚɧɚɩɪɢɦɟɪɜɫɥɭɱɚɟ ɷɦɩɢɪɢɱɟɫɤɢ ɡɚɞɚɧɧɵɯ $3 y $y $2 y x y ɮɭɧɤɰɢɣ x0 y0 $y0 $2 y0 $3 y0 ɑɢɫɥɨ ɩ ɠɟɥɚɬɟɥɶɧɨ ɜɵɛɢɪɚɬɶ ɬɚɤ ɱɬɨɛɵ x1 y1 $y1 $2 y1 $3 y1 ɪɚɡɧɨɫɬɢ $n yi  ɛɵɥɢ x2 y 2 $y 2 $2 y 2 $3 y 2 ɩɪɚɤɬɢɱɟɫɤɢ ɩɨɫɬɨɹɧɧɵɦɢ x3 y3 $y3 $2 y3 ɫɦɩɪɢɦɟɪ  x4 y 4 $y 4 Ɏɨɪɦɭɥɚ   ɢɫɩɨɥɶɡɭɟɬɫɹ ɞɥɹ x5 y5 ɢɧɬɟɪɩɨɥɢɪɨɜɚɧɢɹ ɢ ɷɤɫɬɪɚɩɨɥɢɪɨɜɚɧɢɹɜɬɨɱɤɚɯɯɛɥɢɡɤɢɯɤɧɚɱɚɥɭɬɚɛɥɢɰɵ x0 . ȼɬɨɪɚɹɢɧɬɟɪɩɨɥɹɰɢɨɧɧɚɹɮɨɪɦɭɥɚɇɶɸɬɨɧɚɢɦɟɟɬɜɢɞ y ( x)  Pn ( x)  y n q (q 1) 2 $ yn 2! 2 q $ yn ! $4 y $4 y0 $4 y1 1 q (q 1)!(q n! n 1) $n y0 , (1.4) ɝɞɟ q  ( x xn ) h . ȼ ɮɨɪɦɭɥɟ   ɢɫɩɨɥɶɡɭɟɬɫɹ ɧɢɠɧɹɹ ɧɚɤɥɨɧɧɚɹ ɫɬɪɨɤɚ ɪɚɡɧɨɫɬɟɣ ɫɦɬɚɛɥ Ɉɫɬɚɬɨɱɧɵɣɱɥɟɧ Rn (x) ɮɨɪɦɭɥɵ  ɢɦɟɟɬ ɜɢɞ f ( n 1) (X) n 1 Rn ( x)  h q (q 1)!(q (n 1)! n) . Ɏɨɪɦɭɥɚ   ɢɫɩɨɥɶɡɭɟɬɫɹ ɞɥɹ ɢɧɬɟɪɩɨɥɢɪɨɜɚɧɢɹ ɢ ɷɤɫɬɪɚɩɨɥɢɪɨɜɚɧɢɹɜɬɨɱɤɚɯɯɛɥɢɡɤɢɯɤɤɨɧɰɭɬɚɛɥɢɰɵɬɟɤ xn . ɉɪɢɦɟɪ  ɉɨ ɞɚɧɧɨɣ ɬɚɛɥɢɰɟ ɡɧɚɱɟɧɢɣ ɮɭɧɤɰɢɢ y  lg x ɬɚɛɥ ɧɚɣɬɢ lg1001. ɊɟɲɟɧɢɟɁɧɚɱɟɧɢɟɛɥɢɡɤɨɤɧɚɱɚɥɭɬɚɛɥɢɰɵɩɨɷɬɨɦɭɞɥɹ ɢɧɬɟɪɩɨɥɢɪɨɜɚɧɢɹ ɢɫɩɨɥɶɡɭɟɦ ɮɨɪɦɭɥɭ   ɋɨɫɬɚɜɥɹɟɦ ɬɚɛɥɢɰɭ ɪɚɡɧɨɫɬɟɣ ɬɚɛɥ   ɢ ɡɚɦɟɱɚɟɦ ɱɬɨ ɬɪɟɬɶɢ ɪɚɡɧɨɫɬɢ ɩɪɚɤɬɢɱɟɫɤɢ ɩɨɫɬɨɹɧɧɵȼɷɬɨɣɫɜɹɡɢɜɮɨɪɦɭɥɟ  ɞɨɫɬɚɬɨɱɧɨɜɡɹɬɶn=3. Ɍɚɛɥɢɰɚ x y $y $2 y $3 y 1000 1010 1020 1030 1040 1050 3,0000000 3,0043214 3,0086002 3,0128372 3,0170333 3,0211893 