6. Euler x

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1 n n n n n Landau o Landau n n i

2 6. Euler x ii

3 iii

4 ... x f(x) =x () x =+(x ) () x f(x) =x = {+(x )} = + (x ) + (x ) (3) x x (x ) (x ) (x ) x (x ) + (x ) (4) f(x) =x + (x ) + (x ) (x ) (neglect ) (x ) (x ) (negligible ).. x x (x ) x { + (x )} =(x ) 0 (x ) 0 (x ) 0..3 f(x) =x y = x +(x ) y = +(x ) xy-

5 f(x) =x + (x ) y = + (x ) f(x) =x y = x x = x =. f(x).. x = x f(x) =x 4 +x 3 x 0 (5) x =+(x ) (6)

6 x 5 f(x) = x 4 +x 3 x 0 (7) = 8 + 5(x ) + 35(x ) + 0(x ) 3 +(x ) 4 (8) = 8 + 5(x ) + (9) x x 5(x ) (x ) 5(x ) (x ) 3 (x ) x f(x) (x ) (x ) 0 0(x ) 3 0 (x ) f(x) =x 4 +x 3 x 0 y = x 4 +x 3 x (x ) y = 8 + 5(x ) xy- 3

7 40 y x x = a.. a =.. a = x a x a x a x a x! a x lim x!a =0 (0) x a a.3. x x + (x ) x = + (x ) + lim x! x (x ) =lim x! x =lim x! (x ) =.3. x x 4 +x 3 x (x ) x 4 +x 3 x 0 = 8 + 5(x ) + 4

8 lim x! x 35(x ) + 0(x ) 3 +(x ) 4 = lim x! x n = lim 35(x ) + 0(x ) +(x ) 3o = x!.4 f(x) x = a f(x).4. f(x) = x () x x =+(x ) f(x) = +(x ) () = f(x) = + = A + B(x ) + (3) x f(x) = x A + B(x ) lim x! =0 (4) x A, B (4) lim x! = lim x! x (x ) = lim x! x lim x! (x ) = x 0 x 0 x! (3) lim x! lim x! = lim x! x = = lim x! A + B(x ) + = A + B 0+ lim x! = A A = (3) x =+B(x ) + 5

9 B B = x x = x x x x! B =lim x! x = lim x x! x +lim = +0= x! x B = = (x ) + "(x) x "(x) lim x! "(x) x =lim x! x +(x ) =lim x x! = x x x! (x ) "(x).4. f(x) x ( ). f(x) A + B(x a) x! a f(x) f(x) =A + B(x a)+ x! a x a lim x!a =0 x a f(x) x = a A, B f(x) =A + B(x a)+ (5) lim x!a lim x!a = lim x!a x a =0 (6) x a (x a) = lim x!a x a lim x!a (x a) =0 6

10 lim f(x) = lim x!a x!a A + B(x a)+ = A + B 0+ lim x!a = A f(x) x = a lim x!a f(x) =f(a) A = f(a) (5) f(x) =f(a)+b(x a)+ B B = f(x) x x! a f(a) a x a f(x) B =lim x!a x f(a) a lim x!a f(x) =lim x a x!a x f(a) a f(x) f(a) ( ). lim f(x) x = a x!a x a f 0 (a) f(x) x = a f 0 f(x) (a) = lim x!a x f(a) a f(x) f(a). x a x f(x) x a x a x = a f(x). x = a + h x! a, h! 0 f 0 (a) = lim h!0 f(a + h) f(a) h A + B(x a) f(x) x! a f(x) x = a A = f(a), B= f 0 (a) A = f(a), B= f 0 (a) A + B(x a) f(x) x! a f(x) =f(a)+f 0 (a)(x a)+r(x) ) lim x!a R(x) x a =0 f(x) =f(a)+f 0 (a)(x a)+r(x) ) R(x) x a = f(x) f(a) x a f 0 (a) 7

11 R(x) lim x!a x a =lim f(x) x!a x f(a) a f 0 (a) =f 0 (a) f 0 (a) =0. f(x) A + B(x a) f(x) x! a, f(x) x = a ( ( A = f(a) B = f 0 (a) f(x) =f(a)+f 0 (a)(x a)+r(x) lim x!a R(x) x a =0 3. n ) n n, n a 8

12 .5 f(x) x = a f(a)+f 0 (a)(x a) x! a f(x) f(x) =f(a)+f 0 (a)(x a)+r(x) lim x!a R(x) x a =0 f(x) ; f(a)+f 0 (a)(x a) 9

13 . x f(x) =x 3 3x (7) x =+(x ) (8) x 7 f(x) =x 3 3x = + 9(x ) + 6(x ) +(x ) 3 (9) x x 0 9(x ) (x ) (x ) 3 9(x ) 6(x ) +(x ) 3 9(x ) + 9(x ) x 3 3x x x 3 3x ; + 9(x ) (0) (x ) (x ) 3 (x ) (x ) 3 x 3 3x ; + 9(x ) + 6(x ) () (0).. x f(x) =x 3 3x (x ) + 9(x ) + 6(x ) 0

14 y = + 9(x ) y = x 3 3x x = x = x = y = + 9(x ) + 6(x ) x = x = y = x 3 3x x =. x a ( ) x a ( ) lim =0 x!a x a ( ) lim x!a x a =0 3 ( ). f(x) A + B(x a)+c(x a) x! a

15 f(x) f(x) =A + B(x a)+c(x a) + () x! a (x a) lim x!a (x a) =0 3 x 3 3x = + 9(x ) + 6(x ) +( ) lim x! (x ) =lim x 3 3x + 9(x ) + 6(x ) (x ) 3 x! (x ) =lim x! (x ) =lim (x ) = 0 x!.3.3. f(x) x 3 m 3x f(x) mx f(x) = a m k x m k = a m x m + a m x m + + a x + a 0 (3) k=0 x =+(x ) x =+(x ) x = + (x ) + (x ) x 3 = + 3(x ) + 3(x ) +(x ) 3 x m =+m(x ) + (3) m(m ) (x ) + +(x ) m f(x) =A 0 + A (x ) + A (x ) + A 3 (x ) A m (x ) m

16 A 3 (x ) A m (x ) m lim x! (x ) =lim A 3 (x ) 3 + A 4 (x ) A m (x ) m x! (x ) =lim x! A 3 (x ) + A 4 (x ) + + A m (x ) m =0 A 0 + A (x ) + A (x ) f(x) x! a x! a.3. f(x) f(x) = x x x =+(x ) f(x) = +(x ).3.3 f(x) x = a +(x a) f(x) =A 0 + A (x a)+a (x a) + A 3 (x a) 3 + A m (x a) m (4) A 3 (x a) A m (x a) m f(x) =A 0 + A (x a)+a (x a) +( ) A (x a) + + A m (x a) m f(x) =A 0 + A (x a)+( ) A 0 = f(a) A = f 0 (a) (4) x = a (4) x f(a) =A 0 f 0 (x) =A +A (x a)+3a 3 (x a) + ma m (x a) m (5) 3

17 x = a f 0 (a) =A (5) f 00 (x) = A +3 A 3 (x a) + m(m )A m (x a) m (6) x = a f 00 (a) =A A 0 = f(a) (7) A = f 0 (a) (8) A = f 00 (a) (7)(8) f(x) (9) (9).3.4 f(x) ( ). x = a f 000 (x) f(x) =f(a)+f 0 (a)(x a)+ f 00 (a) (x a) R(x) + R(x) lim x!a (x a) =0 R(x) ( ). x = a f 000 (x) M R(x) apple M x a 3 3! 3 ( ). Z R(x) =(x a) 3 f 000 ( t) (a + t(x a)) dt norip/taylor approx.pdf 4

18 5. p.0. f 00 (a) 3 pp.00-0 pp f(x) =,R(x) =f(x) x f(a)+f 0 (a)(x a)+ f 00 (a) (x a) R(x) a = R(x) lim x! (x ) =0 Z R(x) =(x a) 3 f 000 ( t) (a + t(x a)) dt 0 a + t(x t y a) =y a, x 5

19 .4 f(x) f(x) =f(a)+f 0 (a)(x a)+ f 00 (a) (x a) R(x) + R(x) lim x!a (x a) =0 6

