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3 i 6 3

4 ii

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9 .. {a n } {b n } lim n a n = a lim b n = b n () lim n (αa n + βb n ) = αa + βb (α, β ). () lim n a n b n = ab a n (3) lim = a n b n b ( b n b ).. {a n } {b n } {c n } () a n b n lim n a n lim n b n () a n c n b n lim a n = lim n {c n } n b n.3.4 N

10 .5 () a a < lim n a n = a n () a lim n n! = (3) a > lim n a n (4) lim n n n ( = lim + n = e n n) (5) lim n a n = α lim n a + a + + a n n = lim n n = lim a n n = = α (α ).. 3 n+ () lim () lim ( n + n) n n + 3 n n sin n n n (3) lim (4) lim n n n n ( ) n () a < lim a n = lim = n n 3 lim n () lim n n = lim n n 3 n+ n + 3 n = lim n = 3 ( 3) n + = 3 lim n + n) n = ( n + n)( n + + n) lim n n + + n = lim = n n + + n

11 .. 3 (3) sin n n sin n n n sin n lim n n = (4) a n = n n = n n.5(4) lim a n = n.5(5) n n lim = n n. ( ) n () lim n n + (3) lim n (5) lim n ( n) n ( n) n (6) lim n () lim n( n + n) n (4) lim n n ( n ) ( n ) n. {a n } a =, a n+ = ) (a n + an (n =,, ) () {a n } () lim n a n

12 4 () n =,, a n > n = a n > a n+ = ) (a n + an = ( a n ) > a n {a n } a n > a n a n+ = a n ) (a n + an = ( a a n ) > n {a n } ().3 () {a n } α = lim a n n n α = ( α + ) α α = ± α lim n a n = a n > () α = lim a n {a n } n a n > a n. {a n } a =, a n+ = a n + (n =,, ) () {a n } () lim n a n

13 f(x) g(x) lim x a f(x) = A lim x a g(x) = B () lim x a (αf(x) + βg(x)) = αa + βb (α, β ). () lim x a f(x)g(x) = AB. (3) lim x a f(x) g(x) = A B ( g(x) B )..7 D f(x) g(x) h(x) () D f(x) g(x) x = a D lim f(x) lim g(x) x a x a () f(x) h(x) g(x) x = a D f g lim f(x) = lim g(x) x a x a lim h(x) x a.8 () lim x sin x x () lim x sin x = lim x cos x = =.3 n x n n x () lim () lim x x x x

14 6 () x n = (x )(x n + x n + + x + ) x n lim x x = lim x (xn + x n + + x + ) = n () y = n x x y () lim x n x x.3 x () lim x x 3 8 x 3 (3) lim x x x + = lim y y y n = n () lim x (4) lim x x x + x x x + + x x.4 cos x () lim x x ( () lim + ) x x x ().8() cos x lim x x = lim x (sin x ) x = lim x ( sin x x ) = () n = [x] [x] x x > n x < n + ( + ) n ( < + n ( + n + x) x) x ( < + n) x ( < + n) n+.5(4) ( lim + x n + ) n = lim x ( + n) n+ ( = lim + n = e x n)

15 lim x () lim x cos x x ( + x) x = e ( () lim + ) x x x.3.9 D f(x) lim f(x) = f(a) x a f(x) x = a f(x) x D f(x) D. D f(x) g(x) () f(x) + g(x) () cf(x) c (3) f(x)g(x) (4) f(x) g(x) g(x). y = f(x) z = g(y) f(x) x = x g(y) y = f(x ) z = g(f(x)) x = x

16 8. f(x) [a, b] f(a) f(b) f(a) f(b) k a < c < b c f(c) = k c.3 f(x) [a, b] f(a) f(b) < f(x) = sinh x = ex e x cosh x = ex + e x tanh x = sinh x cosh x cosh x coth x = sinh x.5 e x () lim log( + x) () lim x x x x () x x.4().4() lim ( + x) x = e. x lim log( + x) = lim x x log( + x) x x ( ) = log lim ( + x) x x = log e =.

17 .3. 9 () e x = y x y () e x x = y log(y + )..5 ()lim x e x e x x ( () lim + a x (3)lim( + sin x) x ± x) x x.6 a x (x > ) f(x) = x a (x ) x lim x x = lim (x + ) = a = x f(x) x = a =.6 a b x + 3x + a (x > ) f(x) = x b (x ).7 a > n x n a =

18 f(x) = x n a f(x) f() = a < b > f(b) = b n a > f() f(b) <.3 [, b] f(x) = x n a = [.7 x cos x =, π ].8 () sin x + cos x = π () cosh x sinh x = (3) sinh x = log(x + x + ) (4) tanh x = log + x x ( () y = sin x z = cos x x = sin y x = cos z = π ) ( π sin z sin y = sin z) π y π, π π z π y = π z sin x+cos x = y+z = π () cosh x sinh x = ( ) e x + e x ( e x e x ) = 4 ((ex + + e x ) (e x + e x )) = (3) y = sinh x x = sinh y = ey e y t = e y x = t t t xt = t = x ± x + t > t = x + x + e y = x + x + sinh x = y = log(x + x + )

19 .3. (4) y = tanh x x = tanh y = ey e y e y + e t = y ey t = + x ( ) + x t > t = x x ( ) + x e y = tanh x = y = ( ) + x x log x.8 () tan x + cot x = π. () sin sin 4 5 = π. (3) cosh x = log(x + x ). (4) coth x = log x + x. (5)(cosh x ± sinh x) n = cosh nx ± sinh nx (n ).

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21 3.. f(x) x = x f(x) f(x ) lim x x x x = lim h f(x + h) f(x ) h f(x) x = x f(x) x = x lim h + f (x ) f(x + h) f(x ) f(x + h) f(x ),, lim h h h f(x) x = x f +(x ), f (x ) f(x) x = x f (x ) y = f(x) P(x, f(x ))

22 4.. f(x) x = x x = x f(x) = x x = f(h) = h f(h) f() h = h h = { (h ) (h < ) f +() = f () = f(x) x = f(x) = x x =.. f(x) = x x x =.3 y = f(x) I f(x) I I I I x f (x ) x y = f(x) y, f dy (x), dx, df dx, df(x) dx f(x) f(x). () y = c (c ) () y = x n (n ) (3) y = sin x

23 .. 5 () (c) c c = lim =. h h () (x + h) n (x + h) n = n C x n + n C x n h + + n C n h n (x + h) n x n h = n C x n + n C x n h + + n C n h n n C x n (h ) (x n ) = n C x n = nx n (3) ( sin(x + h) sin x = cos x + h ) sin h sin(x + h) sin x h = cos(x + h ) sin h h ( lim cos x + h ) = cos x, lim h (.8) sin h h h = (sin x) = lim h sin(x + h) sin x h = cos x. y = cos x

24 6.3 () y = e x () y = log x (x > ) () e x+h e x h = ex (e h ) h e h lim = h h (.5()) (e x ) = e x () log(x + h) log x h = log( + h x ) h log( + k) lim k k = = x log( + h x ) h x (.5()) (log x) = x.3 y = log( x) (x < ).4 (log x ) = x (x )..5 f(x) g(x) () (f(x) ± g(x)) = f (x) ± g (x), (kf(x)) = kf (x) () (f(x)g(x)) = f (x)g(x) + f(x)g (x)

25 .. 7 (3) ( ) g(x) = g (x)f(x) g(x)f (x) f(x) (f(x)).6 ( y = f(x) x = x z = g(y) y = f(x ) z = (g f)(x) x = x (g f) (x ) = g (f(x ))f (x ) y = f(x) z = g(y) z = (g f)(x) = g(f(x)).6 z = g (f(x))f (x) (.) g (f(x)) g(y) y g (y) y = f(x) dz dx = dz dy dy dx (.) (.), (.).4 k (x k ) = kx k (x > )

26 8 x k = (e log x ) k = e k log x z = e y y = k log x z y y x dz dx = dz dy dy dx = ey k x = k ek log x x = k xk x = kxk. y = x k (x > ) log y = k log x (.3) x y y = k x (.4) (.5) y = k y x = kxk (.3) y x (.3) y = e k log x y x.4 () tan x () log a x (x ) (3) + x (4) a x (a > ) (5) log x + x + A (A (6) x x (x > ).7 y = f(x) f(x) x = x f (x ) x = f (y) y = y (= f(x )) (f ) (y ) = f (x ) (.5)

27 .. 9 (.5) (f ) (y) = f (x) = f (f (y))..5 (cos x) = dx dy =. dy dx x ( < x < ) y = cos x [, π] y = sin x, < x < π x = cos y (, ) (cos y) = y = ( sin x) = cos x = y x y x = cos y [, ] y = ± y = cos x y = sin x x =, π.5 () (sin x) = x ( < x < ). () (tan x) = + x ( < x < ).

