i 6 3

ii 3 7 8 9 3 6

iii 5 8 5 3 7 8

v...................................................... 5.3....................... 7 3........................ 3.................3.......................... 8 3 35 3............................. 35 3.............................. 58 3.3............................ 7 4 85 4...................... 85 4......................... 9 4.3......................... 4 5 5........................

vi 5........................ 5 5.3......................... 6 3 6......................... 3 6.................... 43 6.3.................. 54 6

.. {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

.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 + + 3 3 + + 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

.. 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) + + 3 3 + + 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 ( + 3 + + n ) ( n ) n. {a n } a =, a n+ = ) (a n + an (n =,, ) () {a n } () lim n a n

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

.. 5..6 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

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)

.3. 7.4 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

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) =.4.5.6 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 =.

.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 =

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 + )

.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 3 5 + 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 ).

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 ))

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

.. 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

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)

.. 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 > )

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)

.. 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 < ).

.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)

...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 3...9 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)

. 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 )

.. 3.9 () 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

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

.. 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!

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 < )

.. 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 ( )

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 ±

.3. 9. 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

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 < )

.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

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 > )

.3. 33.3 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 +

35 3 3. 3. 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.

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.

3.. 37 () (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 () =.

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)

3.. 39 3.3 () (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 = 5 5 + (x + 3) 5+ = 8(x + 3). 4 (3) dx = x + 3 + 3) + = x + 3. + (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 )

4 3 3. () x () (3) ( 3x) 3 (5) (6) 8 4x x log x x x + (4) x + x + 3.4 () (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

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)

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 + 3 8 cos x sin x + 3 8 x.

3.. 43 3.3 () x(x + ) () x sin(x + ) (3) cos x sin 4 x (4) cos 5 x (5) (log x) 3 3.5 () () (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

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 =

3.. 45 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.

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 + )

3.. 47 3.7 () (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 3. 3.6 x + x 3 x 3 () () (3) x + 3 + x + x 8

48 3 3.8 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

3.. 49 3.9 () 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.

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 < ).

3.. 5 3. 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

5 3 3. 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

3.. 53 () 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 t7 + 6 5 t5 + t 3 + 6t + log t t + = 6 7 x 6 x + 6 6 x5 + x + 6 6 x + 6 5 log x 6. x +

54 3 3.7 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.

3.. 55 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

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

3.. 57 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.

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)

3.. 59 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).

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

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)9 9 + 8 (b a) (x a) 8 ] b a = (b a). 36

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

3.. 63 3.8 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)

64 3 3.3 () () 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.

3.. 65 () (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

66 3 3. 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, ).

3.. 67 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

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.

3.. 69 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.

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 =.

3.3. 7 3.3 (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

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

3.3. 73 () 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 = ).

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 3.4 3 () dx () dx (3) x x (4) dx (5) x + x dx x( x) dx () x = x dx = [log x ] = lim log( x) =. x

3.3. 75 () 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

76 3 3.8 () (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

3.3. 77 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=

78 3 3.6 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 <.

3.3. 79 3.3 () 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

8 3 3.5 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

3.3. 8 3.7 () 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 <

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

3.3. 83 (3) (4) tan ax tan bx dx x x a x b dx log x

85 4 4. 4. 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

86 4.5.5.5.5 4.: 4. () -4-4 - - 4.: 4. ()..4.6.8 -. -.4 -.6 -.8 - -. -.4 4.3: 4. (3)

4.. 87 4. 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)

88 4 4.4 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

4.. 89 4.5 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

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) (, )

4.. 9 4.7 (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)

9 4 4.8 () () lim xy log(x + y ), (x, y) (, ) lim (x, y) (, ) xy x + y 4. 4.9 x 3 + y 3 (x, y) (, ) f(x, y) = x + y (x, y) = (, ) 4. 4. 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)

4.. 93 4.7 () () 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

94 4 4. () () (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

4.. 95 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 (, ) = 4. 4.8 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) (, )

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) (, )) 4. 4.3 4.8 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

4.. 97 4.4 () () 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.