20 3 n n 3. (6) x f 000 (x) =A 3 3 +A (x a)+ + A m m(m )(m )(x a) m 3 x = a f 000 (a) = 3! A 3 f(a) =A 0,f 0 (a) =A,f 00 (a) =! A,f 000 (a) = 3! A 3,...,f (m) (a) =m! A m A 0,A,...,A m f(x) x A 0 = f(a), A = f 0 (a), A = f 00 (a)!,a 3 = f 000 (a),...,a m = f (m) (a) 3! m! a f(x) x = a (4) f(x) = mx k=0 f (k) (a) k! (x a) k = f(a)+f 0 (a)(x a)+ f 00 (a) (x a) + + f (m) (a) (x a) m m! m f(x) n<m n f(x) =f(a)+f 0 (a)(x a)+ + f (n) (a) n! (x a) n + + f (m) (a) (x a) m m! lim x!a R(x) = f (n+) (a) (n + )! (x a)n+ + + f (m) (a) (x a) m m! f(x) = f(a)+f 0 (a)(x R(x) (x a) n =lim x!a =lim x!a a)+ + f (n) (a) (x a) n + R(x) n! f (n+) (a) (n+)! (x a) n+ + + f (m) (a) m! (x a) m f (n+) (a) (x a) n (n + )! (x a)+ + f (m) (a) (x a) m n =0 m! 7

21 f(x) =f(a)+f 0 (a)(x a)+ + f (n) (a) (x a) n R(x) + R(x) lim n! x!a (x a) n =0 m n f(x) 3. n f(x) ( ). x = a f (n+) (x) f(x) =f(a)+f 0 (a)(x a)+ + f (n) (a) (x a) n R(x) + R(x) lim n! x!a (x a) n =0 x = a n R(x) 4 ( ). x = a f (n+) (x) M R(x) apple 4 5 ( ). 0 M (n + )! x a n+ (30) Z R(x) =(x a) n+ f (n+) ( t)n (a + t(x a)) dt n! norip/taylor approx.pdf 7. p.0. f (n) (a) 5 pp.00-0 pp

22 3.3 n x =+x + x + + x n + x = (x ) + (x ) +( ) n (x ) n + ( + x) m m(m ) =+mx + x m(m ) (m n + ) + + x n + + x m ( + x) =+ x + ( ) x + + e x =+x + x + + n! xn + cos x = sin x = x m n! ( ) ( n + ) x n + n!! x + 4! x4 +( ) n (n)! xn + 3! x3 + 5! x5 +( ) n (n + )! xn f(x) =a 0 + a (x a)+ + a n (x a) n S(x) + S(x) lim x!a (x a) n =0 ) a 0 = f(a),a = f 0 (a),,a n = f (n) (a) n! f(x) =f(a)+f 0 (a)(x a)+ + f (n) (a) (x a) n R(x) + R(x) lim n! x!a (x a) n =0 0=(a 0 f(a)) +(a f 0 (a)) (x a)+ + + a n f (n) (a) n! S(x) R(x) (x a) n (x a) n (x a) n (x a) n (3) x! a 0=a 0 f(a) (3) a 0 f(a) =0 n 0=(a f 0 (a)) (x a)+ a f 00 (a) (x a) a n f (n) (a) n! S(x) R(x) (x a) n (x a) n (x a) n (x a) n 9

23 x a 0=(a f 0 (a)) + a f 00 (a) (x a)+ + + a n f (n) (a) n! S(x) R(x) (x a) n (x a) n (x a) n (x a) n x! a 0=a f 0 (a) 3.5 0

24 3.6 3 f(x) m f(x) =f(a)+f 0 (a)(x a)+ f 00 (a) (x a) + + f (m) (a) (x a) m m! f(x) n f(x) =f(a)+f 0 (a)(x a)+ + f (n) (a) (x a) n R(x) + R(x) lim n! x!a (x a) n =0 x = a f (n+) (x) M R(x) apple M x a n+ (n + )!

25 4 4. x! f(x) f(x) =f() + f 0 ()(x ) + f(x) = p x = x f() =, f 0 (x) = x,f 0 () = (30) p x =+ (x ) + apple M! x (3) M p x 4 x 3 4 x 3 x x = p =+ ( ) + =.5+ 4 x 3 M x 3 4 apple 4! = 8 =0.5 p.5 p 0.5 x! f(x) f(x) =f() + f 0 ()(x ) + f 00 () (x ) + f 00 (x) = 4 x 3,f 00 () = 4 (30) p x =+ (x ) 8 (x ) + apple M 3! x 3 (33) M p x 3 8 x x 5 x x = p =+ ( ) 8 ( ) + =.375 +

26 3 8 x 5 M x apple 3 8 3! 3 = 6 =0.065 p.375 p p x = M y 3 x 3. p x x = p 3

27 d (t) dt = g sin (t) (34) l (t) t 0 g l (t) =0 f( ) f( ) =f(0) + f 0 (0) +( ) f( ) =sin f 0 ( ) = cos, f(0) = 0, f 0 (0) = 0 sin = +( ) (34) sin (t) (t) d (t) dt = g (t) (35) l (34) (t) =C 0 cos r g r g l t + C sin l t 4. (35) 4

28 4.. Einstein m E = mc c m v m 0 m = q (36) v c v =0 m = m 0 m 0 E = mc E = m 0c q (37) v c f(x) = p x = ( x) f 0 (x) = ( x) 3,f 00 (x) = ( x) 5,f(0) =, f 0 (0) =,f00 (0) = p x =+ x + x +( ) v 0 x c v c v q ; + v c + v 4 c 4 c (37) E = m 0c q ; m 0 c + v m 0v v 4 + m 0 c c m 0 c m 0v v 4 m 0 Newton Einstein c m 0 v 4 c 5. v 000km Newton 5

29 4.3 f(x) =x 3 3x y = f(x) x = + 0(x ) y = + 0(x ) + 0(x ) + 3(x ) y = + 0(x ) + 3(x ) xy f(x) ; f(a)+f 0 (a)(x a)+ f 00 (a) (x a) f 0 (a) =0 f(x) ; f(a)+ f 00 (a) (x a) y = f(a) + f 00 (a) (x a) f 00 (a) > 0 x = a f 00 (a) < 0 x = a f 0 (a) =0,f 00 (a) > 0 ) x = a f(x) f 0 (a) =0,f 00 (a) < 0 ) x = a f(x) 6

30 lim x!a (x a) =0 7

31 6. f(x) = cos x x =0 y = cos x xy 4.4 n n n! n n 6 8

32

33 5 Landau o x = + (x R (x) lim x! x =0 (x ) + R (x){+(x )} lim =lim x! x x! ) + R (x) x 3 = x x = { + (x ) + R (x)} {+(x )} = + 3(x ) + (x ) + R (x){+(x )} apple (x ) + R (x) {+(x )} = 0 + 0( + 0) = 0 x (x ) + R (x){+(x )} x 3 x = x 3 = + 3(x ) + ) n =,, 3, x n =+n(x ) + ) Landau o O f(x) lim x!a g(x) =0 x! a f(x) g(x) f(x) =o (g(x)) (x! a) f(x) = f(x) g(x) g(x) 0 f(x) o g(x) x! 30

34 (x ) = o(x ) ( ) o(x ) = o(x ) x = + (x ) + o(x ) x 3 = x x = { + (x ) + o(x )} {+(x )} = + 3(x ) + (x ) + o(x ){+(x )} = + 3(x ) + o(x ) 7. x = a +(x a) x = a +a(x a)+o(x a) x 3 = a 3 +3a (x a)+o(x a) 5. Landau x! a f(x) = f(a)+f 0 (a)(x a)+o(x a) g(x) = g(a)+g 0 (a)(x a)+o(x a) f(x)g(x) = f(a)g(a)+{f(a)g 0 (a)+f 0 (a)g(a)} (x a)+f 0 (a)g 0 (a)(x a) +f 0 (a)(x a)o(x a)+o(x a)g 0 (a)(x a)+o(x a)o(x a) = f(a)g(a)+{f(a)g 0 (a)+f 0 (a)g(a)} (x a)+o(x a) f(x)g(x) =h(x) f(x)g(x) =h(x) = h(a)+h 0 (a)(x a)+o(x a) h 0 (a) =f(a)g 0 (a)+f 0 (a)g(a) {f(x)g(x)} 0 = f(x)g 0 (x)+f 0 (x)g(x) x! 0 e x = +x + o(x) sin x = x + o(x) e x sin x = ( + x + o(x))(x + o(x)) = x + x + o(x)x +(+x + o(x))o(x) = x + o(x) 3