28 .8 x = ϕ(t) y = ψ(t) x = ϕ(t) y x ϕ(t) ψ(t) ϕ (t) y x dy dx = dy dt dx dt.6 a xy θ x = a(θ sin θ), y = a( cos θ) dy dx dx dθ = a( cos θ), dy dθ = a sin θ dx > ( < θ < π) x = a(θ sin θ) θ π dθ.8 dy dy dx = dθ dx dθ = a sin θ a( cos θ) = sin θ cos θ ( < θ < π)

29 ...6 C : x a + y = (a, b > ) b C : x = a cos θ, y = b sin θ θ C.7 dy dx ( a 3b ), () C : x = a cos 3 t, y = b sin 3 t. () C : x = 3at + t 3, y = 3at + t f(x) [a, b] (a, b) f(a) = f(b) f (c) = c (a < c < b) f(x) x = a, x = b.8 f(x) [a, b] f (a) < A < f (b) f (c) = A c (a < c < b). f(x) [a, b] (a, b) f(b) f(a) b a c (a < c < b) = f (c)

30 . f(x) I () f(x) I I f (x) (f (x) ) () f (x) > (f (x) < ) f(x) I (3) f(x) f (x) =.7 ( π x < sin x < x < x < π ). f(x) = sin x π x f (x) = cos x π f() =, f (x) > ( < x < cos ) π f(x) > ( < x < cos ) π f (x) < ( cos π < x < π ) ( π ), f = f(x) > ( cos π < x < π )

31 () x x3 3! < sin x < x (x > ). () x x < cos x < + x4 (x ). 4! x (3) < log( + x) < x (x >, x ). + x. f(x) f (x) x = x f (x ) f(x) x = x f (x) f (x) f(x) f(x) n f(x) n f (n) (x) y = f(x) n y (n), d n y dx n, d n f(x) dx n f () (x) = f(x).8 y = sin x ( y (n) = sin x + nπ ) n n = n = k

32 4 n = k + { ( f (k+) (x) = sin x + kπ )} ( = cos x + kπ ) ( = sin x + kπ + π ). n = k +. y = cos x ( y (n) = cos x + nπ ). n () cos x cos x () log( + x) (3) (4) x k (x > ) x.3 f(x) n f(x) n f (n) (x) f(x) n C n f(x) C.4 f(x), g(x) n f ± g cf (c ) fg n () (f ± g) (n) = f (n) ± g (n), (cf) (n) = cf (n) n () (fg) (n) = nc r f (r) g (n r) r=.9 y = x e x n

33 .. 5 n y (n) = nc r (x ) (r) (e x ) (n r) r= = n C x e x + n C xe x + n C e x = {x + nx + n(n )}e x.. n ( ) x sin x () e x sin x.5 f(x) I n I a, b f(b) = f(a) + f (a)! (b a) + f (a) (b a) +! + f (n ) (a) (n )! (b a)n + R n c a b R n = f (n) (c) (b a) n n!.6 f(x) x = a I n I x f(x) = f(a) + f (a)! R n = (x a) + f (a) (x a) +! + f (n ) (a) (n )! (x a)n + R n (x a)n f (n) (a + θ(x a)) ( < θ < ) n!

34 6 f(x) R n (n ) f(x) = f(a) + f (a)! (x a) + f (a) (x a) +! + f (n) (a) (x a) n + n! f(x) x = a a = f(x) = f() + f ()! x + f ()! f(x) x + + f (n) () x n + n!. () e x = + x + x! + + xn + ( x < ). n! () sin x = x x3 3! + x5 5! + x n+ ( )n + ( x < ). (n + )! (3) cos x = x! + x4 xn + ( )n 4! (n)! + (4) log( + x) = x x + x3 3 + ( )n xn n ( x < ). + ( x < ). () f(x) = e x.3 f (n) (x) = e x e x = + x +! x + + (n )! xn + R n, R n = xn n! eθx ( < θ < )

35 .. 7.5() R n x n n! e x (n ) () f(x) = sin x.8 ( f (n) (x) = sin x + nπ ) sin x = x x3 3! + x5 5! + ( )(n ) x n (n )! + R n+, (n+)π n sin(θx + R n+ = ( ) (n + )! ) x n+ ( < θ < ).3 R n+ x n+ (n + )!.(3) (n )..4 () x = + x + x + + x n + ( x < ). () ( x) = + x + 3x + + (n + )x n + ( x < ). (3) log + x x = x + x3 3 + x5 5 + xn + ( x < ). n.5 e x x = iθ (i = ) e iθ = cos θ + i sin θ ( )

36 8.3.7 f(x) g(x) [a, b] (a, b) (a, b) g (x) f(b) f(a) g(b) g(a) = f (c) g (c) c (a < c < b).8 () f(x) g(x) x = a a g (x) f(a) = g(a) = f (x) lim x a g (x) = l ( l ) f(x) lim x a g(x) l () f(x), g(x) x = a a g (x) x a g(x) f (x) lim x a g (x) = l ( l ) f(x) lim x a g(x) l (), () x a ±, x ±

37 x sin x x log x () lim () lim x x 3 x)cos (3) lim (α > ) x π (tan x x α cos x () lim x 3x () = lim x sin x 6x = 6. cos x log tan x = log tan x. cos x log tan x lim x π cos x = lim x π sec x tan x sin x cos x = lim x π tan x sin x =. lim x π (tan x)cos x = lim x π ecos x log tan x = e =. (3) lim x x = lim αxα x αx α =..6 e x () lim () lim x x x log x a x b x (3) lim n x + x x (a, b > ) (4) lim x x x..6 x sin x a x b x () lim () lim (a, b > ) x x 3 x x

38 3 () f(x) = sin x.6 f(x) = f() + f () ()! x + f () ()! x + f (3) () 3! x 3 + f (4) () x 4 + R 5, 4! R 5 = x5 5! f (5) (θx) ( < θ < ) (.8 f (n) (x) = sin x + nπ ) f() =, f () () =, f () () =, f (3) () =, f (4) () =, R 5 = (θx x5 5! sin + 5π ) ( < θ < ). sin x = x x3 3! + R 5, R 5 = x5 5! sin (θx + 5π x sin x x 3 = 6 R 5 x 3. R 5 x 3 = x 5! sin (θx + 5π ) x 5! ) ( < θ < ) (x ) x sin x (x ) x 3 6 () (a x ) = (log a)a x (a x ) = (log a) a x a x = + (log a)x + x (log a) a θx ( < θ < )

39 .3. 3 b x = + (log b)x + x (log b) b θ x a x b x x ( < θ < ). = log a log b + x {(log a) a θx (log b) b θ x } x θx θ x a x b x x log a log b (x ).7.6 e x x cos x () lim () lim x x x x.9 f(x) x = c x = c x = c f(x) < f(c) (f(x) > f(c)) f(x) x = c f(c). f(x) x = c x = c f (c) =. f(x) x = c h > (i) c h < x < c f (x) > (< ) (ii) c < x < c + h f (x) < (> ) f(x) x = c

40 3 x c f (x) + f(x). C n f(x) x = a I f (a) = = f (n ) (a) =, f (n) (a) () n f (n) (a) > f(x) x = a () n f (n) (a) < f(x) x = a (3) n f(x) x = a.3 f(x) = 3x 4 4x 3 4x + 48x 5 f (x) = x 3 x 48x + 48 = (x + )(x )(x ), f (x) = 36x 4x 48 = (3x x 4). f (x) = x =,, f ( ) = 44 >, f () = 36 <, f () = 48 >. x = f( ) = 7 x = f() = 8 x = f() =.8 () x 3 + 3x + x 5 () x + x 3 (3) x x (x > )

41 f(x) I y = f(x) (x I) A, B A, B y = f(x) AB f(x) I a < x < b I 3 a, x, b f(x) f(a) x a f(b) f(x) b x.4 f(x) f(x) I I f (x).5 I f(x) I f(x) I (a, f(a)).9 () y = x 3 x x + () (3) y = x log x (4) y = x x 3 +

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43 F (x) = f(x) F (x) f(x) F (x) F (x) f(x) dx = F (x) + C. C 3. () (f(x) + g(x)) dx = f(x) dx + () af(x) dx = a f(x) dx (a ). g(x) dx. (3) f (x)g(x) dx = f(x)g(x) f(x)g (x) dx, f(x) dx = xf(x) xf(x) dx.

44 36 3 (4) f(x) dx = f(g(t))g (t) dt 3.3 () x a dx = a + xa+ (a ). () x dx = log x. (3) (4) (5) x a dx = a log x + a dx = a tan x a a x dx = sin x a (x = g(t)). x a x + a (a > ). (a > ). (a > ). a (6) x dx = ( x a x + a sin x ) a x (7) x + a dx = log + x + a (a ). (a > ). x (8) + a dx = (x x + a + a log x + x + a ) (a ). (9) sin x dx = cos x. () cos x dx = sin x. () tan x dx = log cos x.

45 () (3) (4) (5) (6) (7) (8) (9) cot x dx = log sin x. tan( cos x dx = log x + π 4 ). tan sin x dx = log x. log x dx = x log x x. e ax ax a sin bx b cos bx sin bx dx = e. a + b e ax ax a cos bx + b sin bx cos bx dx = e. a + b sinh x dx = cosh x. cosh x dx = sinh x. 3. () f(x) = (x + ) + x F (x) F () = () f(x) = F (x) F () =. x 5 (3) f(x) = cos x + e x F (x) F () =.

46 38 3 () F (x) = f(x) dx = 3 (x + )3 + x + C. F () = 3 + C = F (x) = 3 (x + )3 + x 3. () F (x) = f(x) dx = log x 5 + C. F () = log 4 + C = F (x) = log x 5 log 4. (3) F (x) = f(x) dx = sin x + e x + C. F () = + C = F (x) = sin x + e x. 3. F () = () (x + ) 5 () (3) log(x + ) (x + 3) 3. () f(x) F (x) f(px + q) (p ) F (px + q) p () f(x) F (x) f(x) dx = log F (x). F (x) () ( ) F (px + q) = p p F (px + q)(px + q) = f(px + q) p F (px + q) () log F (x) f(x) F (x)

47 () (x + 3) 5 () (3) (x + 3) 5 (5) e 3x+ (6) (7) 4(x + ) + (9) x + 4x x + 3 (4) sin (x + ) x + 4x + 3 (8) x + x () (x + 3) 5 dx = 5 + (x + 3)5+ = (x + 3)6. () (x + 3) dx = (x + 3) 5+ = 8(x + 3). 4 (3) dx = x ) + = x (x (4) sin (x + ) dx = ( cos (x + )) = cos (x + ). (5) e 3x+ dx = 3 e3x+. (6) 4(x + ) + dx = (x + ) + dx = tan (x + ). (7) x + 4x + 3 = ( x + ) + x + 4x + 3 dx = tan (x + ). (8) x + x = (x + ) x + x dx = log (x + ) (x + ) + = log x x +. (9) x + 4x = 4 (x ) x + 4x dx = x dx = sin. (x )

48 () x () (3) ( 3x) 3 (5) (6) 8 4x x log x x x + (4) x + x () (x )(x 3) () x log x (3) x sin x (4) e ax cos bx (5) cos x (6) cos 4 x () (x 3) (x )(x 3) dx = (x ) (x 3) (x 3) dx = (x )(x 3) (x 3) = 58 (x )(x 3). (x )(x 3) = (x 3) + (x 3) () (log x) = x x log x dx = 3 x3 log x = 3 x3 log x 9 x3 = 9 x3 (3 log x ). 3 x3 x dx

49 3.. 4 (3) x n (n ) n x sin x dx = x ( cos x) x( cos x) dx = x cos x + x cos x dx = x cos x + x sin x sin x dx = x cos x + x sin x + cos x = ( x + ) cos x + x sin x. (4) e ax sin bx, cos bx I = e ax cos bx dx = a eax cos bx + b e ax sin bx dx a = a eax cos bx + b a eax sin bx b e ax b cos bx dx a = a eax cos bx + b a eax sin bx b a I I I = a cos bx + b sin bx a + b e ax. 3.3(7) e ax sin bx 3.3(6)

50 4 3 (5) cos x = (sin x) I = cos x(sin x) dx = cos x sin x ( sin x) sin x dx = cos x sin x + ( cos x) dx = cos x sin x + x I. cos x sin x + x I = cos x cos cos x + x = (6) (5) I = cos 4 x dx = cos 3 x(sin x) dx = cos 3 x sin x 3 cos x( sin x) sin x dx = cos 3 x sin x + 3 cos x sin x dx = cos 3 x sin x + 3 cos x( cos x) dx = cos 3 x sin x + 3 cos x dx 3I. (5) I = 4 cos3 x sin x cos x sin x x.