0,0043241 0,0042788 0,0042370 0,0041961 0,0041560 -0,0000426 -0,0000418 -0,0000409 -0,0000401 0,0000008 0,0000009 0,0000008 q (q 1) 2 $ y0 2! q (q 1)(q 3! y ( x)  y0 q $ y0 2) $3 y0 . Ⱦɥɹx ɢɦɟɟɦ q  (1001 1000) / 10  0,1 . lg1001  3,0000000 0,1 – 0,0043214 0,1 – 0,9 0,0000426 2 0,1 – 0,9 – 1,9 0,0000008  3,0004341 . 2 Ɉɰɟɧɢɦɨɫɬɚɬɨɱɧɵɣɱɥɟɧɉɨɮɨɪɦɭɥɟ  ɩɪɢn ɢɦɟɟɦ f ( 4 ) ( X) 4 h q (q 1)(q 4! R3 ( x)  2)(q 3) , ɝɞɟȟ ȼɧɚɲɟɦɫɥɭɱɚɟ f ( 4) ( x)  3! 3! x (1000) 4 lg e ɩɨɷɬɨɦɭ f ( 4) (X) b 4 lg e. Ɉɤɨɧɱɚɬɟɥɶɧɨɩɨɥɭɱɚɟɦ R3 (1001) b 0,1 – 0,9 – 1,9 – 2,9 – 10 4 lg e 4 – (1000) 4 z 0,5 – 10 9. Ɉɫɬɚɬɨɱɧɵɣ ɱɥɟɧ ɦɨɠɟɬ ɩɨɜɥɢɹɬɶ ɬɨɥɶɤɨ ɧɚ ɞɟɜɹɬɵɣ ɞɟɫɹɬɢɱɧɵɣ ɡɧɚɤ Ɂɚɦɟɬɢɦ ɱɬɨ ɩɨɥɭɱɟɧɧɨɟ ɡɧɚɱɟɧɢɟ lg1001 ɩɨɥɧɨɫɬɶɸ ɫɨɜɩɚɞɚɟɬ ɫɨ ɡɧɚɱɟɧɢɟɦ ɜ ɫɟɦɢɡɧɚɱɧɨɣ ɬɚɛɥɢɰɟ ɥɨɝɚɪɢɮɦɨɜ Ɇɟɬɨɞ ɧɚɢɦɟɧɶɲɢɯ ɤɜɚɞɪɚɬɨɜ  ɋɭɳɧɨɫɬɶ ɞɚɧɧɨɝɨ ɦɟɬɨɞɚ m ɫɨɫɬɨɢɬ ɜ ɩɨɫɬɪɨɟɧɢɢ ɩɨɥɢɧɨɦɚ Pm ( x)  ¤a j x j ɫɬɟɩɟɧɢ m  n , j 0 ɫɭɦɦɚ ɤɜɚɞɪɚɬɨɜ ɨɬɤɥɨɧɟɧɢɣ ɤɨɬɨɪɨɝɨ ɨɬ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɯ ɡɧɚɱɟɧɢɣ ɮɭɧɤɰɢɢ yi  f ( xi ) ɦɢɧɢɦɚɥɶɧɚ Ɂɚ ɦɟɪɭ ɤɚɱɟɫɬɜɚ ɚɩɩɪɨɤɫɢɦɚɰɢɢ ɮɭɧɤɰɢɢ f (x) ɩɨɥɢɧɨɦɨɦ Pm (x) ɜ ɬɨɱɤɚɯ xi ɩɪɢɧɢɦɚɸɬɫɭɦɦɭ n S (a0 , a1 ,..., am )  ¤ W ( xi ) ; f ( xi ) Pm ( xi ) =, i 1 ɝɞɟ W ( x) r 0 – ɡɚɪɚɧɟɟɜɵɛɪɚɧɧɚɹ©ɜɟɫɨɜɚɹªɮɭɧɤɰɢɹ Ⱦɥɹ ɨɬɵɫɤɚɧɢɹ ɤɨɷɮɮɢɰɢɟɧɬɨɜ a0 , a1 ,..., am ɩɨɥɢɧɨɦɚ ɢɫɩɨɥɶɡɭɟɬɫɹ ɧɟɨɛɯɨɞɢɦɨɟ ɭɫɥɨɜɢɟ ɦɢɧɢɦɭɦɚ ɮɭɧɤɰɢɢ ɦɧɨɝɢɯ uS  0  ɤɨɬɨɪɨɟ ɩɪɢ W ( x)  1 ɢ ɩɨɫɥɟ ɧɟɤɨɬɨɪɵɯ ɩɟɪɟɦɟɧɧɵɯ ua j ɩɪɟɨɛɪɚɡɨɜɚɧɢɣɫɜɨɞɢɬɫɹɤɧɨɪɦɚɥɶɧɨɣɫɢɫɬɟɦɟɭɪɚɜɧɟɧɢɣ n a0 ¤ xi0 i 1 n a1 ¤ x1i i 1 n ! am ¤ xim i 1 n  ¤ yi xi0 , i 1 . . . . . . . . . . . . . . . . . n a0 ¤ xik i 1 n a1 ¤ xik 1 i 1 n ! am ¤ xik m i 1 n  ¤ yi xik , (1.5) i 1 . . . . . . . . . . . . . . . . . n a0 ¤ i 1 xim n a1 ¤ i 1 xim 1 n ! am ¤ i 1 xi2m n  ¤ yi xim . i 1 ɉɪɢ ɛɨɥɶɲɢɯ m ɫɢɫɬɟɦɚ   ɫɬɚɧɨɜɢɬɫɹ ©ɩɥɨɯɨɨɛɭɫɥɨɜɥɟɧɧɨɣª ɢ ɧɚ ɪɟɡɭɥɶɬɚɬ ɪɟɲɟɧɢɹ ɧɚɱɢɧɚɸɬ ɨɤɚɡɵɜɚɬɶ ɫɢɥɶɧɨɟ ɜɥɢɹɧɢɟ ɨɲɢɛɤɢ ɨɤɪɭɝɥɟɧɢɹ ɧɟɬɨɱɧɨɟ ɡɚɞɚɧɢɟ ɢɫɯɨɞɧɵɯ ɞɚɧɧɵɯ ɢ ɬɩ ɉɨɷɬɨɦɭ ɨɛɵɱɧɨ ɫɬɚɧɞɚɪɬɧɵɦ ɦɟɬɨɞɨɦ ɧɚɢɦɟɧɶɲɢɯ ɤɜɚɞɪɚɬɨɜ ɩɨɥɶɡɭɸɬɫɹ ɞɥɹ ɬɨɝɨ ɱɬɨɛɵ ɚɩɩɪɨɤɫɢɦɢɪɨɜɚɬɶ ɜ ɫɪɟɞɧɟɦ ɛɨɥɶɲɨɣ ɦɚɫɫɢɜ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɯ ɞɚɧɧɵɯ ɩɪɨɫɬɵɦ ɩɨɥɢɧɨɦɢɚɥɶɧɵɦɜɵɪɚɠɟɧɢɟɦ m^2w3). ɉɪɢɦɟɪ ɉɨɞɚɧɧɨɣɬɚɛɥɢɰɟɡɧɚɱɟɧɢɣɮɭɧɤɰɢɢy = f(x  ɬɚɛɥ  ɩɨɫɬɪɨɢɬɶɢɧɬɟɪɩɨɥɹɰɢɨɧɧɭɸɤɪɢɜɭɸɢɧɚɣɬɢɡɧɚɱɟɧɢɟɮɭɧɤɰɢɢ ɜɬɨɱɤɟx=1,5. Ɍɚɛɥɢɰɚ i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 xi -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10 yi 29 22 