35 x! a e x = e a + e a (x a)+o(x a) sin x = sin a + (cos a)(x a)+o(x) e x sin x = (e a + e a (x a)+o(x a))(sin a + (cos a)(x a)+o(x)) = e a sin a +(e a sin a + e a cos a) (x a)+o(x a) (e x sin x) 0 = e x sin x + e x cos x 8. e x cos x x = a 3

36 5.3 5 f(x) lim x!a g(x) =0 x! a f(x) g(x) f(x) =o (g(x)) (x! a) o Landau o g(x) o Landau o 9. Landau o pp

37 6 6. n n f(x) n f(x) =f(a)+f 0 (a)(x a)+ f 00 (a) (x a) + + f (n) (a) (x a) n + R n+ (x) n! x a e x x =0 n y = e x y = y =+x y =+x + x y =+x + x + x3 3! y =+x + x + x3 3! + x4 4! y =+x + x + x3 3! + x4 4! + x5 5! y =+x + x + x3 3! + x4 4! + x5 5! + x6 6! x =0 n x sin x 34

38 y =sinx y = x y = x x 3 3! y = x x 3 3! + x5 5! y = x x 3 3! + x5 5! y = x x 3 3! + x5 5! y = x x 3 3! + x5 5! y = x x 3 3! + x5 5! x 7 7! x 7 7! + x9 9! x 7 7! + x9 9! x 7 7! + x9 9! x! x! + x3 3! n x n (n ) e x = X n=0 x n n! x =+x + + x3 xn + + 3! n! + 35

39 f(x) =f(a)+f 0 (a)(x n apple M = a)+ f 00 (a) M (n + )! x a n+, f (n+) (a + t(x a)) 0 apple t apple f(x) =e x,a=0 f (n) (x) =e x (x a) + + f (n) (a) (x a) n + n, n! e x =+x + x xn n, n! n apple M (n + )! x n+, M = e tx 0 apple t apple apple e x x N N x n apple M (n + )! x n+ apple e x n+ x x N x apple e N! N + e x = X n=0 (n + )! x n+ x x N = e N! N x x N! e N! lim n = 0 n! x n n! x N + 0=0(n!) x =+x + + x3 xn + + 3! n! + x N + x n + 9. sin x x =0 (n + ) x n! 0 6. Euler cos x = sin x = X ( ) n xn (n)! = x! + x4 4! X ( ) n xn+ (n + )! = x x 3 n=0 n=0 3! + x5 5! 36 x 6 6! xn + +( )n + (38) (n)! x 7 7! xn+ + +( )n + (39) (n + )!

40 e ix = X (ix) n n=0 =+ix = n! =+ix + (ix) x x + x4 4! i x3 3! + x4 x 6 6! + + (ix)3 3! 4! + ix5 5! + i x + (ix)4 4! x 6 6! + x 3 3! + x5 5! + (ix)5 5! + (ix)6 6! + = cos x + i sin x (40) e ix z e z X e z z n z = =+z + n! + z3 zn + + 3! n! + n=0 z z z e ix = cos x + i sin x Euler (4) 6.. Euler z w X e z+w (z + w) n = = n! n=0 X nx = n = n=0 k=0 X n=0 P n k=0 (n k)! k! zn k w k = X (n,k){(n,k)n N 0applekapplen<} e z+w = e z e w (4) n! (n k)! k! zn k w k n! X nx n=0 k=0 z n k (n k)! w k k! z n k = X nx n=0 k=0 w k k! (n k)! X X z n k = (n k)! k=0 n=k n! (n k)! k! z n n! w k k! = k w k k = l n = k, k +,k+, l =0,,, e z+w = X k=0 w k k! X z n k (n k)! = X n=k k=0 w k k! X l=0 X k=0 z l l! = ew e z = e z e w w k k! X n=k z n k (n k)! (4) cos( + )+i sin( + ) = e i( + ) = e i e i =(cos + i sin ) (cos + i sin ) = (cos cos sin sin )+i(cos sin +sin cos ) 37

41 , cosine sine 6.. e ix = cos x i sin x (43) i i x x cosine sine (4) (43) i cos x = eix + e ix sin x = eix e ix i 0. Z e x sin xdx 0 Z cos x sin xdx 0 Z e x x sin xdx 0 38

42 6.3 x f(x) = x f 0 (x) = ( x),f00 (x) = ( x) 3,f000 (x) = 3 ( x) 4,,f(n) (x) = f(0) =, f 0 (0) =, f 00 (0) =, f 000 (0) = 3,,f (n) (0) = n!, x =0 n x =+x + x + x x n + n n!, ( x) n+ ( x)( + x + x + + x n ) =(+x + x + + x n ) x ( + x + x + + x n ) =(+x + x + + x n ) (x + x + + x n + x n+ ) = x n+ +x + x + + x n = xn+ x x n + x x =0 n R n+(x) R n+ (x) = x +x + x + x x n = x x n+ x = xn+ x x n! x > x < 0 x =+x + x + x 3 + ( x < ) e x cos x, sin x x lim R n+(x) =0 n! x < lim x R n+(x) =0 n! 39

43 y x - y = x y = y =+x - y =+x + x y =+x + x + x 3 y =+x + x + x 3 + x 4 y =+x + x + x 3 + x 4 + x 5 y =+x + x + x 3 + x 4 + x 5 + x x = a f(x) =f(a)+f 0 (a)(x a)+ f 00 (a) a x (x a) + + f (n) (a) (x a) n + R n+ (x) n! a <r lim R n+(x) =0 n! 40

44 a f(x) = X n=0 f (n) (a) (x a) n n! = f(a)+f 0 (a)(x a)+ f 00 (a) (x a) + + f (n) (a) (x a) n + n! f(x) x = a. e x a e x ( 0 (x apple 0) f(x) = e x (x >0) f(h) lim h!+0 f(h) lim h! 0 f(0) h f(0) h = lim h!+0 = lim h! 0 h e h 0 0 h = lim h!+0 =0 h e h t = lim t! e t =0 f 0 (0) = 0 f(x) f 0 (x) = ( 0 (x apple 0) x e x (x >0) f 00 (x), f 000 (x), f(x) x =0 f (n) (0) = 0 x =0 X n=0 f (n) (0) n! x n = f(0) + f 0 (0)x + f 00 (0) x + + f (n) (0) x n + n! =0+0 x + 0 x n! xn + =0 x =0 x <r r x =0 f(x) =e x 0 4

45 6.5 6 x = a n n x a x n! e x = cos x = sin x = X n=0 x n n! x =+x + + x3 xn + + 3! n! + X ( ) n xn (n)! = x! + x4 4! X ( ) n xn+ (n + )! = x x 3 n=0 n=0 3! + x5 5! x 6 6! + +( )n xn (n)! + x 7 7! x = X x n =+x + x + x 3 + ( x < ) n=0 + +( )n xn+ (n + )! + z e z = X n=0 z n n! z =+z + + z3 zn + + 3! n! + e z+w = e z e w Euler Euler e i = cos + i sin cos x = eix + e ix sin x = eix e ix i 4

46 7 7. x f(x) x, y f(x, y) xy y = f(x) xyz z = f(x, y) f(x) =ax + b f(x, y) =ax + by + c xy y = ax + b xyz z = ax + by + c f(x, y) =x + y f(x, y) =x + y (44) f(x) =x f( ) x x f( ) = ( ) = f(x, y) =x + y f(, 3) x + y x y 3 f(, 3) = ( ) +3 = f(x) =x xy y = x f(x, y) =x + y xyz z = x + y xyz z = x + y f(x, y) =x + y z = x + y xyz xyz y =0 xz z = x + y y =0 z = x + y xz z = x + y y =0 () z = x y =0 43

47 xz z = x z = x + y yz yz z = y 4 z 4 z 3 3 x y xyz xy z = k k z = x + y z = k x + y = k z = k 8 >< k<0 k =0 (0, 0) >: k>0 (0, 0) p k x + y = k 44

48 y k k k x k k z = k k xy y x z = x + y 45

49 f(x, y) = 3 x +y (45) z = 3 x +y z = x + y xz y =0 z = 3 x +y y =0 xz z = 3 x 46