51 () x(x + ) () x sin(x + ) (3) cos x sin 4 x (4) cos 5 x (5) (log x) () () (x a) n, ax + b (x + px + q) n (n, p q < ). () (.) dx = log x a, x a (.) (x a) dx = (n ). n n (x a) n () ax + b (x + px + q) = a(x + p) n ((x + p) + (q p )) + pa + b n ((x + p) + (q p )). n ax + b (x + px + q) dx = a n t dt + ( pa + q) (t + c ) n (t + c ) n dt. t = x + p, c = q p

52 44 3 (.) (.) t t + c dx = log(t + c ) (n = ), t (t + c ) dx = n (n ) (n ). (t + c ) n (.3) t + c dx = t c tan (n = ), c (.4) (t + c ) dx = { t n c (n ) (t + c ) n + n 3 } dt n (t + c ) n (n ). 3.4 () () + x 3 x (x ) 3.6 () () (3) x + x + 3x + x(x )(x ) x 3 + x x + (4) (x ) (x + ) () x + 3x + = (x + )(x + ) = A x + + B x +. (A + B)x + (A + B) (x + )(x + ) A + B =, A + B =

53 A =, B = x + 3x + dx = x + dx x + dx = log x + x +. A, B (x + ) x + = A + B(x + ) x + x = A = (x + ) x = x + = A B = = ( ) + () x(x )(x ) = A x + B x + C x A =, B =, C = () A = ( )( ) =, B = ()( ) =, C = ()( ) = x(x )(x ) dx = log x(x ) (x ). x + (3) x 3 + x = x + x (x + ) = A x + B x + C x + A =, B =, C = x + dx = log x x 3 + x x + x.

54 46 3 B x B = + = A + x x = x B x = x x(x + ) = A x + B x + x + (4) (x ) (x + ) = A x + B (x ) + Cx + D x + A =, B =, C =, D = C, D x + x = i x + (x ) (x + ) = { x + (x ) + x x + }. x + x + (x ) (x + ) dx = 4 log x + (x ) x tan x. 3.5 () (3) x (x ) + x 3 () (x )(x )(x 3) (4) 3x x 3 (x + )

55 () (x + ) x + () x x (3) x x 6 + () t = x + x = t, dx = t dt (x + ) x + dx = (t )t(t)dt = 4 5 t5 3 t3 = 5 (6x + )(x + ) x +. () t = x x = t +, dx = t dt x t + dx = ( t) dt = x t 3 t3 t = 3 (x + ) x. (3) t = x 3 dt = 3x dx x x 6 + dx = 3 t + dt = 3 tan t = 3 tan x x + x 3 x 3 () () (3) x x + x 8

56 f(x, y) x, y ( f x, ) ax + bx + c a > t = ax + bx + c + ax ( t c at f at + b, + bt + ) ac ( at + bt + ac) at + b ( dt. at + b) t ax = ax + bx + c x = t c at + b ax + bx + c = t ax = t a(t c) at + b = dx = ( at + bt + ac) ( at + b) dt at + bt + ac at + b

57 () x + bx + c () x + bx + c (3) x x + bx + c () t = x + bx + c + x x (t + bt + c) + bx + c dx = dt (t + b) 3 = (s α) ds 6 s 3 = ) (s α 4α log s 3 s = 4 (x + b) x + bx + c 8 (b 4c) log x + bx + c + x + b s = t + b = x + bx + c + x + b, α = b 4c () t = x + bx + c + x x + bx + c dx = t + b dt = log t + b = log x + bx + c + x + b.

58 5 3 (3) t = x + bx + c + x x x + bx + c dx = t c dt log t c c t + c (c > ) = (c = ) t = c tan t c (c < ) x + bx + c + x c log c x + bx + c + x + c x + bx + c + x (c > ) (c = ) c tan x + bx + c + x c ( c < ).

59 f(x, y) x, y ( f x, ) ax + bx + c a <, ax + bx + c = a(x α)(β x), α < β x α t = β x ( ) βt + α a(β α)t (β α)t f t +, t + (t + ) dt. x = βt + α t + ax + bx + c = a(β α)t a(β x)t =. t + dx = (β α)t (t + ) dt

60 f(x, y) x, y ( ) ax + b f x, n (ad bc ) cx + d t = n ax + b cx + d ( ) ax + b f x, n cx + d dx = f ( ) dt n + b n(ad bc)t n ct n a, t dt (ct n a) t n = ax + b cx + d, dx = x = dtn + b ct n a n(ad bc)tn (ct n a) dt 3. x + x () () x 3 x

61 () t = x + x x = t + 4t, dx = t (t ) dt x + x dx = 4t (t ) dt. 4t (t ) = t + t + (t ) (t + ). x + x dx = log t + t + t + t + = log x + x + x. () t = 6 x x = t 3, 3 x = t, dx = 6t 5 dt. x t 8 3 dx = 6 x t dt { } = 6 t 6 + t 4 + t + + (t ) dt (t + ) = 6 7 t t5 + t 3 + 6t + log t t + = 6 7 x 6 x x5 + x x log x 6. x +

62 x x () () x x 3.3 f(x, y) x, y f(sin x, cos x) t = tan x f(sin x, cos x) dx = f ( t + t, ) t + t + t dt t = tan x = cos x cos x, cos x = + t. sin, cos sin x = tan x cos x = cos x = cos x ( tan x t + t ) = t + t.

63 dt = cos x dx, dx = + t dt 3.4 () () + sin x + sin x + cos x (3) cos 4 x () t = tan x + sin x dx = ( + t) dt = + t = + tan x. () t = tan x + sin x + cos x dx = + t dt = log + t = log + tan x. (3) sin x cos x t = tan x

64 56 3 sin x = cos 4 x dx = t + t, cos x = + t, dx = + t dt ( + t ) dt = t + 3 t3 = tan x + 3 tan3 x. 3.8 () () (3) sin x sin x a cos x + b sin x (ab > ) 3.5 n n () I n = dx (x + c ) n I n = { } x n 3 + c n (x + c ) n n I n. () I n = sin n x dx (3) I n = I n = n sinn x cos x + n n I n. x n sin x dx () I n = x n (n sin x x cos x) n(n )I n. ( ) x = (n ) (x + c ) n (x + c ) n

65 I n = = x x + (n ) (x + c ) n (x + c ) dx n x (x + c ) + (n )I n n (n )c I n I n () sin x = cos x I n = I n sin n x cos x dx = I n (sin n x) cos x dx n = I n n sinn x cos x n I n. I n (3) I n = x n ( cos x) dx = x n cos x + n x n cos x dx = x n cos x + nx n sin x n(n ) x n sin x dx. 3.9 () () (3) n x n e x dx = x n e x + n x n e x dx. cos n x dx = n cosn x sin x + n cos n x dx. n tan n x dx = n tann x tan n x dx.

66 58 3 (4) (5) x n cos x dx = x n (n cos x + x sin x) n(n ) (log x) n dx = x(log x) n n (log x) n dx. x n cos x dx. 3. f(x) [a, b] n : a = x < x < x < < x n = b ξ = (ξ, ξ,, ξ n ) (ξ i [x i, x i ]) n R( n, ξ) = f(ξ i )(x i x i ) i= n ξ n = max x i x i i n R( n, ξ) f(x) [a, b] ( ) b a f(x) dx = lim n R( n, ξ). x i = a + i (b a) (i =,, n) n ξ i [x i, x i ] (i =,,, n)

67 b a b a n f(x) dx = lim f (a + in ) n n (b a) i= b a n = lim f (a + in ) n n (b a). 3.4 f(x) [a, b] f(x) [a, b] 3.5 () b () (3) a b a 3.6 i= f(x), g(x) [a, b] {f(x) + g(x)} dx = αf(x) dx = α f(x) dx = c a b a f(x) dx f(x) dx + f(x) dx + (α R). b c b a f(x) dx g(x) dx. (a < c < b). f(x), g(x) [a, b] () f(x) g(x) = () b a 3.7 b a f(x) dx b f(x) dx f(x) dx. a b a g(x) dx. f(x), g(x) [a, b] () x (a < x < b) F (x) = x a f(t) dt f(x) F (x) = f(x).