98 4 4. 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.

4.. 99 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

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 )

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 )

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 = =

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).

4 4 4.3 4.5 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 + 4.9 () () f(x, y) (a, b) f(x, y) = x + xy 3y x + 5y +, f(x, y) = e x+y.

4.3. 5 4.7 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

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) <

4.3. 7 () H(a, b) < f (a, b) 4.7 4.4 () () () 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

8 4 4. () () (3) f(x, y) = x 3 + y 3 3xy, f(x, y) = x + y + y, f(x, y) = (x + y)e x y. 4.8 4S 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

4.3. 9 ( ) ( ) 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) =

4 ( (x, ( y) = (, ) (, ) λ = (x, y) = 3, ), ) λ = (x, y) 3 3 3 3 g(x, y) = f(x, y) (x, y) (x, ( ( y) = (, ) (, ) (x, y) = 3, ), ) 3 3 3 3 4.4 f(x, y) = x + y (x, y) (x + y ) = (x y ) f(x, y)

5 5. 5. 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

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

5.. 3 5. 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

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 > )

5.. 5 (4) dx +x x f(x, y) dy 5. 5.4 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

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

5.. 7 5.5 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

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

5.. 9 5.7 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

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

5.3. 5.7 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) 5.3 5.9 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=

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

5.3. 3 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

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

5.3. 5 5. 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

6 5 5.9 () 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

5.3. 7 (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

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 > }

5.3. 9 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.

3 5 3.3(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.

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

3 6 n 6. 6. 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

6.. 33 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 3 6.3 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

34 6 y = C y + C y L(y) = L(C y + C y ) = C L(y ) + C L(y ) = C + C =. e x 6.9 6. 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

6.. 35 C ye R p(x) dx = e R p(x) dx q(x) dx + C e R p(x) dx 6.4 6.5 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) 6.7 6.6 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 6.8 6.

36 6 6.9 6. y + p(x)y = q(x) y + p(x)y = q(x) 6. y, y y + p(x)y = c y = cy y = cy 6. 6.3 6. 6.3 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

6.. 37 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

38 6 y + p(x)y = q(x) 6.3 6.3. 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)

6.. 39 y z z = y a z = ( a)y a y z z + ( a)p(x)z = ( a)q(x) 6. 6. p(x), q(x) ( a)p(x), ( a)q(x) 6. 6.5 6.6 xy + y = x 3 y 3 xy + y = y log x 6.7 6.4 Bernoulli Riccati y + p(x) + q(x)y + r(x)y = 6. 6.4 Bernoulli p(x) Riccati a = Bernoulli 6. y p(x) Riccati

4 6 6.5 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

6.. 4 6. 6.4 6.5 Bernoulli Riccati 6.8 xy (x + x) + (x + )y y = y = x 6.9 6.5 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 4 6.9 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

4 6 6. Riccati y, y 6.3 6.5 Riccati y y = z r(x) z z ( ) z + q(x) r (x) z + p(x)r(x)z = r(x) Riccati Riccati 6.5 6. 6. 3 6. 6.4 Bernoulli 6.5 Riccati 6.4 Bernoulli 6.5 Riccati

6.. 43 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)

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 ]

6.. 45 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

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

6.. 47 (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) 6.5 6.5 6.6 6.6 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

48 6 6.7 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 6.6 3 (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

6.. 49 6.8 6.7 (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 3 6.8 6.8 6.8 3

5 6 6.9 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 6.6 6.8 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

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 6.9 6.3 y + 6y + 9y = y 5y + 6y = 6.3 6.9 e 3x, xe 3x Wronskian 6.3 6.3 e x, e 3x Wronskian 6.9 a 4b L(y) y + ay + by =

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

6.. 53 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 6.33 6. 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

54 6 6.3 L(y) y + ay + by = f(x) 6. 6. 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}

6.3. 55 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 6.36 6.37 6.38 L(y) = y 4y + 3y = sin x L(y) = y y 3y = x L(y) = y 3y + y = e x