50 z z 3 x x yz x =0 z = 3 x +y x =0 yz z =y z z y y 47

51 xy z = k z = k 3 x +y = k y x k xy 3 y 3 3 x 3 48

52 z = 3 x +y. z = 3 x +y 0 apple x apple, 0 apple y apple (,, ) (00, 00, 00) 49

53 5 3 z 0 50

54 7.3. a, b, c x, y f(x, y) =ax + by + c a = 3,b=,c=0 a, b, c f(x, y) =ax + by + c z = ax + by + c xz y =0 z = ax + by + c y =0 z = ax + c xz a z z ax c x z = ax + by + c yz x =0 z = ax + by + c x =0 z = by + c yz b z z by c y z = ax + by + c z = k xy xy ax + by + c k =0 k xy k z = ax + by + c 5

55 y k 3 k k k 0 k x f(x) =ax + b y = ax + b y (0,b) a f(x, y) =ax + by + c z = ax + by + c z (0, 0,c) x a y b (, ) a y = a(x )+ (,, ) x a z = a(x )+b(y )+ y b 3. () 0 () z = A + Bx + Cy (cos, sin ) (cos, sin ) 5

56 7.4 7 f(x, y) xyz z = f(x, y) xy f(x, y) =k f(x, y) k f(x, y) z = f(x, y) f(x, y) =ax + by + c xyz z = ax + by + c a x b y (,, ) x a z = a(x )+b(y )+ y b 53

57 8 8. x =+(x ), y=+(y ) 44 f(x, y) = x + y = {+(x )} + {+(y )} = + (x ) + (x ) + + (y ) + (y ) = + (x ) + (y ) + (x ) +(y ) (x, y) (, ) (x, y) (, ) p (x ) +(y ) x = p (x ) apple p (x ) +(y ) y = p (y ) apple p (x ) +(y ) x y (x ) (y ) (x ) (y ) (x ) (y ) (x, y) (, ) (x ) +(y ) + (x ) + (y ) f(x, y) =x + y (x, y) (, ) x + y ; + (x ) + (y ) 0. ; nearly equal 7 x x ; + (x ) y = x x = y = + (x (x, y) (, ) ) x + y ; + (x ) + (y ) 54

58 8.. z = x + y z = + (x ) + (y ) z = k k k = x + y k = + (x ) + (y ) ( x + y k 0 p! k = k k<0 k x + y p =0 +! k p k y y x x 55

59 y x ( ) = +! k p ( p k) = + k (k ) = 8 k = (k + ) 8 0( k = ) k = k +4k k ( ) ( k = ) f(x, y) =x + y z = x + y x + y (x, y) = (, ) + (x ) + (y ) z = + (x ) + (y ) (x, y) =(, ). 56

60 8.. pp pp (x, y) (a, b) f(x, y) lim f(x, y) = (x,y)!(a,b) (x, y)! (a, b) f(x, y) xy (x 0,y 0 ), (x,y ), (x,y ), x x 0,x,x, y y 0,y,y, 4 ( ). (x n,y n ) (a, b) lim (x n,y n )=(a, b), (x n,y n ) (a, b) = p (x n a) +(y n b)! 0(n!) n! 4 5 ( ). lim f(x, y) =, (a, b) (x n,y n ) lim f(x n,y n )= (x,y)!(a,b) n! 5 (x, y) A 6. lim (x,y)!(a,b), (x,y)a 6 f(x, y) =, (a, b) A (x n,y n ) lim n! f(x n,y n )= 57

61 8..3 f(x, y) (a, b) lim f(x, y) =f(a, b) (x,y)!(a,b) x x, y e x x, y f(x) A + B(x x! a f(x) ; A + B(x a) a) f(x) x! a f(x) =A + B(x 7 ( ). f(x, y) a)+r(x) lim x!a R(x) x a =0 (x, y)! (a, b) f(x, y) ; A + B(x a)+c(y b) A + B(x a)+c(y b) f(x, y) (x, y)! (a, b) R(x, y) f(x, y) =A + B(x a)+c(y b)+r(x, y) lim (x,y)!(a,b) (x, y) (a, b) =0 7 (x, y)! (a, b) f(x, y) A + B(x a)+c(y b) 58

62 R(x, y) (x, y) (a, b) A + B(x a)+c(y b) f(x, y) (x, y)! (a, b). f(x, y) =x + y (x, y)! (, ) x + y ; + (x ) + (y ) x + y = + (x ) + (y ) + R(x, y) R(x, y) (x, y) (, ) = x + y ( + (x ) + (y )) p (x ) +(y ) = (x ) +(y ) p (x ) +(y ) = p (x ) +(y )! 0((x, y)! (, )) f(x) x! a f(x) ; A + B(x a) ) A = f(a),b = f 0 (a).4. f(x) ; f(a)+f 0 (a)(x a) (x! a) (x, y)! (a, b) f(x, y) ; A + B(x a)+c(y b) ) A =?,B =?,C =? (x, y)! (a, b) f(x, y) ; A + B(x a)+c(y b) y b x! a f(x, b) ; A + B(x a) f(x, b) =g(x) A = g(a) =f(a, b),b = g 0 (a) 3. (x, y)! (a, b) f(x, y) ; A + B(x a)+c(y b) R(x, y) f(x, y) =A + B(x a)+c(y b)+r(x, y) lim (x,y)!(a,b) (x, y) (a, b) =0 R(x, y) lim (x,y)!(a,b) (x, y) (a, b) =0 (x, y)! (a, b) R(x, y) (x, y) (a, b)! 0 y b x! a R(x, y) (x, y) (a, b)! 0 f(x, b) =A + B(x a)+c(b b)+r(x, b) lim x!a R(x, b) (x, b) (a, b) =0 59

63 f(x, b) =A + B(x a)+r(x, b) lim x!a R(x, b) x a =0 f(x, b) =g(x) g(x) g(x) =A + B(x a)+r(x, b) lim x!a R(x, b) x a =0 A = g(a) =f(a, b),b = g 0 (a) y = b x = a f(a, y) = h(y) A = h(b) =f(a, b),c = h 0 (b) 8.4 y f(x, y) (x, y) f(x, y) f(x + h, y) f(x, y) (x, y) := h!0 h f(x, y) (x, y). f(x, y) =x + x y z = (x, y) z = @(f(x, (f(x, y)) = x + x y 3 + y 4 y 3 ) ) ( ) + ( ) =x + y 3 x + y 4 0=x +xy (x + x y 3 + y 4 ) (x + x y 3 + y 4 60

64 5. f(x, () a, b f(x, b) =g(x),f(a, y) =h(y) g 0 (x, b),h0 (y) (a, (3) z = g(x) z = x + x y 3 + y 4 y = b a =,b = z = g(x) x = a z = g(x) xz 6

65 8.5 f(x, b) =g(x) g 0 (x, b) y b y x y b B = g 0 (a, b) C = h0 (a) (a, f(x, y) (x, y)! (a, b) f(x, y) ; A + B(x a)+c(y b) ) A = f(a, (a, b),c = 6

66 8.6 f(x) x! a f(a)+f 0 (a)(x a) y = f(a)+f 0 (a)(x a) y = f(x) x = a f(x, y) (x, y)! (a, b) a)+ (a, b) z = a)+ (a, b)(y z = f(x, y) (x, y) =(a, b) x f(x) x, y f(x, y) xy y = f(x) xyz z = f(x, y) f(x) ; f(a)+f 0 (a)(x a) (x! a) y = f(x) x = a y = f(a)+f 0 (a)(x a) f(x, y) ; f(a, b) (a, b)(x @x (a, b)(y b) ((x, y)! (a, b)) z = f(x, y) (x, y) =(a, b) z = f(a, (a, b)(y 6. () f(x, y) =x + x y 3 + y 4 (x, y)! (, ) x + x y 3 + y 4 ; () z = x + x y 3 + y 4 (x, y) =(, ) =0 63