68 6 3 () F (x) f(x) b a f(x) dx = [F (x)] b a = F (b) F (a). 3.8 ϕ(x) [α, β] C ϕ([α, β]) f(x) b a f(x) dx = β ϕ(α) = a, ϕ(β) = b α f(ϕ(t))ϕ (t) dt. 3.9 f(x), g(x) [a, b] C b b f (x)g(x) dx = [f(x)g(x)] b a f(x)g (x) dx. a a 3. f(x) [a, b] x [a, b] f(x ) > b a f(x) dx > 3.6 n () lim k () lim n n 3 () lim k= () lim n 4n n k=n n 3 n k= k = lim n 4n n k=n k = lim n n n 4n k=n k/n = k n k= 4 (3) lim n n k= n n + k ( ) k = x dx = n 3. dx = log 4. x

69 3.. 6 (3) lim n n k= = [tan x] = π 4. n n + k = lim n n n k= + (k/n) = + x dx 3. () lim n n 4 (4) lim n n n k 3 k= n k= 3 k n () lim n n k= n + k (3) lim n n k= n + k 3.7 () (3) b a (x a) 7 (b x) dx () dx (4) x 3 + x 3 x x + dx 4 (x )(x + ) dx () b a (x a) 7 (b x) dx = b a ( (x a) 9 (b a)(x a) 8 + (b a) (x a) 7) dx [ = (x a) (b a)(x a) (b a) (x a) 8 ] b a = (b a). 36

70 6 3 () t = x + (3) (t ) t dt = t x(x + ) dx = [ 5 t5 4 ] 3 t3 + t = 5 (4 6). = ( x x ) dx x + [ log x log(x + ) = 3 log log 5. ] (4) 4 (x )(x + ) = x (x + ) x + 3. () (4) (7) 3 [ 4 (x )(x + ) dx = log(x ) + ] 3 log(x + ) x + = log 3 6. (3x + ) 4 x + dx () dx (3) x 3 x x + dx (5) x (x + ) dx dx (6) x + x + a ( a x) dx x + x + x + dx

71 F (x) = () (4) x g(x) () () d dx (3) x f(t) dt () d dx x x f(t) dt (3) f(t) dt x f(t) dt g(x) x (x t)f(t) dt f(t) dt = [F (t)] x = F () F (x) d dx x f(t) dt = d F (x) = f(x). dx f(t) dt = d dx F (x ) = f(x )(x ) = xf(x ). x (x t)f(t) dt = d d (xf (x)) dx dx x tf(t) dt = F (x) + xf(x) xf(x) = F (x). (4) d dx g(x) f(t) dt = d dx (F (g(x)) F ()) = f(g(x))g (x). 3. F (x) = () d dx () dn dx n (3) d dx x x g (x) g (x) (x t) f(t) dt. x (x t) n f(t) dt n f(t) dt f(t) dt g (x), g (x)

72 () () c π. f(x) dx = c f(sin x) dx = f(c x) dx. π f(cos x) dx. (3) f(x) (f( x) = f(x)) c (4) f(x) (f( x) = f(x)) (5) π xf(sin x) dx = π π f(sin x) dx. c c f(x)dx = c c f(x) dx =. f(x) dx. 3.9 () x dx () (3) π sin x dx sin mx sin nx dx m, n () x = sin t, t π x = cos t, dx = cos t dt π/ x dx = cos t dt π/ = ( + cos t) dt = [t + ] π/ sin t = π 4.

73 () (sin x) = x sin x dx = [ x sin x ] x dx x = π + [ x ] = π. (3) sin mx sin nx = {cos(m n)x cos(m + n)x} (i) m ±n π π sin mx sin nx dx = (cos(m n)x cos(m + n)x) dx = [ ] π sin(m n)x sin(m + n)x =. m n m + n (ii) m = n( ) π (iii) m = n( ) 3.4 () (3) π π π sin mx dx = ( cos mx) dx = [ ] π sin mx x = π m. sin mx sin( m)x dx = log x dx () π π sin mx dx = π. cos mx cos nx dx n, m cos mx sin nx dx n, m

74 n π () sin n x dx = () π (3) I(n, m) = sin n x cos x dx = n +. π (n )(n 3) 3 n(n ) 4 (n )(n 3) 4 n(n ) 5 3 π sin n x cos m x dx (m ) I(n, m) = I(n, m ) I(n +, m ) (n, ) (n 3, ). () I n = π sin n x dx I n = [ cos x sin n x ] π/ + (n ) = (n ) π π ( sin x) sin n x dx = (n )I n (n )I n. cos x sin n x dx I n = n n I n π sin n x dx = (n )(n 3) 3 I n(n ) 4 (n )(n 3) 4 I n(n ) 5 3 (n, ) (n 3, ).

75 I = π/, I = () t = sin x (3) 3.5 () π I(n, m) = π sin n x( sin x) cos m x dx = I(n, m ) I(n +, m ). cos n x dx () π sin 5 x cos 3 x dx (3) π sin 3 x cos 5 x dx 3. () f(x) [a, b] g(x) [a, b] ξ (a, b) b a f(x)g(x) dx = f(ξ) b a g(x) dx. () f(x) [a, b] C g(x) [a, b] ξ (a, b) b a ξ b f(x)g(x) dx = f(a) g(x) dx + f(b) g(x)dx. a ξ () g(x) g(x) = ξ (a, b) b a g(x) dx

76 68 3 g(x) min f(x) a x b b g(x) dx a b f(ξ) = a b g(x)dx a ξ (a, b) () G(x) = x a b a f(x)g(x) dx max a x b f(x) b a f(x)g(x) dx g(t) dt f(x)g(x) dx = [f(x)g(x)] b a b a f (x)g(x) dx f(b)g(b) f(x) f (x) b a f (x)g(x) dx = G(ξ) b ξ (a, b) f(b)g(b) G(ξ)(f(b) f(a)) = f(a) a f (x) dx = G(ξ)(f(b) f(a)) ξ a g(x) dx + f(b) b ξ g(x) dx. 3. f(x) [a, b] g(x) c > b lim n a f(x)g(nx) dx = c c g(x) dx b a f(x) dx.

77 g(x) = g(x) g(x) [a, b] c k (k =, ±, ±, ) n a = x < x < x < < x m < x m = b n(b a) m x i x i = c c n (i =,, m ) b a f(x)g(nx) dx = m k= xk x k f(x)g(nx) dx k =,, m j x k x k = xk (j )c n x k g(nx) dx = jc/n (j )c/n g(nx) dx = n jc (j )c g(x) dx = n ξ k (x k, x k ) (k =,,, m) = b a m f(ξ k ) k= m = = c k= f(x)g(nx) dx xk g(nx) dx x k xk f(ξ k ) g(nx) dx + f(ξ ) x k m c f(ξ k ) n k= c x a c = jc n, g(x) dx b g(nx) dx + f(ξ m ) g(nx) dx x m g(x) dx + c n f(ξ ) g(x) dx + a n f(ξ m) b g(x) dx.

78 7 3 a c, b c n 3 g(x) g + (x) = max{g(x), }, g (x) = min{g(x), } g(x) = g + (x) + g (x) 3.6 () lim b lim n a b n a b = lim = c = c a b a b n a b n a b n a b n () lim (3) lim (4) lim f(x)g(nx) dx f(x)g + (nx) dx + lim f(x) dx f(x) dx a c c n b g + (x) dx + c g(x) dx. f(x) sin nx dx = π f(x) cos nx dx = π f(x) sin nx dx =. b a b sin nx cos mx dx = lim m a a b a f(x) dx. f(x) dx. b a f(x)g (nx) dx f(x) dx c g (x) dx sin nx cos mx dx =.

79 (a, b] f(x) lim y a+ b a b y f(x) dx ± f(x) dx b a f(x) dx = lim y a+ b y f(x) dx. f(x) F (x) b a f(x)dx = [F (x)] b a = F (b) lim x a+ F (x). b a f(x) dx f(a) f(a) f(x) [a, b] [a, b) x = b (a, b) (, b] f(x) lim y b b y f(x) dx ± ) f(x) dx

80 7 3 b f(x) dx = b lim y y f(x) dx. f(x) F (x) b f(x) dx = [F (x)] b = F (b) lim y F (y). b f(x) dx [a, ) (, ) a < c < b f(x) x = c (a, c), (c, b) c a f(x) dx, b c f(x) dx (a, b) b a f(x) dx = c a f(x) dx + b c f(x) dx. 3. f(x), g(x) (a, b) a = b = () g(x) f(x) b a f(x) dx b a g(x) dx

81 () g(x) f(x) (3) g(x) f(x) b f(x) dx b a a b f(x) dx = b a a g(x) dx g(x) dx = 3. f(x), g(x) (a, b) a = b = () b a f(x) dx () g(x) f(x) (3) g(x) f(x) b b a f(x) dx g(x) dx b a a b g(x) dx = b a a f(x) dx f(x) dx = 3.3 () α x α dx () x α dx () y > y x α dx = α (y α ) (α ) log y (α = ).

82 74 3 y (α ) x α dx = α (α > ). () < y < x α dx = α ( y α ) (α ) y log y (α = ). y ( α) x α dx = α (α < ). 3.7 α x α dx () dx () dx (3) x x (4) dx (5) x + x dx x( x) dx () x = x dx = [log x ] = lim log( x) =. x

83 () x = 3 dx = [ x ] 3 = ( lim x x ) =. x + (3) < a < b < x = sin t, < t < π dx = cos t sin t dt b a x( x) dx = sin b sin a dt = (sin b sin a). lim a + sin a =, lim b sin b = π π (4) x = x dx = x dx + x dx. x dx = [log x] = lim log x =. x + dx = x (5) + x dx = [ tan x ]. lim x tan x = π. lim x tan x = π π

84 () (4) 3.9 () e x sin x dx () dx (3) 4 x x x dx cos x dx (5) dx (n ) x x n log x log x dx (n ) () sin xe x dx xn 3.5 s > Γ (s) = x s e x dx. s > x s e x dx = x s e x dx + x s e x dx. < x e x > < x s e x < x s. x s e x dx < x s dx = [ s xs ] = s

85 lim x xs+ e x = M > x x s e x Mx [ x s e x dx M x dx = M ] = M <. x 3. () Γ () =, Γ (s + ) = sγ (s) (s > ). () Γ (n + ) = n! ( n ). 3. lim n n! x n e x dx. 3. () s I s = ( + x) s e x dx () I s = + si s n n! (3) n I n = k! k=

86 s >, t > B(s, t) = x s ( x) t dx s >, t > s < t < < c < x s ( x) t dx = c x s ( x) t dx + c < x c x s ( x) t max{, ( c) t }x s. x s ( x) t dx c x s ( x) t dx max{, ( c) t } c x s dx = max{, ( c) t } s cs c < x < x s ( x) t max{, c s }( x) t x s ( x) t dx max{, c s } c c ( x) t dx = max{, c s } t ( c)t <.