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

6.3. 57 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 }

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

6.3. 59 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) 6.9 6.39 6. 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 6. 6. 6.4 x y xy + y = x x L(y) y x y + x y = W (x) = W [y, y ] L(y) = y y x

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

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

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 3 5 + 4 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 =

64 y = log(x + x ) (4) y = coth x x = coth y = ey + e y t = e y ey e y t = x + ( ) x + t > t = x x coth x = y = ( ) x + log x (5) (cosh x sinh x) n = (e x ) n = e nx = cosh nx sinh nx. f(h) f() h = h.. (cos x) = sin x cos(x + h) cos x ( = sin x + h ) sin h.).3 (log( x)) = (x < ). x log( (x + h)) log( x) = ( h h log + h ). x (.4 () sec x = ) tan x = sin x cos x cos x..5(3) ) () x log a log log x a x = log a. x (3).4 + x (4) (log a)a x a x = e x log a (5).3 x + A (6) ( + log x)x x x x = e x log x

65 [.5 () y = sin x π, π ] ( y = cos x π < x < π ) ( () y = tan x π, π ) y = ( cos x π < x < π ).6 b 3a.7 () b a tan t (t, π, π, 3π ) () t( t3 ) ( ) t ± t.8 g(x) = f(x) Ax (x [a, b]) ) (x x3 f() =, f (x) = 3!.9 () f(x) = sin x cos x ( x f (x) > (x > ). f() = f(x) > (x > ). (), (3). ). f () =, f (x) = sin x + x > (x > ). n (n =,, ). () { ( 3 n cos 3x + nπ ) ( + cos x + nπ )} cos x cos x = ( ) cos 3x + cos x.. () ( )n (n )! (n ). ( + x) n (3) n! { } ( )n + ( x) n+ ( + x) n+

66 x = ( x + ). + x (4) k(k ) (k n + )x k n. (. () x sin x+ nπ ) ( nx cos x+ nπ ) ( n(n ) sin x+ nπ ).8 () e x ( ( ) n sin x + nπ ) 4 (e x sin x) = e x sin x + e x cos x = e x ( x + π ). 4 (.3 (3) f(x) = cos x. f (n) (x) = cos x + nπ ) f(x) = x! + x4 xn + ( )n 4! (n)! + R n+, n+ cos(θx + (n + )π) R n+ = ( ) x n+ ( < θ < ) (n + )! R n+ x n+ (n + )! (n )..4 () ( x)( + x + + x n ) = x n+, x = + x + + xn + xn+ x. () ( x) ( + x + 3x + + (n + )x n ) (3) log + x x = { } log( + x) log( x)..(4).5 = e iθ = + iθ + (iθ)! + (iθ)3 3! + (iθ)4 4! ) ( θ! + θ4 θ3 ) + i(θ 4! 3! + θ5 5!. + (iθ)5 5! +.6 () () (3) log a b (4)

67.7 () () ex = + x + x! + x3 3! eθx ( < θ < ). cos x = x! + x4 4! cos (θx + 4π.8 () ( () f ) = 3 f() =. 3 9 (3) f(e) = e e. ) ( < θ < )..9 3. () F (x) = 6 (x + )6 + C F () = 6 + C = F (x) = 6 {(x + )6 }. () F (x) = (x + 3) + C F () = 3 + C = F (x) = (x + 3) + 3 = x 3(x + 3). (3) F (x) = (x + ) log(x + ) x + C F () = C = F (x) = (x + ) log(x + ) x. 3. () 3 ( x) 3 () (4) 3 (x ) x + (5) sin x (6) (log x) = log log x. x 6( 3x) (3) tan (x )

68 3.3 () (x + ) () cos(x + ) (3) 5 sin5 x (4) cos 5 x = ( sin x) cos x = ( sin + sin 4 x) cos x sin x 3 sin3 x + 5 sin5 x (5) (log x) 3 = (x) (log x) 3 x(log x) 3 3x(log x) + 6x log x 6x. x + 3 x x x + 3.4 () 3 = 3(x + ) x 6(x x + ) + (x x + ). () x + x x + (x ) 3.5 () () 3 log x + 3 x x + + x tan. 3 3 () () log x x x x. (3) x x + x 3 log (x )(x 3) (x ). (4) 4 x + 4 x x + 4 3 x + 4 log x + x + 8x. x

