Ma348lec8-1

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ax 2 + bx + c = 0 x 1 , 2 = b ± b 2 4 ac 2 a ax 3 + bx 2 + cx + d = 0 f ( x ) = 0 x x 3 + x 2 3 x 3 = 0 y = x 3 + x 2 3 x 3 x f y = 0 x y f ( x ) = sin 10 x + cos 3 x
f ( x ) = 0 a x b f ( a ) f ( b ) < 0 x r f ( x ) = 0 ϵ [ x l , x u ] f ( x l ) f ( x u ) < 0 x r = 1 2 ( x l + x u ) f ( x r ) = 0 x r f ( x l ) f ( x r ) < 0 x u = x r f ( x l ) f ( x r ) > 0 x l = x r | x u x l | < 2 ϵ L 0 = ∆ x 0 = x 0 u x 0 l 0 E 0 a = L 0 E 1 a = L 0 2 E 2 a = L 0 2 2 E n a = L 0 2 n x r n L 0 / 2 n n x r E a,d n = log 2 L 0 E a,d = ln( L 0 /E a,d ) ln 2 ϵ a ϵ a = x new r x old r x new r × 100%
x new r x old r x new r x old r = x u x l 2 x old r = x u + x l 2 ϵ a = x u x l x u + x l × 100% . f ( x ) = x 6 2 [0 , 2] x (0) l = 0 , x (0) u = 2; f ( x (0) l ) f ( x (0) u ) < 0; x (0) r = 1 2 ( x (0) l + x (0) u ) = 1 . f ( x (0) l ) f ( x (0) r ) > 0; x (1) l = 1 , x (1) u = 2; x (1) r = 1 2 ( x (1) l + x (1) u ) = 3 2 . f ( x (1) l ) f ( x (1) r ) < 0; x (2) l = 1 , x (2) u = 3 2 ; x (2) r = 1 2 ( x (2) l + x (2) u ) = 5 4 . E 3 a = x 0 2 3 = x (0) u x (0) l 2 3 = 2 0 2 3 = 0 . 25 E a,d = 10 4 L 0 = ∆ x 0 = 2 n = ln( L 0 /E a,d ) ln 2 = ln(2 × 10 4 ) ln 2 = 14 . 3 15
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