87 () B(s, ) =. () B(s, t) = B(t, s). s (3) B(s, t) = t B(s +, t ). s Γ (s)γ (t) (4) B(s, t) = (6) B(s, t) = ( (8) Γ n + ). (5) B(s, t) = Γ (s + t) π sin s θ cos t θ dθ = (n)! n n! π. x s ( + x) ( ) (7) Γ = π s+t dx 3.4 s >, t I(s, t) = x s ( log x) t dx. () () I(s, t) = t I(s, t ) s + Γ (t + ) (3) I(s, t) = (s + ) t+ n! (4) n I(n, n) = (n + ) n+ (5) x x = e x log x x x dx = k= k k (6) x x dx

88 µ σ < x <, < µ < σ > f(x) = ) (x µ) exp (. πσ σ () 3.6 a n = () f(x) dx () π xf(x) dx (3) x f(x) dx sin n x dx (n ) a n dx () ( + x ) n ( x ) n dx

89 () f(x) (, ) f() = lim x + f(x), f( ) = lim f(x) x a, b > f(ax) f(bx) x dx = (f() f( )) log b a () f( ) = lim f(x) x f(ax) f(bx) x dx = f() log b a. (3) f() = lim f(x) x + f(x) x f(x) x dx dx f(ax) f(bx) x dx = f( ) log b a. () < a < b < m < M <

90 8 3 M m f(ax) f(bx) x dx = = = M m am am bm am f(ax) dx x f(x) x f(x) x dx dx M m bm bm bm am f(bx) x f(x) x f(x) x ξ m (am ξ m bm) ξ M (am ξ M bm) dx dx dx M m f(ax) f(bx) x dx = f(ξ m ) bm am bm x dx f(ξ M) am = (f(ξ m ) f(ξ M )) log b a. x dx m +, M () f(ax) f(bx) x dx = lim m + m bm = lim m + () (3) () 3.7 () () am f(ax) f(bx) x f(x) x < a < b sin ax sin bx dx x cos ax cos bx dx x dx dx

91 (3) (4) tan ax tan bx dx x x a x b dx log x

92

93 xy () x = t y = t t < () x = t y = t < t < (3) x = t y = αt < t < α () t y = x x < 4. () t y = x < x < 4. (3) t y = αx < x < 4.3 α = 3

94 : 4. () : 4. () : 4. (3)

95 xy () x = t y = t t () x = t y = t t 4. xy x = t cos π t, y = t sin π t, < t. 4. f(x, y) = x + y + 3 () f(, ) f(, ) () z = f(x, y) () f(, ) = 3 f(, ) = 5 () x y + z = 3 3 (,, 3) (, 3 ), (3,, ) 4.3 f(x, y) = x + y () f(, ) f(r cos θ, r sin θ) () z = f(x, y)

96 z = f(x, y) () z = f(x, y) = x y () z = f(x, y) = x4 4 x3 3 x + y 4.3 u(x, y) = x y, v(x, y) = xy Φ(x, y) = (u(x, y), v(x, y)) () Φ(, ) Φ(, ) () y = 3x x Φ (3) Φ () Φ(, ) = (, ) Φ(, ) = ( 4, ) () y = 3t x = t t u = u(t, 3t) = t, v = v(t, 3t) = 3 t v = 3u u (3) x = r cos θ y = r sin θ r > θ < π u = u(r cos θ, r sin θ) = r cos θ, v = v(r cos θ, r sin θ) = r sin θ u + v = r 4

97 u(x, y) = x y, v(x, y) = x + y x + y Φ(x, y) = (u(x, y), v(x, y)) () Φ(, ) Φ(, ) () y = x < x Φ (3) y = 3x < x Φ (4) x + y = y Φ (5) x + y = 4 Φ R P Q d(p, Q) P = (p x, p y ) Q = (q x, q y ) d(p, Q) = (p x q x ) + (p y q y ) 4. R {P n } P ε > N n > N n d(p n, P) < ε 4.4 ( P n = ) P = (, ) ε > n > N n, n n d(p n, P ) < ε N

98 9 4 n > 4.6 d(p n, P ) = n + n < ε ε N > { ( )} cos nπ P n = n, sin nπ n ε N P = (, ) 4.5 (x, y) (, ) f(x, y) = x y x + y lim (x, y) (, ) f(x, y) r > θ < π x = r cos θ y = r sin θ d((, ), (r cos θ, r sin θ)) = r (x, y) (, ) r f(r cos θ, r sin θ) = r cos θr sin θ r cos +r sin θ = r cos θ sin θ < r θ r f(r cos θ, r sin θ) lim f(x, y) = (x, y) (, )

99 (x, y) (, ) (, ) f(x, y) = x y (x ) (y ) x (x ) + y (y ) lim (x, y) (, ) f(x, y) lim f(x, y) (x, y) (, ) 4.6 (x, y) (, ) f(x, y) = x y x + y lim (x, y) (, ) f(x, y) α y = αx t x = t y = αt t f(t, αt) = t αt t + αt = α + α lim f(x, y) = α (x, y) (, ) + α α lim (x, y) (, ) f(x, y)

100 () () lim xy log(x + y ), (x, y) (, ) lim (x, y) (, ) xy x + y x 3 + y 3 (x, y) (, ) f(x, y) = x + y (x, y) = (, ) f(x, y) (a, b) f(x, y) (a, b) x f(a + h, b) f(a, b) lim h h x f x (a, b) f(x, y) (a, b) y f(a, b + h) f(a, b) lim h h y f y (a, b)

101 () () f(x, y) = xy f(x, y) = x + y. () f(h, ) f(, ) lim h h = lim h h = x f x (, ) = f(, h) f(, ) lim h h = lim h h = y f y (, ) = () f(h, ) f(, ) h h lim h + h = = h h = h h lim h h h = x f(, h) f(, ) h = h y h = h h

102 () () (3) f(x, y) = x 3 + x y 3 + x y + xy + y, f(x. y) = x y y x, f(x, y) = e x sin y. 4.3 f(x, y) (a, b) f(x, y) (a, b) (cos θ, sin θ) f(a + t cos θ, b + t sin θ) f(a, b) lim t t 4.8 x 3 y 3 (x, y) (, ) f(x, y) = x + y (x, y) = (, ) f(x, y) = (x ( y)(x + xy + y ) = (x y) + xy ) x + y x + y

103 f(r cos θ, r sin θ) f(, ) r ( ) = r(cos θ sin θ) + r cos θ sin θ r r = (cos θ sin θ)( + cos θ sin θ) df (, ) (cos θ, sin θ) = (cos θ sin θ)( + cos θ sin θ) df (, ) (cos, sin ) = df (, ) (cos π, sin π) = f x (, ) = ( df (, ) cos π, sin π ) ( = df (, ) cos 3π, sin 3π ) = f y (, ) = f(x, y) 4.4 f(x, y) (a, b) f(x, y) (a, b) A B f(a + h, b + k) f(a, b) = Ah + Bk + ε h + k lim ε = (h, k) (, )

104 96 4 f x (a, b) = A f y (a, b) = B f x (a, b)dx + f y (a, b)dy f (a, b) df 4.9 f(x, y) = x + y (x, y) = (a, b) f(a + h, b + k) f(a, b) ah bk h + k = (a + h) + (b + k) a b ah bk h + k = h + k ((h, k) (, )) f(x, y) f(x, y) (a, b) (a, b) df (a, b) (cos θ, sin θ) = f x (a, b) cos θ + f y (a, b) sin θ

105 () () f(x, y) = log( + x + y ), f(x, y) = x e x cos y. 4.5 z = f(x, y) D I x = x(t) y = y(t) (x(t), y(t) D (t I) f(x(t), y(t)) I dz dt = f x(x(t), y(t))x (t) + f y (x(t), y(t))y (t) 4. f(x, y) C (a, b) x(t) = a + t cos θ y(t) = b + t sin θ z(t) = f(x(t), y(t)) df (, ) (cos θ, sin θ) = lim t z(t) z() t = f x (a, b) dx dt + f y(a, b) dy dt = f x (a, b) cos θ + f y (a, b) sin θ.

106 f(x, y) = e xy x = log u + v y = tan v u f u f v f x = ye xy f y = xe xy x u = u u + v y u = v u + v f u = u yexy u + v v xexy u + v = utan v u v log u + v u + v x v = y v = v u + v u u + v e log u + v tan v u f v = v yexy u + v + u xexy u + v vtan v = u + u log u + v u + v e log u + v tan v u 4. x = r cos θ y = r sin θ.