67 8.7 8 f(x, y) (a, b), lim f(x, y) =f(a, b) (x,y)!(a,b) A + B(x a) +C(y b) (x, y)! (a, b) f(x, y) R(x, y) f(x, y) =A + B(x a)+c(y b)+r(x, y) lim =0 (x,y)!(a,b) (x, y) (a, b) z = A + B(x a)+c(y b) (x, y) =(a, b) z = f(x, y), A + B(x a)+c(y b) (x, y)! (a, b) f(x, y) y) = y f(x, y) x (x, y) = x f(x, y) z = A + B(x a)+c(y b) (x, y) =(a, b) z = f(x, y) ) A = f(a, @x(a, b),c b) A + B(x a)+c(y b) (x, y)! (a, b) f(x, y) ) A = f(a, @x(a, b),c (a, b) (a, b) x (a, (a, z = f(x, y) (x, y) =(a, b) z = f(a, (a, (a, b)(y 64

68 ( ). f(x, y) (x, y)! (a, b) f(x, y) ; A + B(x a)+c(y b) A, B, C f(x, y) (x, y) =(a, b) f(x, y) (x, y) =(a, b) A, B, C R(x, y) :=f(x, y) {A + B(x a)+c(y b)} lim (x,y)!(a,b) R(x, y) (x, y) (a, b) = p.0 z = f(x, y) x y xy >< f(x, y) = x + y (x, y) 6= (0, 0) >: 0 (x, y) =(0, = y x + y 4x y (x + = x 4xy x + y (x + y f(h, 0) f(0, 0) 0 0 (0, 0) = lim = lim = lim (0, 0) = h!0 h h!0 h 65

69 f(x, y) f(x, y) f(x, y) =f(0, 0) = 0 t 6= 0 f(t, t) = lim (x,y)!(0,0) lim f(x, y) =0 (x,y)!(0,0) lim f(t, t) = t!0,t6= >< y 3 f(x, y) = x + y (x, y) 6= (0, 0) >: 0 (x, y) =(0, 0) xy x y 66

70 xy V @f V ) f 3 0. p @f pp y. p.66 9 ( ). V f(x, y) V continuously di erentiable C x x, y =0 x, y ex x, y f(x, y) =e x +y 67

71 9.4 9 f xy V ) V f V 6( f(x, y) 68

72 0 0.. f(x) =x 3 3x 3 apple x apple 3 f(x) 3 <x< 3 f 0 ( ) =0 6 f 0 ( ) =0 f(x) 0=f 0 (x) =3x 3 = 3(x + )(x ), x = ± x = ± f(x) x = 3, 3 x = ± f(x) 4 f( 3) = 8,f( ) =,f() =,f( 3 )= 9 8 f(x) 8 3 apple x apple 3 6 ( ). a< <b f(x) a<x<b f(x) x = ) f 0 ( ) =0, x = f(x) 6 f 0 (x) f 0 ( ) 6= 0 f 0 ( ) > 0 f 0 ( ) < 0 f 0 ( ) > 0 f 0 (x) x = f 0 (x) > 0 f(x) x> f(x) >f( ) f( ) x< f(x) <f( ) f 0 ( ) < 0 f( ) f 0 ( ) =0 6 ) 4 (Weierstraß ). [a, b] f(x) 4 4. pp pp

73 . 0 apple x f(x) =x f(x) 3. apple x f(x) = 3 6. y = f(x) 0 apple y< x 4. 0 <xapple 0 <xapple f(x) = x f(x) apple x< 0 apple x< f(x) =x y = f(x) 0 apple y< < 0 (x =0) 6. 0 apple x apple f(x) = : f(x) (0 <xapple ) x 6 ( x (0 apple x<) 7. 0 apple x apple f(x) = y = f(x) 0 apple y< 0 (x =) f(x) =x 4 4x + 4x apple x apple 3 f(x) 70

74 0. 0 ( ). xy R V V R r V V V 0 8. p.68 3 p.66 p.6 8 ( ). V = {(x, y) x + y apple } V V {(x, y) x + y =} W = {(x, y) x + y apple x 6= } W {(x, y) x + y =} W (, 0) 8 ( ). xy R V V {(x, y) a apple x apple b c apple y apple d} a, b, c, d 9 ( ). V = {(x, y) x 0 apple y apple x } V V V = Z dx x = 9. V V {(x, y) (x = 0 apple y apple ) (x apple y =0) (x apple y = x )} V 7

75 5 (Weierstraß ). xy V f(x, y) pp p V = R V f(x, y) =x f(x, y) V = {(x, y) 0 <xapple, 0 apple y apple } V f(x, y) = x f(x, y) < 0 (x =0). V = {(x, y) 0 apple x apple, 0 apple y apple } f(x, y) = : f(x, y) (0 <xapple ) x ( ). xy R V f(x, y) 8 >< (, ) @x >: (, )=0 ) (, ) )=0 f(, ) V (, ) V V V (, ) V 0 " y = " apple x apple + " V g(x) =f(x, ) ( " apple x apple + " ) g(x) x (, )=g0 ( (, )=0 7 ) 0.4 D D D f(x, (x, y), (x, () x, (x, y) =0 (x, y) =0. D (x, y) 7

76 (3) D 8 < x = '(t) (a apple t apple b) D f(x, y) : y = (t) f('(t), (t)) (a apple t apple b) (4) D f(x, y) D Weierstraß 5 f(x, y) D f(, ),f(, ) f(x, y) D D (, ), (, ) (, ) D 7 () D f(, ) D (3) (, ) ()(3) 3. f(x, y) =x y D = {(x, y) x + y apple } () 8 = y () =x, (x, y) =(0, 0) = y (0, 0) D f(0, 0) = 0 (3) D x + y = 8 < x = cos t (0 apple t apple ) : y =sint f(x, y) =f(cos t, sin t) = cos t sin t = cos t (0 apple t apple ) g(t) g 0 (t) = sint OK 3 t 0 g(t) & % & % g 0 (t) (3) (4) D f(x, y) Weierstraß f(x, y) D f(x, y) D () (0, 0) D (3) ± f(0, 0) = 0 ± f(x, y) D 73 f(x, y) D

77 0. f(x, y) =x + xy + y + x + y D : y applex apple D xy D 74

78 0.5 0 [a, b] f(x) (Weierstraß f(x) a<x<b f 0 (x) =0 x x = a x = b xy V f(x, y) (Weierstraß f(x, @f (x, y) y) =0 (x, y) V 75

79 . f(x) d f dx = d df (= f 00 (x)) dx dx f(x) d 3 f dx 3 = d d f dx dx = d d df (= f 000 (x)) dx dx dx f(x) f(x, y) =x + xy @(x + xy + y + xy + y @y ) + ) ) x @() + y + x + f(x, y) 76

80 ( ) x m m y n ) x xm ny n = (xm ) = ny n m y n ) y yn mx m = (yn ) = mx m ny m y n m y @x 6 (Clairault). xy V @y 34. pp @y f(x, y) ( x 3 y f(x, y) = x +y (x, y) 6= (0, 0) 0 (x, y) (0, 0), @x, f(x, y) C f(x, y) C f(x, y) @y f(x, @ f(x, y) C 3 f(x, y) @x,, f f 77

81 @3 @ 3 3 f(x, y) n C n f(x, y) n n n x k y l k + l apple k+l l f(x, y) n n C. f(x, y) =e x cos y xy. f(x, y) x = a +(x a),y = b +(y b) x a y b NX nx f(x, y) = b n k,k (x a) n k (y b) k (46) n=0 k=0 = b 00 + b 0 (x a)+b 0 (y b) + b 0 (x a) + b (x a)(y b)+b 0 (y b) + + b N0 (x a) N + + b N k,k (x a) N k (y b) k + + b 0N (y b) N. x = a, y = b f(a, b) =b 00 78

82 (46) (x, y) = X nx b n k,k (n k)(x a) n k (y b) k (47) n= k=0 = b 0 + b 0 (x a)+b (y b) + b 30 3(x a) + b (x a)(y b)+b (y b) + + b N0 N(x a) N + + b N k,k (N k)(x a) N k (y b) k + + b,n (y b) N, x = a, y (a, b) =b 0 (46) (x, y) = X n= k= = b 0 nx b n k,k (x a) n k k(y b) k (48) + b (x a)+b 0 (y b) + b (x a) + b (x a) (y b)+b 03 3(y b) + + b N, (x a) N + + b N k,k (x a) N k k(y b) k + + b 0N N(y b) N, x = a, y (a, b) =b 0 79

