3.4.1 3.4.2 3.4.3 3.4.4 问题 §3.4Cauchy 中值定理和未定式的极限 3.4.1 Cauchy值 定理 1(Cauchy 中值定理)设 f(x)和 g(x)在 [a,b]上连续，在 (a,b)内可 微.而且对任一点x∈(a,b),g'(x)≠0.则在(a,b)内，必存在一点ξ，使得 证明设辅助函数 F(x)=f(x)-f(a)-(g(x)-g(a)) 易知，F(x)在 [a,b]上连续，在(a,b)内可微，且 F(b)- F(a)=0.即，F(x) 满足 Rolle 定理的三个条件. ‖返回全屏关闭退出 1/30
3.4.1 3.4.2 3.4.3 3.4.4 ¯K §3.4 Cauchy ¥½nÚ½ª4 3.4.1 Cauchy ¥½n ½n 1 (Cauchy ¥½n) f(x) Ú g(x) 3 [a, b] þëY, 3 (a, b) S . é?: x ∈ (a, b), g0 (x) 6= 0. K3 (a, b) S, 73: ξ, ¦ f(b) − f(a) g(b) − g(a) = f 0 (ξ) g 0(ξ) . y² 9ϼê F(x) = f(x) − f(a) − f(b) − f(a) g(b) − g(a) g(x) − g(a) ´, F(x) 3 [a, b] þëY, 3 (a, b) S, F(b) − F(a) = 0. =, F(x) ÷v Rolle ½nn^. 1/30 kJ Ik J I £ ¶ '4 òÑ
3.4.1 3.4.2 3.4.3 3.4.4 ¯K â Rolle ½n, 3: ξ ∈ (a, b), ¦ F 0 (ξ) = 0, = f 0 (ξ) − f(b) − f(a) g(b) − g(a) g 0 (ξ) = 0. â^k g 0 (ξ) 6= 0, 9 g(a) 6= g(b). u´k f(b) − f(a) g(b) − g(a) = f 0 (ξ) g 0(ξ) . d=½n(Ø. y. 2/30 kJ Ik J I £ ¶ '4 òÑ
3.4.1 3.4.2 3.4.3 3.4.4 ¯K Cauchy ¥½nAÛ¿Â f(t) Ú g(t) 3 t ∈ [a, b] þëY, 3 (a, b) S. ëê§ x = g(t), y = f(t), (t ∈ [a, b]) ¤ ( ½ L, T ü à : ´ A = (g(a), f(a)) Ú B = (g(b), f(b)), ëùüà: Ç´ f(b)−f(a) g(b)−g(a) . â^3 L þØà: z:Ñ´k. Cauchy ¥½nAÛ¿ÂÒ´ þk: C = (g(ξ), f(ξ)) Ç f 0 (ξ) g 0(ξ) TÐ u f(b)−f(a) g(b)−g(a) . x y A B C O 3/30 kJ Ik J I £ ¶ '4 òÑ
3.4.1 3.4.2 3.4.3 3.4.4 ¯K ~ 1 0 < a < b, ¼ê f(x) 3 [a, b] ëY, 3 (a, b) . ¦y: 3 ξ ∈ (a, b) ¦ af(b) − bf(a) a − b = f(ξ) − ξf0 (ξ). y² ¤y²ªf±¤ f(b) b − f(a) a 1 b − 1 a = f(x) x 0 1 x 0 x=ξ . Ïdé¼ê f(x) x Ú 1 x A^ Cauchy ¥½n, =(Ø. 4/30 kJ Ik J I £ ¶ '4 òÑ
3.4.1 3.4.2 3.4.3 3.4.4 ¯K ~ 2 f(x), g(x) 3 [a, b] þëY, 3 (a, b) þ, Ù¥ g 0 (x) 3«m (a, b) ¥Ã":. ¦y: 3 ξ ∈ (a, b), ¦ f 0 (ξ) g 0(ξ) = f(ξ) − f(a) g(b) − g(ξ) . y² ¼ê F(x) = f(x) − f(a) g(b) − g(x) . â^, F(x) 3 [a, b] þëY, 3 (a, b) þ, w,k F(a) = F(b) = 0. d Rolle ½n, 3 ξ ∈ (a, b), ¦ F 0 (ξ) = 0, =, f 0 (ξ) g(b) − g(ξ) − g 0 (ξ) f(ξ) − f(a) = 0. du g 0 (x) 3«m (a, b) ¥Ã":, k g 0 (ξ) 6= 0, 9 g(b) − g(ξ) 6= 0. Ï f 0 (ξ) g 0(ξ) = f(ξ) − f(a) g(b) − g(ξ) . 5/30 kJ Ik J I £ ¶ '4 òÑ