107 x x ( ) r θ y y = cos θ r sin θ. sin θ r cos θ r θ (x, y) (r, θ) = r cos θ + r sin θ = r. 4.5 a b c d () x = au + bv y = cu + dv, () x = u cos a v sin a y = u sin a + v cos a, (3) x = uv y = u uv. 4.6 (u, v) = (f(x, y), g(x, y)) U C U (x, y ) (x, y ) V (u, v ) = (f(x, y ), g(x, y )) W W u = f(x, y) v = g(x, y) W C

108 4 4.3 (u, v) = Φ(x, y) = (x + y, xy) () Φ (x, y) () () Φ (3) (x, y) (u, v) Φ (4) (3) () (5) () (u, v ) Φ xy (6) (3) (u, v ) Φ xy (7) (u, v ) (3) Φ () () u u (u, v) (x, y) = x y v v = x y = x y y = x (u, v) = Φ(x, x) = (x, x )

109 4.. v = u 4 (3) u v u = x + y v = xy x y y = u x v = x(u x) u 4v (4) () (3) (5) x v = x(u x) u 4v = x = u ( u, u ) (6) (5) u 4v > ( u ± u 4v u ) u 4v, (7) (u, v ) v = u R < r < R r 4 (u, v ) (6) ( Φ u + u 4v u ) u 4v (u, v) =, Φ (u, v) = ( u u 4v, u + u 4v )

110 4 4.6 (u, v) = Φ(x, y) = (y xy, xy) () Φ (x, y) () () Φ (3) (x, y) (u, v) Φ (4) (3) (u, v ) Φ xy (5) (u, v ) (3) Φ 4.4 xy x y (x, y) (, ) f(x, y) = x + y (x, y) = (, ) f x (, y) f y (x, ) f x (, ) f y (, ) f xy (, ) f yx (, ) y f x (, y) = lim x f(x, y) f(, y) x y = f x (, ) = lim x f(x, y) f(, ) x f xy (, ) = lim y f x (, y) f x (, ) y = lim x y x y x + y = y = lim x x = lim y y y = =

111 4.. 3 x f y (x, ) = lim y f(x, y) f(x, ) x x = f y (, ) = lim y f(x, y) f(, ) y f yx (, ) = lim x f y (x, ) f y (, ) x = lim y x x y x + y = x = lim y y = lim x x x = = 4.7 () () (3) 4.8 () () f f x f y f xx f yy f xy f yx f(x, y) = xy x y, f(x, y) = e xy, f(x, y) = sin(x + y ). f f x + f y f(x, y) = log(x + y ), f(x, y) = e x (x cos y y sin y).

112 f(x, y) = e x sin y f x (x, y) = e x sin y f xx (x, y) = e x sin y f y (x, y) = e x cos y f yy (x, y) = e x sin y f xy (x, y) = f yx (x, y) = e x cos y f(x, y) = f(, ) + f x (, )x + f y (, )y + {f xx(, )x + f xy (, )xy + f yy (, )y } + = y + xy () () f(x, y) (a, b) f(x, y) = x + xy 3y x + 5y +, f(x, y) = e x+y.

113 f(x, y) (a, b) C f(a, b) = f y (a, b) a I x C y = ϕ(x) () b = ϕ(a), () f(x, ϕ(x)) = (x I), (3) ϕ (x) = f x(x, ϕ(x)) f y (x, ϕ(x)). 4.6 x 3 + y 3 3xy = f(x, y) = x 3 + y 3 3xy f x = 3x 3y f y = 3y 3x f x = f y = x = y = x 4 (x, y) = (, ) (, ) (, ) (, ) (a, b) (a, b) (, ) f y (a, b) y x dy dx = f x f y = a b b a

114 6 4 (a, b) y b = a b (x a), b a (a b)x + (b a)y = ab f x (a, b) 4. () () y x dy dx x 3 y 3 + y x =, x y y x =. 4.8 f(x, y) (a, b) C (a, b) f f x (a, b) = f y (a, b) = 4.9 f(x, y) (a, b) C f (a, b) 4. f(x, y) (a, b) C (a, b) f H(a, b) = f xx (a, b)f yy (a, b) {f xy (a, b)} () H(a, b) > f (a, b) f xx (a, b) > f xx (a, b) <

115 () H(a, b) < f (a, b) () () () f x = x f y = y (, ) f xx = f yy = f xy = H(, ) = f xx (, )f yy (, ) {f xy (, )} = 4 < f () f x = x 3 x x = x(x )(x + ) f y = y (, ) (, ) (, ) f xx = 3x x f yy = f xy = H(, ) = f xx (, )f yy (, ) {f xy (, )} = 4 < (, ) H(, ) = f xx (, )f yy (, ) {f xy (, )} = 3 > f xx (, ) = 3 > (, ) f(, ) = 5 H(, ) = f xx (, )f yy (, ) {f xy (, )} = 6 > f xx (, ) = 6 > (, ) f(, ) = 8 3

116 () () (3) f(x, y) = x 3 + y 3 3xy, f(x, y) = x + y + y, f(x, y) = (x + y)e x y S x y z x+y +z = S < x < S < y < S f(x, y) f(x, y) = xy(s x y) f x = y(s x y) xy f y = x(s x y) xy f x = f y = x = y = S 3 f xx = y f yy = x f xy = S (x + y) H ( ) ( S 3, S S = f xx 3 3, S 3 = ( 3 S ) ) ( S f yy 3, S 3 ( S 3 ) > S 3 ) { ( )} S f xy 3, S 3

117 ( ) ( ) S f xx 3, S S < 3 3, S 3 f(s, y) = f(, y) = f(x, S) = f(x, ) = 4. a x y z x + y + 3z = a x 3 y z 4.3 f(x, y) = 3x + 3xy + y (x, y) x + y = f(x, y) 4. f(x, y) g(x, y) D C (x, y) g(x, y) = f(x, y) Φ(x, y, λ) = f(x, y) + λg(x, y) Φ x (x, y, λ ) = Φ y (x, y, λ ) λ 4.9 f(x, y) = x + y (x, y) x + xy + y = f(x, y) g(x, y) = x +xy+y Φ(x, y, λ) = f(x, y) λg(x, y) Φ x (x, y, λ) = x λ(x + y) = Φ y (x, y, λ) = y λ(x + y) = g(x, y) =

118 4 ( (x, ( y) = (, ) (, ) λ = (x, y) = 3, ), ) λ = (x, y) g(x, y) = f(x, y) (x, y) (x, ( ( y) = (, ) (, ) (x, y) = 3, ), ) f(x, y) = x + y (x, y) (x + y ) = (x y ) f(x, y)

119 ϕ (x) ϕ (x) [a, b] ϕ (x) ϕ (x), x [a, b] D = {(x, y) a x b, ϕ (x) y ϕ (x)} [c, d] ψ (y) ψ (y) ψ (y) ψ (y), y [c, d] D = {(x, y) c y d, ψ (y) x ψ (y)} 5. 3 y = x = y = π x D

120 5 { D = (x, y) { = (x, y) x, y π } x y π, π y x 5. D = {(x, y) y x } 5. ( ) () D = {(x, y) a x b, ϕ (x) y ϕ (x)} f(x, y) b ( ϕ (x) ) f(x, y) dx dy = f(x, y) dy dx. D a ϕ (x) () D = {(x, y) c y d, ψ (y) x ψ (y)} f(x, y) d ( ψ (y) ) f(x, y) dx dy = f(x, y) dx dy. D b a dx 5. ϕ (x) ϕ (x) c f(x, y) dy, d c ψ (y) dy ψ (y) ψ (y) }. f(x, y) dx D 3 y = x = y = π x I = D cos y dx dy x

121 D = { (x, y) x, y π x } 5. ( π ) x I = cos y x dy dx = = [ x = [ x sin y x x dx ] =. ] π x cos y (, ) D x cos y x x x y 5. () () (3) D D D dx (x + y) dx dy (D : x, y ) e x y dx dy (D : y x ) x dx dy (D : x + y x) 5.3

122 4 5 () () (3) (4) D D D D x y dx dy (D : y x ) sin(x + y) dx dy ( D : x, y, x + y π ) e y x dx dy (D : x, y x) x y dx dy (D : x + y ) 5.3 dx x f(x, y) dy = dy y f(x, y) dx. = f(x, y) dx dy D = {(x, y) x, y x} D D = {(x, y) y, y x } = f(x, y) dx dy 5.4 () () (3) a a D e x dx f(x, y) dy dx dy βx αx y+a a y a f(x, y) dy ( < α < β) f(x, y) dx (a > )

123 5.. 5 (4) dx +x x f(x, y) dy xy a = (a, a ) b = (b, b ) D D = {sa + tb s, t } D D ( ) D = det a a b b a b θ ( < θ < π) cos θ = a b a b

124 6 5 D = a b sin θ = a b ( ) a b a b = a b (a b) = (a + a )(b + b ) (a b + a b ) = a b + a b a a b b = a b a b. 5.5 k h > ( a A = c ) b d xy (, ) (ak, ck) (ak + bh, ck + dh) (bh, dh) D D = det A kh

125 uv xy x = au + bv y = cu + dv (x, y) = ad bc (u, v) uv (, ) (k, ) (k, h) (, h) (k h > ) E (, ) (ak, ck) (ak + bh, ck + dh) (bh, dh) D D dx dy = E (x, y) (u, v) du dv 5.5 dx dy = D = ad bc kh D E (x, y) (u, v) du dv = ad bc du dv E = ad bc du dv = ad bc kh E 5.3 ( E = {(u, v) a u b, c v c} uv x = ϕ(u, v) y = ψ(u, v)) E

126 8 5 C (u, v) (x, y) E xy D (ϕ, ψ) (u, v), (u, v) E (u, v) (x, y) E D D f(x, y) f(x, y) dx dy = f(ϕ(u, v), φ(u, v)) (ϕ, ψ) (u, v) du dv D E 5.6 D : (a b > ) x a + y b x = au y = bv E : u + v D (x, y) (u, v) = ab = ab E π 5.3 D = dx dy = ab du dv = ab du dv = abπ D E E

127 x = u + v, y = u v dx x {(x + y) + x y} dy 4.5() (x, y) (u, v) = ( ) = xy D = {(x, y) x, y x} E = {(u, v) u + v, u v u v} E D E = {(u, v) u, u v u} 5.3 {(x + y) + x y}dx dy = {(u) + v} du dv D = 4 = 4 = 4. E du u u 4u 3 du (u + v)dv

128 5 5.6 x = u + v, y = u v dx x (x + y)e (x y) dy 5.8 dx dy (D : x + y 6) 5 x y D rθ E = {(r, θ) r 4, θ π} (x, y) D 4. (r, θ) = r 5.3 D dx dy 5 x y = = = E π π r dr dθ 5 r 4 dθ r 5 r dr [ 5 r ] 4 dθ = π(5 3) = 4π. xy (, ) {(r, θ) r >, θ = } {(r, θ) r >, θ = π} {(x, y) x, y = } 5.8