83 (47) N (x, y) = X nx b n k,k (n k)(n k )(x a) n k (y b) k (49) n= k=0 = b 0 + b 30 3 (x a)+b (y b) + b (x a) + b 3 3 (x a)(y b)+b (y b) + + b N0 N(N )(x a) N + + b N k,k (N k)(n k )(x a) N k (y b) k + + b,n (y b) N x = a, y = b (47) N f (x, y) = X NX n= k= = (a, b) = b 0 b n k,k (n k)(n k )(x a) n k (y b) k (50) + b (x a)+b (y b) + b 3 3(x a) + b (x a) (y b)+b 3 (y b) + + b N, N(x a) N + + b N k,k (N k)(x a) N k k(y b) k + + b,n (y b) N x = a, y = f (a, b) = b. (48) x k+l l (a, b) =k! l! b kl. 80

84 b kl k+l f k! k (a, f (46) f(x, y) = f(a, (a, f (a, a) + + N! + + N! (a, b)(y f (a, b)(x a)(y f (a, N N (a, b)(x a)n + N f (a, b)n (N N N k (a, b)(x a)n k (y b) k.3 f(x, y) 7. f(x, y) (a, b) 7 lim f(x, y) = NX N! n=0 k=0 X nx n=0 k=0 f(x, y) = X n=0 k=0 nx b n k,k (x a) n k (y b) k nx b n k,k r n r (n k)!k! = f(a, (a, f (a, a) + + n n (a, b)(x a)n n n k (a, b)(x a)n k (y b) k (a, b)(y f (a, b)(x a)(y f (a, n n (a, b)(y b)n + (n k)!k! n n k (a, b)(x a)n k (y b) k

85 35. f(x, y) f(x, y) (a, b) f(x, y) (x, y) =(a, b) 8. f(x, y) (x, y) =(a, b) C n - n f(x, y) = nx mx m=0 k=0 (m k)!k! = f(a, (a, f (a, a) m m k (a, b)(x a)m k (y b) k + R(x, y) (a, b)(y f (a, b)(x a)(y f (a, b)(y n f n (a, n f a)n + + (n n k (a, b)(x a)n k (y b) k + n f n (a, b)(y b)n + R(x, y) R(x, y) x!! n a R(x, y) 0 lim0!! n = y b BxC Ba x a A y b y b p pp R(x, y) f(x, y) (x, y) =(a, b) n 5. f(x, y) =x 5 + x y 3 f(x, y) (x, y) =(, ) x 5 + x y 3 = 8 3 f =5x4 +xy 3 = 0x 3 +y = 60x 3 3 =3x >: 3 =6x = xy y 3 =6x f (, ) = ) =7 (, ) = 3 3 f f(, ) ) ) =3 >: 3 f (, ) (, ) =6 3 (, ) =6 3 8

86 x 5 + x y 3 = + 7(x ) + 3(y ) + (x ) + 6(x )(y ) + 3(y ) +0(x ) 3 + 3(x ) (y ) + 6(x )(y ) +(y ) , 7, 3 3. f(x, y) = f(x, y) (x, y) =(0, 0) x + y + = x + y + 83

87 .4 n n C n C n l f(x, y) (x, y) =(a, b) f(x, y) = X nx n=0 k=0 (n k)!k! (x, y) =(a, n n k (a, b)(x a)n k (y b) k f(x, y) (x, y) =(a, b) n f(x, y) ; nx mx m=0 k=0 (m k)!k! (x, y) =(a, b) m m k (a, b)(x a)m k (y b) k 84

88 . x y x y y = x ( ). f(x, y) (, ) (, ) f(x, y) (x, y) =(, ) 38. f(x, y) (, ) (, ) f(x, y) apple f(, ) (x, y) =(, ) 39. f(x, y) = y (0, 0) f(x, y) f(x, 0) = 0 y =0 40. maximum local maximum 7 8 ( ). f(x, y) xy R V 8 f(x, y) (, ) ) (, f(x, y) f(x, y) ) x, (x, y) =0 (x, y) =0 (, )=0 (, )=0 85

89 f 0 (a) =0,f 00 (a) > 0 ) x = a f(x) f 0 (a) =0,f 00 (a) < 0 ) x = a f(x) f 0 (a) =f 00 (a) =0 f 0 (a) =0 x = a f(x) ; f(a)+ f 00 (a) (x a) x = a f(x) x = a x = a f(x) x = a f(x, y) (a, (a, b) (a, f(x, y) ; f(a, f (a, a) f (a, b)(x a)(y (a, b)(y b) (5) f(x, y) f(x, y) 4. f(x, y) (x, y) =(0, ) f(x, y) =(x y )e (x +y ) 86

90 ... f(x, y) (a, b) (5) z = f(a, f (a, a) f (a, b)(x a)(y f (a, b)(y f(a, b) (a, b) f (a, b) (a, b) =C z = d + A(x a) + B(x a)(y b)+c(y b) z = Ax + Bxy + Cy x a y b z d xy Ax + Bxy + Cy = k 5. 4 f(x, y) =(x y )e (x +y ) (x, y) =(0, ) z = Ax + Bxy + Cy z = Ax + Bxy + Cy A, B, C.. 87

91 ..3 F, F 0 d FP + F 0 P = d P F, F 0 y P F F x..4 (, 0), (, 0) >0 (x, y) p (x + ) + y, p (x ) + y d p (x + ) + y + p (x ) + y = d 0 < <d p (x + ) + y + p (x ) + y = d, x d! + y p d 4! = norip/conic.pdf p d d = a, 4 = b x a + y b = (5) (5) 88

92 y b a a x b x a + y b = 8 < :a>b (± p a b, 0) a a<b (0, ± p b a ) b a, b..5 (5) ( x = ax y = by (53) X + Y = (54) (5) (54) x a y b (x, y) X a y b x a, y b = (53) x a + y b ( a, 0) (a, 0) (0, b) (0,b) 6. x 9 + y 4 x y + = 3 = xy 89

93 y -3 3 x - 3 x 9 + y 4 =..6 F, F 0 d FP F 0 P = d P F, F 0 y P F F x 4 90

94 ..7 (, 0), (, 0) >0 (x, y) p (x + ) + y, p (x ) + y d p p (x + ) + y (x ) + y = d 0 <d< p (x + ) + y p (x ) + y = d, x d! y p 4 d! = norip/conic.pdf p d 4 = a, d = b x a y b = (55) x a y b = (± p a + b, 0) a, b..8 x y + = x + y = x a y b a b x y = x y = x a a b y b x y = x a y a = x y = a a = b 9

95 ..9 a b! c d! cos! 0! = cos! 0 sin sin cos +!! sin = sin + cos a a = c c b a = d c b cos = d 0 sin b 0 sin = d cos a c b cos = d sin sin cos (x, y) (x 0,y 0 ) x 0 cos y 0 = sin sin x cos y F (x, y) =0 (x, y) (x cos( ) y sin( ),xsin( )+ y cos( )) = (x cos + y sin, x sin + y cos ) F (x, y) =0 F (x cos + y sin, x sin + y cos ) =0..0 y = c x c>0 xy = c (x cos + y sin ) ( x sin + y cos ) = c, (cos sin )x + (cos sin )xy + (cos sin )y = c, sin x + (cos )xy + 9 sin y = c (56)

96 cos =0 = 4 sin = sin = (56) x y x = c p c y p c =..8 y = c x px p y c c = 4 y x 5 y = c x p x c y p c = 93

97 .3 z = x y x y + + a b a b = c ( < c < ) x y 8> (c <0) + = c (c = 0) a b <>: x a p c + y b p c = (c >0) x y z = + a b x y 6. a =,b=3 z = + z =0,z = a b,z = xy y x

98 x y x z = a b a x y = c a b y b = c ( <c<) x z = a y b 8 y >< b p x c a p c = (c <0) x a ± y b =0 (c = 0) >: x a p y c b c p = (c >0) 7. a =,b =3 z =,z =,z =0,z =,z = xy y x a y b z = 4 0 x

99 .4 ax + bxy + cy = x a! b b c y a b b c x y (57) ax + bxy + cy x cos = y sin sin X cos Y x y = X Y cos sin sin cos ax + bxy + cy = X A B B C Y cos = sin ax + bxy + cy = X cos sin Y sin a b cos sin a b cos A B B C b c b c cos sin cos sin sin cos sin X cos Y X = AX + BXY + CY Y z = ax + bxy + cy z z = Ax + Bxy + Cy (59) B = cos sin a b b c sin = b cos sin +(a cos c)sin cos B = b(cos sin )+(a c) sin cos = b cos +(a b cos c)sin = a c sin a, b, c!! b cos a c sin B =0 z = ax + bxy + cy z z = Ax + Cy 96 (58) (59)