3.4.1 3.4.2 3.4.3 3.4.4 ¯K ½n 2 (Cauchy ¥½ní2) f(x) Ú g(x) 3 [a, b] þk n ¼ ê, é?¿ x ∈ (a, b) k g (n) (x) 6= 0. K3 ξ ∈ (a, b) S, ¦ f(b) − Pn−1 k=0 f (k) (a) k! (b − a) k g(b) − Pn−1 k=0 g (k) (a) k! (b − a) k = f (n) (ξ) g (n)(ξ) . y² Eü¼ê F(x) = f(x) − X n−1 k=0 f (k) (a) k! (x − a) k G(x) = g(x) − X n−1 k=0 g (k) (a) k! (x − a) k . K F(x) Ú G(x) 3 [a, b] þk n ¼ê, F(a) = F 0 (a) = · · · = F (n−1)(a) = 0, G(a) = G0 (a) = · · · = G(n−1)(a) = 0. 6/30 kJ Ik J I £ ¶ '4 òÑ
3.4.1 3.4.2 3.4.3 3.4.4 ¯K du g (n) (x) 6= 0, EA^ Rolle ½n G(k) (x) (k = 0, 1, · · · , n) 3 (a, b) SÃ":. 2EA^ Cauchy ¥½n, k F(b) G(b) = F(b) − F(a) G(b) − G(a) = F 0 (ξ1) G0(ξ1) = F 0 (ξ1) − F 0 (a) G0(ξ1) − G0(a) = F 00(ξ2) G00(ξ2) = · · · = F (n−1)(ξn−1) − F (n−1)(a) G(n−1)(ξn−1) − G(n−1)(a) = F (n) (ξ) G(n)(ξ) = f (n) (ξ) g (n)(ξ) Ù¥ a < ξ < ξn−1 < · · · < ξ2 < ξ1 < b. 7/30 kJ Ik J I £ ¶ '4 òÑ
3.4.1 3.4.2 3.4.3 3.4.4 ¯K 3.4.2 0 0 .½ª ½n 3 ( 0 0 . L’Hospital {K) f(x) Ú g(x) 3 x0 NC, g 0 (x) 6= 0, ÷v lim x→x0 f(x) = lim x→x0 g(x) = 0. XJ lim x→x0 f 0 (x) g 0(x) = l, @ok lim x→x0 f(x) g(x) = l, ùp l ±´k¢ê, ±´ ∞. y² du lim x→x0 f(x) = lim x→x0 g(x) = 0, x → x0 , f(x) g(x) 4 ¼ê f Ú g 3 x0 Ã', Ïd·Øb f(x0) = g(x0) = 0, ù, ¼ê f Ú g 3 x0 ÑëY. x ´«m (x0 − δ, x0 + δ) ¥?¿:£x 6= x0¤, 3± x Ú x0 à:4«mþ, f Ú g ÷v Cauchy ¥½n^, u´30u x 8/30 kJ Ik J I £ ¶ '4 òÑ
3.4.1 3.4.2 3.4.3 3.4.4 ¯K Ú x0 m: ξ, ¦ f(x) g(x) = f(x) − f(x0) g(x) − g(x0) = f 0 (ξ) g 0(ξ) . Ï |ξ − x0| < |x − x0|, ¤± x → x0 , ξ → x0. d½nb, = lim x→x0 f(x) g(x) = lim ξ→x0 f 0 (ξ) g 0(ξ) = l. 51: ½n¥4L§Uüÿ4£=x → x0 ± 0¤, d(ØÓ ¤á, ,¡éu4L§ x → +∞, x → −∞ ½ x → ∞ 0 0 . ½ª, kaq L’Hospital {K. ± x → ∞ ~, XJ lim x→∞ f(x) = lim x→∞ g(x) = 0, lim x→∞ f 0 (x) g 0(x) = l, K lim x→∞ f(x) g(x) = l. 9/30 kJ Ik J I £ ¶ '4 òÑ
3.4.1 3.4.2 3.4.3 3.4.4 ¯K y²L§¥ y = 1 x , K x → ∞ , y → 0, lim x→∞ f(x) g(x) = lim y→0 f 1 y g 1 y = lim y→0 1 y2f 0 1 y 1 y2g 0 1 y = lim x→∞ f 0 (x) g 0(x) = l. 52: 3¦^ L’Hospital {K, XJ f 0 (x) g 0(x) ´ 0 0 .½ª, =Ø lim x→x0 f(x) = lim x→x0 g(x) = 0, lim x→x0 f 0 (x) = lim x→x0 g 0 (x) = 0, K±U YÄ ê, XJ lim ξ→x0 f 00(ξ) g 00(ξ) = l, K lim ξ→x0 f(ξ) g(ξ) = l. Ø+¦^Ag, cJ^ ´cö7L´ 0 0 .½ª, ´ö4½3, üö"Ø. 53: 3¦^ L’Hospital {Kc, Ðk^ dáO{ò© f½©1¤{ü¼ê. 10/30 kJ Ik J I £ ¶ '4 òÑ