129 a > () a x y dx dy (D : x + y a ) () (3) D D D e x +y dx dy (D : x + y a, x, y) xy dx dy (D : x + y ax, y) dx dy (D : y < x ) x y D f(x, y) = x y x = y D n =,, D n : n x, y x n D n D {D n } D D D n D n+, D n = D n=

130 5 n I n = f(x, y)dx dy D n I n = = = = = = n n D n n dx dx dy x y x n [ (x y) { [ 3 x 3 { 3 x ( n (x y) dy ] x n ) } ( ) ] x n n ( ) n + 3 I = lim n I n = 4 3 dx dx ( n ) } 3 f(x, y) D D {D n } f(x, y)dx dy = lim f(x, y)dx dy D n D n

131 f(x, y) D 5.8 () () (3) D D D dx dy (D : x, y ) ( + x + y) 3 dx dy x + y (D : x >, y x ) e xy dx dy (D : x, < a y b) 5.4 f(x, y, z) V V (x, y, z) x a x a x V V x = {(y, z) (x, y, z) V } V f(x, y, z)dx dy dz = a a { V x f(x, y, z)dy dz } dx 5. dx dy dz (V : x + y + z+, x, y, z ) V ( + x + y + z) 3

132 4 5 V (x, y, z) x x x V V x = {(y, z) y + z x, y, z } V x = {(y, z) y x, z x y} dx dy dz ( + x + y + z) = 3 V = = = = = { dx dx dx [ { V x x x x = log 5 6. dy dz ( + x + y + z) 3 x y } dx dz dy ( + x + y + z) 3 [ ] z= x y dy ( + x + y + z) z= { ( + x + y + z) } dy 8 ( + x + y + z) 8 y ] y= x y= 4 } 8 ( x) + dx ( + x) dx

133 x dx dy dz (V : x + y + z a ) V x = r sin θ cos ϕ, y = r sin θ sin ϕ, z = r cos θ (r > ) x x x r θ ϕ sin θ cos ϕ r cos θ cos ϕ r sin θ sin ϕ y y y det = det sin θ sin ϕ r cos θ sin ϕ r sin θ cos ϕ cos θ r sin θ r z r θ z θ ϕ z ϕ = r sin θ rθϕ E = {(r, θ, ϕ) r a, θ π, ϕ π} V x dx dy dz = (r sin θ cos ϕ) r sin θ dr dθ dϕ V = a E = 4π 5 a5 r 4 dr π π sin 3 θ dθ cos ϕ dϕ

134 () dx () (3) V V x dy x+y e x+y z dz dx dy dz x + y + z (V : x + y + z a ) (x + y + z)dx dy dz (V : x + y + z a, z ) 5.5 D xy f(x, y) g(x, y) D f(x, y) g(x, y), (x, y) D z = f(x, y) z = g(x, y) V = {(x, y, z) (x, y) D, f(x, y) z g(x, y)} V V = {g(x, y) f(x, y)} dx dy. D 5. a (a > ) V : x + y + z a

135 (A) 5.5 D : x a, y a x V = 8 a x y dx dy D xy V = 8 π/ = 4 3 πa3 dθ a a r r dr (B) 5.4 V (x, y, z) x a x a x V V x = {(y, z) y + z a x }, a x V = dx dy dz = = = a a a a a a V = 4 3 πa3 dx dy dz V x V x dx π(a x ) dx

136 8 5 (C) V = dx dy dz = = a V E r dr r sin θ dr dθ dϕ π π sin θ dθ = a3 3 π = 4 3 πa3. E = {(r, θ, ϕ) r a, θ π, ϕ π} 5. a + y b + z c (a, b, c > ) x dϕ 5.3 B(p, q) = Γ (p)γ (q) Γ (p + q). ( ) a > D(a) = {(x, y) < x a, < y a}, E(a) = {(x, y) x + y a, x >, y > }

137 E(a) D(a) E( a) f(x, y) = 4e x y x p y q (x, y > ) f(x, y) > f(x, y) dx dy f(x, y) dx dy (5.) E(a) D(a) E( a) f(x, y) dx dy 5., 5. f(x, y) dx dy = e x x p e y y q dx dy D(a) = ( D(a) a ) ( e x x p dx Γ (p)γ (q) (a ) I(a) = f(x, y) dx dy E(a) = 4 E (a) a e r (r cos θ) p (r sin θ) q r dr dθ. ) e y y q dy E (a) = {(r, θ) < r a, < θ < π } 5.

138 (6) I(a) = = ( ( a a e r r p+q dr) ( π/ ) e r r (p+q) dr B(p, q) cos p θ sin q θ dθ ) Γ (p + q)b(p, q) (a ). (5.) a Γ (p + q)b(p, q) Γ (p)γ (q) Γ (p + q)b(p, q) 5. f(x), g(x) (, a], (, b] a b f(x) dx, g(y) dy (a, b > ). f(x)g(y) dx dy D D = {(x, y) < x a, < y b} a b f(x)g(y) dx dy = f(x) dx g(y) dy. 5. D e x x p dx = Γ (p). 5.3 e x dx = π.

139 3 6 e x x y x y n y, y,, y (n) F (x, y, y, y,, y (n) ) = y n n F n + F y y (k) ( k n). F = x y n y. n n

140 3 6 n f, g x, y y x x, y dy dx = y = f(x)g(y) C g(y) dy = f(x) dx + C y ϕ(y) ϕ(y) dy = ϕ(y) dy dx dx

141 d ϕ(y) dy = dy d ϕ(y) dy = dy dx dx dy dx ϕ(y) g(y) dy dx = f(x) x C dy f(x) dx = g(x) dx dx + C = g(y) dy + C 6. xy = y 6. y = 3y y + y = L(y) y + p(x)y = q(x) L(y) y + p(x)y = q(x) = L(y) y, y L(y) = C, C

142 34 6 y = C y + C y L(y) = L(C y + C y ) = C L(y ) + C L(y ) = C + C =. e x y + p(x)y = q(x) { } y = e R p(x)dx e R p(x)dx q(x)dx + C y + p(x)y = y = Ce R p(x) dx C d dx er p(x) dx = p(x)e R p(x) dx d { ye R } p(x) dx = {y + p(x)y}e R p(x)dx = e R p(x) dx q(x) dx

143 C ye R p(x) dx = e R p(x) dx q(x) dx + C e R p(x) dx a, b y + ay = b y + xy = x 6.6 y + p(x)y = q(x) y C y = Cf(x) + g(x) C y Cf(x) y + p(x)y = g(x) y = Cf(x) + g(x) C C i ( i 3) y i = C i f(x) + g(x) ( i 3) y y 3 y y 3 = C C 3 C C 3 y y

144 y + p(x)y = q(x) y + p(x)y = q(x) 6. y, y y + p(x)y = c y = cy y = cy y + p(x)y = q(x) y y = z + y y z z + p(x)z = C y = Ce R p(x) dx + y y + p(x)y = q(x) y = z + y y + p(x)y = q(x) z + p(x)z = 6. C z = Ce R p(x) dx

145 y = z + y C y = Ce R p(x) dx + y 6. q(x) 6. q(x) Riccati ( ) 6. y + x y = 3x 6.3 y + p(x)y = q(x) y, y x = x y (x ) = y (x ) y y x = x y (x ) < y (x ) y, y I y (x) < y (x) 6.4 y + p(x)y = q(x) y, y y = C(y y ) + y C

146 38 6 y + p(x)y = q(x) Bernoulli( ) Riccati Bernoulli y + p(x)y = q(x)y a a a =, 6. a, 6.4 a a, Bernoulli ( ) y a = e R ( a)p(x) dx e R ( a)p(x) dx ( a)q(x)dx + C C Bernoulli y a y a y + p(x)y a = q(x)

147 y z z = y a z = ( a)y a y z z + ( a)p(x)z = ( a)q(x) p(x), q(x) ( a)p(x), ( a)q(x) xy + y = x 3 y 3 xy + y = y log x Bernoulli Riccati y + p(x) + q(x)y + r(x)y = Bernoulli p(x) Riccati a = Bernoulli 6. y p(x) Riccati

148 Riccati y + p(x) + q(x)y + r(x)y = y = y y = z + y y z Bernoulli z + {q(x) + r(x)y (x)}z = z y ( ) (y y ) = e R {q(x)+r(x)y (x)} dx e R {q(x)+r(x)y (x)} dx dx + C Riccati C y = z + y Riccati (z + y ) + p(x) + q(x)(z y ) + r(x)(z + y ) = y Riccati z Bernoulli z + {q(x) + r(x)y (x)}z = z 6.4 a = (y y ) = z = e R {q(x)+r(x)y (x)} dx ( ) e R {q(x)+r(x)y (x)} dx dx + C C

149 Bernoulli Riccati 6.8 xy (x + x) + (x + )y y = y = x Riccati y C y = Cf (x) + f (x) Cf 3 (x) + f 4 (x) f (x)f 4 (x) f (x)f 3 (x) C Riccati f f 4 f f 3 6. i C C i y i = C if (x) + f (x) C i f 3 (x) + f 4 (x) y y 3 y y 3 / y y 4 = C / C 3 C C 4 y y 4 C C 3 C C 4 (y, y, y 3, y 4 ) (y, y, y 3, y 4 ) (C, C, C 3, C 4 ) 6. Riccati 3 y, y, y 3

150 Riccati y, y Riccati y y = z r(x) z z ( ) z + q(x) r (x) z + p(x)r(x)z = r(x) Riccati Riccati Bernoulli 6.5 Riccati 6.4 Bernoulli 6.5 Riccati

151 Bernoulli 6. a, b L(y) y + ay + by = f(x) L(y) y + ay + by = L(y) y + p(x)y + q(x)y = g(x) L(y) y + p(x)y + q(x)y = L(y) (Wronskian)

152 44 6 n I n {f i } n i= I n c, c,, c n I c f (x) + c f (x) + + c n f n (x) {f i } n i= n c, c, c n I c f (x) + c f (x) + + c n f n (x) c = c = = c n = {f i } n i= Wronskian( ) W [f,, f n ] f j i i j ( W [f,, f n ] = det f (i) j ) (x) i, j i n j n f () j (x) = f j (x) f (x) f (x) f n (x) f (x) f (x) f n(x) W [f,, f n ] =.... f (n ) (x) f (n ) f n (n ) (x) {f i } n i= Wronskian W [f,, f n ] I {f,, f n } = W [f,, f n ]