100 8. x + xy + y = Ax + Cy = x + xy + y = xy.5 x y (i) A, C > 0 = p A, = p C z = + (ii) A, C < 0 = p A, = p C z = x y (iii) A<0 <C = p A, = p C z = x + y x (iv) C<0 <A = p A, = p C z = y 97

101 A, C AC > 0 z = Ax + Cy AC < 0 z = Ax + Cy z = ax + bxy + cy z z = Ax + Bxy + Cy AC B A B 4 =det cos sin a b B =det cos sin b C sin cos c sin cos cos sin a b =det det cos sin a b b det =det b sin cos sin cos b = ac 4 a! b b c AC = ac b 4 A B B C c! c B =0 ac b 4 > 0 z = ax + bxy + cy ac b 4 < 0 z = ax + bxy + cy (60) 9. f(x, y) =x +3xy + y f(x, y) =x p 3xy +y 3 f(x, y) =x +4xy +4y.6 3 ( ). f(x, y) H f (x, y) (x, (x, f (x, y) (x, f (x, (x, y) H f f (x, y) 4. H f D D f(x, y) (a, b) (5) z = f(a, f (a, a) z = f (a, b)(x a)(y (a, b)(y b) (a, b)x f (a, b)xy f (a, 98

102 (a, (a, f (a, (a, b) 4 (a, b) > 0 f (a, b) < 0 (6) 9. f(x, y) xy D D (a, (a, b) =0,H f (a, b) > f (a, b) > 0 ) f(x, y) @f (a, (a, b) =0,H f (a, b) > f (a, b) < 0 ) f(x, y) @f (a, (a, b) =0,H f (a, b) < 0 ) f(x, y) (a, b) 9 4. p n p.59 (a, (a, b) =0,H f (a, b) =0 f(x, y) (a, b)

103 7. f(x, y) =(x y )e (x +y ) (x, y) (x, y) () =x( x + y )e x =y( x + y )e x y (x, y) +y ex ( 0=x( x + y ) 0=y( x + y ), (x =0 x + y = 0) (y =0 x + y = 0), (x =0 y = 0) (x =0 x + y = 0) ( x + y =0 y = 0) ( x + y =0 x + y = 0), (x =0 y = 0) (x =0 +y = 0) ( x =0 y = 0) ( x + y =0 = 0), (x =0 y = 0) (x =0 y = ±) (x = ± y = 0) (0, 0), (0, ±), (±, 0) = f =4xy(x x + y )e x y 4x e x y 4x ( x + y )e x y y )e x = ( x + y )e x y +4y e x y 4y ( x + y )e x (x, H f (x, y) (x, f (x, f (0, 0) (0, 0) = f (0, f (x, y) H f (0, 0) = ( ) 0 = 4 < 0 (0, 0) 00

104 (0, ±) = f (0, ±) = (0, ±) = 4e H f (0, ±) = (4e ) 0 = 6e > 0 (0, (0, ±) = 4e > 0 (0, ±) f(x, y) (±, 0) = f, (±, 0) = f (±, 0) 4e H f (±, 0) = ( 4e ) 0 = 6e > 0 (±, f (±, 0) 4e f(x, y) f(x, y) (±, 0) (0, ±) < 0 (±, 0) 30. f(x, y) =x xy + y 3x +3 0

105 .7 ) f(x, y) (a, b) z = f(a, f (a, a) f (a, b)(x a)(y f (a, b)(y z (a, b)x f (a, b)xy (a, b)y x (a, (a, b) f x (a, b) y (a, b) H f (x, y) (x, (x, f (x, y) (x, f (x, (x, f (x, y) (a, b) 0 H f (a, b) > f (a, b) > 0 ) f(x, y) (a, H f (a, b) > f (a, b) < 0 ) f(x, y) (a, H f (a, b) < 0 ) f(x, y) (a, b) 0

106 R n R m m = n = 3. st s =,s=0,s=,s=,t=,t=0,t=,t= xy R 3 s! 7! x! =!! s R t y 3 t 3. st 7 s =,s=0,s=,s=,t=!!! 3,t=0,t= 3 xy R 3 s 7! x = s cos t t y s sin t R 03

107 3.. f : R 3 s! 7! x! = s cos t! R (s, t) =(, 0) t y s sin t s s = cos t, sin t t =0 s =+(s ) cos t =+0 t + " (t) sin t =0+ t + " (t) " (t) lim =0 t!0 t " (t) lim =0 t!0 t (6) (63) s cos t = {+(s )}{+" (t)} =+(s ) + " (t)+(s )" (t) s sin t = {+(s )}{t + " (t)} = t +(s )t + " (t)+(s )" (t) " 3 (s, t) = " (t)+(s )" (t) " 4 (s, t) =(s )t + " (t)+(s )" (t) ( s cos t =+(s s sin t = t + " 4 (t) lim (s,t)!(,0) lim (s,t)!(,0) ) + " 3 (t) " 3 (t) p (s ) + t =0 " 4 (t) p (s ) + t =0 (64) s cos t, s sin t (s, t) =(, 0) (64) 44. (6) g(t) = cost g 0 (t) = sin t g(0) = cos 0 =,g 0 (0) = sin 0 = 0 g(t) =g(0) + g 0 "(t) (0)t + "(t) lim =0 p.8 44 t!0 t f (s, t) = s cos t f (s, t) = s sin t 04

108 cos t = sin t s sin t s cos t s cos t = f (s, t) =f (, (, 0)(s ) (, 0)t =+ (s ) + 0 t + s sin t = f (s, t) =f (, (, 0)(s ) (, 0)t =0+0 (s ) + t + (s, t)! (, 0) s cos t 0 s = + + s sin t 0 0 t f (s, t) =(, 3 ) s s = cos t, sin t t = 3 s =+(s ) sin 3 t cos t = cos 3 sin t =sin 3 + cos t n s cos t = {+(s )} cos 3 = cos 3 + cos 3 + " 5 (t) + " 6 (t) sin t 3 t (s ) sin 3 s sin t = {+(s )} nsin 3 + cos t 3 =sin 3 + sin (s ) + cos 3 3 " 7 (t) = " 5 (t)+(s )" 5 (t) " 8 (t) = " 6 (t)+(s )" 6 (t)+ lim (s,t)!(, 3 ) lim (s,t)!(, 3 ) sin 3 cos 3 p (s " 7 (t) ) +(t 3 " 8 (t) p (s ) +(t 3 lim t! 3 lim t! 3 t " 5 (t) t 3 " 6 (t) t 3 =0 =0 o + " 5 (t) 3 + " 7 (t) 3 o + " 6 (t) 3 3 (s ) (s ) ) =0 ) =0 t t + " 8 (t)

109 s cos t, s sin t (s, t) =(, 3 ) ( s cos t = cos 3 + cos 3 (s ) sin 3 t 3 + s sin t =sin 3 + sin 3 (s ) + cos 3 t 3 + s cos t = f (s, t) =f (, (, 3 )(s (, 3 ) t = cos 3 + cos (s ) sin t s sin t = f (s, t) =f (, (, 3 )(s (, 3 ) t =sin 3 + sin (s ) + cos t s cos t cos = 3 cos s sin t sin + 3 sin 3 s 3 sin 3 cos + 3 t y x f : R n 3 B A 7! x n y m R m C A 8 >< >: y = f (x,,x n ) y m = f m (x,,x n ) 06

110 (a,,a n ) 8 f (x,,x n ) = f (a,,a (a,,a n )(x a n (a,,a n )(x n a n ) +( ) >< f m (x,,x n ) = f m (a,,a n >: (a,,a n )(x a )+ n (a,,a n )(x n a n ) +( ) m 0 0 f (x,,x n ) f (a,,a (a,,a n n (a,,a n ) 0 B = C B + C B f m (x,,x n ) f m (a,,a n (a,,a n ) n (a,,a n ) 0 ( ) + B A ( ) m 4 ( ). @x (x,,x n m (x,,x n ) J f (x,,x n ):= B @x (x,,x n ) n (x,,x n ) x a 45. I J f (x,,x n ) f 0 (x,,x n ) p.7 det J f (x,,x n ) J f (x,,x n ) det f 0 (x,,x n ) p. x n a n C A 07