15 9 5 3 -2 -3 -2 -1 2 6 10 16 23 29 37 46 58 Ɋɟɲɟɧɢɟ. ɉɨɫɬɪɨɢɦ ɚɩɩɪɨɤɫɢɦɢɪɭɸɳɭɸ ɮɭɧɤɰɢɸ ɜ ɜɢɞɟ ɤɜɚɞɪɚɬɧɨɝɨɩɨɥɢɧɨɦɚ P2 ( x)  a0 a1 x a2 x 2 , ɢɫɩɨɥɶɡɭɹɦɟɬɨɞɧɚɢɦɟɧɶɲɢɯɤɜɚɞɪɚɬɨɜ n ɂɦɟɟɦ n  20, ¤ xi0  20, m  2, i 1 n ¤ xi  10, i 1 n n ¤ xi2  670, i 1 ¤ xi4  40666, n ¤ xi3  1000, i 1 n ¤ yi xi0  302, i 1 i 1 n n i 1 i 1 ¤ yi xi  1146, ¤ y i xi2  19742. Ɍɨɝɞɚ ɫɢɫɬɟɦɚ ɭɪɚɜɧɟɧɢɣ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ a j ( j  0, 1, 2) ɩɪɢɦɟɬɜɢɞ «20 a0 10 a1 670 a2  302 , ®® ¬10 a0 670 a1 1000 a2  1146 , ® ®­670 a0 1000 a1 40666 a2  19742 , ɨɬɤɭɞɚ a0  1,66; a1  0,99; a2  0,49. Ɂɧɚɱɟɧɢɟɮɭɧɤɰɢɢ y (1,5) z P2 (1,5)  0,915 . ɉɨɫɬɪɨɟɧɧɚɹ ɢɧɬɟɪɩɨɥɹɰɢɨɧɧɚɹ ɤɪɢɜɚɹ ɫɩɥɨɲɧɚɹ ɥɢɧɢɹ  ɢ ɬɚɛɥɢɱɧɵɟ ɡɧɚɱɟɧɢɹ ɮɭɧɤɰɢɢ ɬɨɱɤɢ  ɩɪɢɜɟɞɟɧɵ ɧɚ ɪɢɫ  ɉɪɨɝɪɚɦɦɚɪɟɚɥɢɡɭɸɳɚɹɞɚɧɧɵɣɦɟɬɨɞɨɩɢɫɚɧɚɜɩɪɢɥ Ɂɚɞɚɧɢɹɤɪɚɛɨɬɟ  Ɋɚɡɪɚɛɨɬɚɬɶ ɫɯɟɦɵ ɢɧɬɟɪɩɨɥɢɪɨɜɚɧɢɹ ɮɭɧɤɰɢɣ ɦɟɬɨɞɚɦɢ Ʌɚɝɪɚɧɠɚɇɶɸɬɨɧɚɧɚɢɦɟɧɶɲɢɯɤɜɚɞɪɚɬɨɜ  ɇɚɩɢɫɚɬɶ ɨɬɥɚɞɢɬɶ ɢ ɜɵɩɨɥɧɢɬɶ ɩɪɨɝɪɚɦɦɵ ɢɧɬɟɪɩɨɥɢɪɨɜɚɧɢɹ ɮɭɧɤɰɢɣ ɬɚɛɥ   ɂɧɬɟɪɩɨɥɢɪɨɜɚɧɢɟ ɩɪɨɜɟɫɬɢ ɥɸɛɵɦ ɢɡ ɜɵɲɟɧɚɡɜɚɧɧɵɯ ɦɟɬɨɞɨɜ ɉɨɫɬɪɨɢɬɶ ɢɧɬɟɪɩɨɥɹɰɢɨɧɧɭɸ ɤɪɢɜɭɸɢɧɚɣɬɢɡɧɚɱɟɧɢɟɮɭɧɤɰɢɢɜɭɤɚɡɚɧɧɨɣɬɨɱɤɟ ɜɫɨɨɬɜɟɬɫɬɜɢɢ ɫɜɚɪɢɚɧɬɨɦɡɚɞɚɧɢɹ  Ɍɚɛɥɢɰɚ xi Ɂɧɚɱɟɧɢɹ yi  f ( xi ) y 40 30 20 10 -5 5 Ɋɢɫ 1.2 x