153 W [f,, f n ] = {f,, f n } {f,, f n } W [f,, f n ] n = f (x) f (x) f (x) = { { x (x > ) (x ), f (x) = (x > ) x (x ) 6.4 f f W [f, f ] W [f,, f n ] {f,, f n } y + ay + by = d dx = D, d dx = D y = dy dx = Dy, y = d y dx = D y

154 46 6 L(y) L(y) = y + ay + by = D y + ady + by = (D + ad + b)y L(y) λ K(λ) = λ + aλ + b = L(y) D 6.6 L(y) = K(λ) = λ + aλ + b = λ, λ a 4b > L(y) = (D λ )(D λ )y λ K(λ) = a 4b = L(y) = (D λ ) y λ + λ = a, λ λ = b

155 (D λ )(D λ )y = (D λ ){(D λ )y} = (D λ )(Dy λ y) = D y λ Dy λ y + λ λ y = y (λ + λ )y + λ λ y = y + ay + by = L(y) λ, λ λ λ (D λ )(D λ )y = (D λ )(D λ )y (D λ )e λx = (D λ ) e λx = (D λ )(D λ )e λx = D e x

156 λ, λ, λ λ λ (D λ )e λx y = e λx Dy (D λ ) e λx y = e λx D y (D λ )(D λ )e λx y = e λx De λx e λx Dy D (D λ )e λ x y = De λ x y λ e λ x y = λ e λ x y + e λ x Dy λ e λ x y = e λ x Dy (D λ )(D λ )e λ x y = (D λ )e λ x Dy = (D λ )e λ x e λ x e λ x Dy = e λ x De λ x e λ x Dy 6.7 (D λ ) e λ x y = e λ x D y 6.7

157 (D λ )y = e λx De λx y (D λ ) y = e λx D e λx y (D λ )(D λ )y = e λx De λx e λx De λx y 6.7 (D λ )y = (D λ )e λ x e λ x y = e λ x De λ x y

158 a, b L(y) = y + ay + by = K(λ) = λ + aλ + bλ = a 4b = K(λ) = (λ λ ) L(y) = C, C y = (C + C x)e λ x a 4b > K(λ) = (λ λ )(λ λ ) L(y) = C, C y = C e λ x + C e λ x C, C L(y) = (D λ ) y = e λx D e λx y = D e λx y = e λx y = C + C x y = (C + C x)e λ x C, C

159 6.. 5 L(y) = (D λ )(D λ )y = e λ x De λ e λ x e λ x y = De λ x e λ x De λ x y = e λ x e λ x De λ x y = C De λ x y = C e (λ λ )x e λ x y = C (λ λ ) e(λ λ )x + C y = C λ λ e λ x + C e λ x C C λ λ C y = C e λ x + C e λ x y + 6y + 9y = y 5y + 6y = e 3x, xe 3x Wronskian e x, e 3x Wronskian 6.9 a 4b L(y) y + ay + by =

160 5 6 a 4b < 6. a, b a 4b < L(y) y + ay + by = y C, C y = (C cos qx + C sin qx)e px p, q a, b p = a, q = 4b a q > λ + aλ + b = λ = a ± 4b a i = p ± iq α = p + iq 6.6 L(y) = D y + ady + by = (D α)(d α)y = L(y) = 6.9 A, B y = Ae αx + Be αx

161 A, B C, C A = C + C i, B = A = C C i y = Ae αx + Ae αx = ReAe αx = Re{(C + C i)(cos qx i sin qx)e px } = (C cos qx + C sin qx)e px C, C C, C C, C y = (C cos qx + C sin qx)e px y = (C cos qx + C sin qx)e px L(y) = 6.34 y y + 5y = 6.35 q {e px cos qx, e px sin qx} Wronskian

162 L(y) y + ay + by = f(x) L(y) y 3y + y = e 4x sin x C, C y = C e x + C e x + e4x ( cos x + sin x) e 4x sin x y y = e 4x (c cos x + c sin x) c, c y y y y = e 4x {(ac + c ) cos x + ( c + 4c ) sin x} y = e 4x {(5c + 8c ) cos x + ( 8c + 5c ) sin x}

163 y, y, y e 4x {5(c + c ) cos x + 5( c + c ) sin x} = e 4x sin x c + c =, 5( c + c ) = y = e4x ( cos x + sin x) L(y) = λ 3λ + λ = (λ )(λ ) = L(y) = y C, C y = C e x + C e x L(y) = e 4x sin x y y = C e x + C e x + e4x ( cos x + sin x) y L(y) = e 4x sin x C, C y L(y) = y 4y + 3y = sin x L(y) = y y 3y = x L(y) = y 3y + y = e x

164 56 6 Wronskian 6. L(y) y + p(x)y + q(x)y = r(x) L(y) = y, y Wronskian W (x) = W [y, y ] W (x) C W (x) = Ce R p(x) dx W (x) I W (x) W (x) L(y) = y, y Wronskian W [y, y ] L(y) = r(x) y 3 = y r(x)y (x) W (x) r(x)y (x) dx + y (x) dx W (x) L(y) = r(x) y = C y + C y + y 3 C, C

165 W (x) y, y L(y) = y = p(x)y q(x)y, y = p(x)y q(x)y d dx W (x) = d dx y y y y = y y y y + y y y y y y = p(x)y q(x)y p(x)y q(x)y = p(x) y y y y q(x) y y y y = p(x)w (x) W (x) = Ce R p(x) dx C C = W C W (x) L(y) = r(x) C (x), C (x) y = C (x)y + C (x)y L(y) = r(x) C, C y y = C (x)y + C (x)y + {C (x)y + C (x)y }

166 58 6 C (x)y + C (x)y = y = C (x)y + C (x)y + {C (x)y + C (x)y } C (x)y + C (x)y = r(x) C (x) C (x) C (x)y + C (x)y =, C (x)y + C (x)y = r(x) C (x), C (x) W (x) = W [y, y ] r(x)y (x) r(x)y (x) C (x) = dx, C (x) = dx W (x) W (x) L(y) = r(x) y 3 = C (x)y + C (x)y = y r(x)y (x) W (x) dx + y r(x)y (x) W (x) L(y) = r(x) y = C y + C y + y 3 C, C dx

167 L(y) = y, y W [y, y ] y = C y + C y C, C C, C y = C (x)y + C (x)y L(y) = r(x) C (x), C (x) y 3 L(y) = r(x) 6.4 L(y) y x x y + x y = x x 6.4 L(y) y 5y + 4y = x x y xy + y = x x L(y) y x y + x y = W (x) = W [y, y ] L(y) = y y x

168

169 6. () n + ( )n n + n + = () n( n + n n) = = = n + + n + n + n(n + ) (3) n = = (4) n n(n + )(n + ) = = ( 6 3 (5).5(4) n) n ( ) n ( ) n n n = = n n ( = + ) n ( + ) = e ( n n (6) ) n ( = ) n ( + n (5) = e n n n) e =. () n =,, a n < n = a n < a n+ = a n + = a n an + + < {a n } a = 3 > a a n+ > a n

170 6 a n+ a n+ = a n+ a n an+ + + a n + > {a n } () () {a n } α = lim n a n α = α + α > α =.3 () x 3 8 = (x )(x x +x+4) lim x x 3 8 = lim x x + x + 4 = x x + () lim x x ( x x + )( x + x + ) = lim x (x )( x + x + ) = lim = x x + x + 3 (3) lim x = (k ) lim xk 3 x = lim x + = x x x x + + x (4) lim = lim x ( x = lim x + x ) + x x x x 3 x x + x + + x x x = ( ) cos x sin x sin x.4 () lim = lim = lim = x x x x x x ( () x = t lim + x ( = lim x x) ) t t t ( ) t ( t = lim = lim + ) t = e t t t t x

171 63.5 () ( ex e x = ex + e x x x x.) ( () e a ( + a ) { x ( = + a } a a ) x x)x (3) e ( log( + sin x) sin x log( + sin x) x =.) x sin x x + 3x + a.6 lim lim(x + 3x + a) = x x x lim(x + 3x + a) = 4 + a a = 4 x x + 3x 4 lim = lim(x + 4) = 5 f(x) x x x b = 5.7( f(x) = x cos x f(x) f() = < π ) f = π ( π ) > f() f <.3 [, π ] [ f(x) = x cos x =, π ].8 ( () y = tan x z = cot x x = tan y x = cot z = π ) ( π ) tan z tan y = tan z π < y < π y = π z tan x + cot x = y + z = π () y = sin 3 5 z = sin 4 5 sin y = 3 5 sin z = 4 5 sin y + sin z = sin y = sin z = cos z ( π ) sin y = ± cos z = ± sin z sin sin 5 = y + z = π (3) y = cosh x x = cosh y = ey + e y t = e y t xt + = t = x ± x t t = x + x e y = x + x cosh x =

(3) (2),,. ( 20) ( s200103) 0.7 x C,, x 2 + y 2 + ax = 0 a.. D,. D, y C, C (x, y) (y 0) C m. (2) D y = y(x) (x ± y 0), (x, y) D, m, m = 1., D. (x 2 y

(3) (2),,. ( 20) ( s200103) 0.7 x C,, x 2 + y 2 + ax = 0 a.. D,. D, y C, C (x, y) (y 0) C m. (2) D y = y(x) (x ± y 0), (x, y) D, m, m = 1., D. (x 2 y [ ] 7 0.1 2 2 + y = t sin t IC ( 9) ( s090101) 0.2 y = d2 y 2, y = x 3 y + y 2 = 0 (2) y + 2y 3y = e 2x 0.3 1 ( y ) = f x C u = y x ( 15) ( s150102) [ ] y/x du x = Cexp f(u) u (2) x y = xey/x ( 16) ( s160101)

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