111 m x,,x m x =(x,,x n @x n (x) J f (x) = B @x (x) n (x) J f n C J f n C A m n m n 3..4 R! R f(x) =f(a)+f 0 (a)(x a)+( ), ( ) lim x!a x a =0 J f (a) =(f 0 (a)) ( ) R R f a f f 0 (a) 08

112 3..5 R! R f(x, y) = f(a, (a, = @x(a, b) (a, b)+( (a, b) x a +( ) y b lim (x,y)!(a,b) ( ) =0 x a y b J f (a, @x(a, (a, b) ( ) 33. R 3 (x, y) 7! f(x, y) =x + y R (a, b) J f (a, b)! x a f(x, y) ; f(a, b)+j f (a, b) f(x, y) =x + y y b 3..6 R! R f (x) = f (x) = f (a)+f(a)(x 0 a)+( ) f (a)+f(a)(x 0 a)+( ) f (a) f 0 + (a) ( ) f (a) f(a) 0 (x a)+ ( ) lim x!a x ( ) =0 a ( ) J f (a) = f 0 (a) f(a) 0 ( ) 09

113 3..7 R! R f (x, y) = f (a, (a, b)(y b)+( ) f (x, y) f (a, (a, b)(y b)+( x (a, b) a = f (a, b) f (a, b) (a, b) y b +! ( ) ( ) lim (x,y)!(a,b) x y a b ( ) =0 ( ) J f (a, (a, b) ( ) 34. f : R 3 r! 7! f! (r, ) f (r, ) f! (r, ) ; f! (a, b) + J f (a, b) f (r, ) f (a, b) = r cos! R (a, b) J f (a, b) r sin! r a b f! (r, ) = r cos! f (r, ) r sin y x f : R n 3 B A 7! x n y m 0 R m, g : R m 3 C B y y m 0 7! B A z z l C A Rl a =(a,,a n ) b =(b,,b m ) y = f(x) =f(a)+j f (a)(x z = g(y) =g(b)+j g (b)(y a)+( ) lim x!a x a ( ) =0 b)+( ) lim y!b y b ( ) =0 0

114 b = f(a) (g f)(x) = g(f(x)) = g(b)+j g (b)(f(x) b) +( ) = g(b)+j g (b) (f(a)+j f (a)(x a)+( ) ) b +( ) = g(b)+j g (b) J f (a)(x a)+( ) +( ) = g(b)+j g (b)j f (a)(x a)+j g (b)( ) +( ) = g(b)+j g (b)j f (a)(x a)+( ) 3 ( ) 3 = J g (b)( ) +( ) lim x!a x a ( ) 3 =0 (g f)(x) =(g f)(a)+j g f (a)(x a)+( ) 4 lim x!a x a ( ) 4 =0 J g f = J g J f f J f g J g f g g f) f) n f J g f 0 (a) m n (a) = C B C A (b) l m m (a) n @x C A C n C A l = m = n = dz dz dy = dx dy dx

115 @x 35. f(x) =(+x),g(y) =siny () x! 0 f(x) f(0) + f 0 (0)x f y = f(x) f y = f(0) + f 0 (0)x xy () y! g(y) g() + g 0 ()(y ) g z = g(y) g z = g() + g 0 ()(y (3) sin x = g() + g 0 ()(x ) + " (x) ) yz " (x) lim x! x =0 ( + t) = f(0) + f 0 (0)t + " (t) " (t) lim =0 t!0 t sin ( + t) =(sin) + (cos ) t + (cos )" (t)+" (( + t) ) (cos )" (t)+" (( + t) ) lim =0 t!0 t (4) b sin x = g(b)+g 0 (b)(x b)+" (x) " (x) lim x!b x b =0 a ( + t) = f(a)+f 0 (a)(t a)+" (t) " (t) lim t!a t a =0 b =(+a) sin ( + t) =(sinb) + (cos b) ( + a)(t a) + (cos b)" (t)+" (( + t) ) (g (cos b)" (t)+" (( + t) ) lim =0 t!a t f) 0 (a)) = g 0 (f(a))f 0 (a)

116 3.3 z =0 ( = x + t = x t x, t ( x = + z = h(x, t) g t = g(, ) =h( +, ) g(x + t, x t) =h(x, t) f h = g f : R 3 x 7! t = f f g : R 3 J x + t R 7! z = g(, ) R J h : R 3 x 7! z = h(x, t) R t 3

117 J chain rule J h = J g f = J g @t ( @ @x ( = x + t = @ @ z @ = @ @ z @ A 4

118 =0, z = '( ) ('( ) Z, z = '( )d + ( ) ( ( ) ) 36. f f : R 3, z = ( )+ ( ) ( ( ) ), z = (x t)+ (x + t) (, )! 7! x t! R J f!! + x = J f t g : R 3! 7! z = g(, ) R J g h : R 3 x! 7! z = h(x, t) R J h t g = h f J g = J h @z 5

119 3.3. z ( x = r cos y = r sin z = g(x, y) h(r, ) =g(r cos, r sin ) f h = g f : R 3 r 7! x = y f f g : R 3 J r cos R x 7! z = g(x, y) R y J h = g f : R 3 r 7! z = h(r, ) R J chain rule J h = J g f = J g @ 6

120 ( @x @ ( x = r cos y = cos = sin r sin sin r sin r cos @x = cos r sin r cos (67) x, y @r cos r sin sin r r cos @ cos sin r sin = cos sin r sin cos r 7

121 = @r @z r cos cos r sin @z r @r r cos r sin @ z sin r cos sin sin z sin z cos r sin sin + sin cos z cos + z sin r cos r cos r @z cos sin r r sin cos r r cos r! sin r 8

122 3.3.3 d x ax dx d x ax dx d x ax dx 6= a x x ax 6= a x (log a)x ax log x A = e log A x ax = e ax log x = e ex log a log x e x log x z = s at s = x, t = x s R 3 x 7! R t R s 3 7! z R t dz ds dx dt ds dx dx d(x ax ) dx at ds dx dt = dx = a t s at + a t (log a)s at log s = a x x ax + a x (log a)x ax log x a t s at d(a t ) dt s at log s dax dx = ax log a Z x d e xy dy dx z = Z s a, b C f(t, y) d dt 0 e ty dy, s = x, t = x Z b a f(t, y)dy = Z b f(t, 9

123 3.4 3 f : R n! R m f J f (a) R n 3 7! J f (a) R m f(x) =f(a)+j f (a)(x a)+( ) f : R! R f 0 (a) R 3 7! f 0 (a) R a f(x) =f(a)+f 0 (a)(x a)+( ) chain rule J g f (a) =J g (f(a))j f (a) (g f) 0 (a) =g 0 (f(a))f 0 (a) 0

f(x) = x (1) f (1) (2) f (2) f(x) x = a y y = f(x) f (a) y = f(x) A(a, f(a)) f(a + h) f(x) = A f(a) A x (3, 3) O a a + h x 1 f(x) x = a

f(x) = x (1) f (1) (2) f (2) f(x) x = a y y = f(x) f (a) y = f(x) A(a, f(a)) f(a + h) f(x) = A f(a) A x (3, 3) O a a + h x 1 f(x) x = a 3 3.1 3.1.1 A f(a + h) f(a) f(x) lim f(x) x = a h 0 h f(x) x = a f 0 (a) f 0 (a) = lim h!0 f(a + h) f(a) h = lim x!a f(x) f(a) x a a + h = x h = x a h 0 x a 3.1 f(x) = x x = 3 f 0 (3) f (3) = lim h 0 (

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2009 IA 5 I 22, 23, 24, 25, 26, (1) Arcsin 1 ( 2 (4) Arccos 1 ) 2 3 (2) Arcsin( 1) (3) Arccos 2 (5) Arctan 1 (6) Arctan ( 3 ) 3 2. n (1) ta 009 IA 5 I, 3, 4, 5, 6, 7 6 3. () Arcsin ( (4) Arccos ) 3 () Arcsin( ) (3) Arccos (5) Arctan (6) Arctan ( 3 ) 3. n () tan x (nπ π/, nπ + π/) f n (x) f n (x) fn (x) Arctan x () sin x [nπ π/, nπ +π/] g n

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