69 3.6 () t = x + 3 x = t 3, dx = t dt x + dx = (t ) dt = x + 3 3 t3 t = 3 x x + 3. () t = + x x = t, xdx = t dt x 3 dx = + x (t ) dt = 3 t3 t = 3 (x ) + x. (3) t = x 4 dt = 4x 3 dx x 3 x 8 + dx = 4 t + dt = 4 tan t = 4 tan x 4. 3.7 () ( x x + log x + ) x x () x + x x log x log x + x 3.8 () tan x () log tan x (3) ( ) b ab tan a tan x 3.9 3. () (3) (4) x 3 dx = 4. () [log dx = x + x + + x [ ] 3 x dx = log 3 3x = log 3. dx = log 3. + x ] = log( + ).

7 [ 3. () (3x + )5 5 [ (3) ax 4 ax x + 3 x ] ] a = 35 [ 5. () x ] = 7 x 8. = a 6. [ (4) t = x + [ ( ( (5) 3 tan 3 x + ))] = π 3 3. (6) [ log(x + x + ) ] = log 7 [ 3. (7) log(x + ) log x ] = log 3 x 4 +. 3. () G(x) = x x 5 t 5 3 t 3 ] = 4 5 ( + ). F (t) dt (x t) f(t) dt = [(x t) F (t)] x + = [(x t)g(t)] x + = x G(t)dt. x x (x t)f (t) dt G(t) dt F (x) () () n!f (x) (3) f(g (x))g (x) f(g (x))g (x) 3.3 () y = c x () y = π x (3) c f(x)dx y = x

7 (4) y = x (5) y = π x 3.4 ()[x log x x] = log. ()m ±n. m = ±n ( ) (3) [ 4n sin nx + ] π x = π. cos mx sin nx = {sin(m + n)x sin(m n)x} m + n m + n m n π π cos mx sin nx dx = {sin(m + n)x + sin(n m)x} dx = [ cos(m + n)x cos(n m)x m + n n m n = n m. 3.5 () () (3) π π 3.6 π cos n x dx = π (sin 5 x sin 7 x) cos x dx = 4. sin 3 x cos 5 x dx = (), (3),(4) π sin n x dx. sin 5 x cos 3 x dx = 4. () c = π π π ] π sin x dx = π. 3.7 x α dx = x α dx + x α dx.

7 3.8 () x = dx = x 4 dx + x 4 dx. x 4 e x () lim = [, ] x + x sin x (3) lim = x = x + x b a sin x [ x dx = cos x x ] b a b a cos x x dx ( < a < b) cos x x x b (4) (3) x = cos x x x [a, ) (a > ) (5) n < x x n log x n x log x.

73 x log x dx = [log log x] = n n = log x x log x [ 3.9 () ] n x (n ) log x + x n dx = n (n ). () [ ] sin x cos x sin xe x dx = e x =. 3. () Γ () = Γ (s + ) = () () 3. 3. lim n x n e x dx = n! x s e x dx = [ x s e x] + s x s e x dx. x n e x dx x n e x dx x n e x dx = lim n n! () x = t I s = e x n e x dx = n! e x dx e t s e t dt x n e x dx =. x n e x dx

74 () I s = [ ( + x) s e x] + s ( + x) s e x dx = + si s. (3) n n = n I n+ = + (n + )I n = (n + )! n (n + )! + k= (n + )! k! = n+ k= (n + )!. k! 3.3 () () y = x (3) (4) ( 5.3) (5) x = y + y (6) x = sin θ (7) s = t = ( (6) B, ) = π ( ) Γ () = (4) Γ = π ( ) ( ) Γ > Γ = π (8) ( Γ n + ) ( = n ) ( n ) ( ) ( ) Γ 3 3.4 () lim x s ( log x) t = [, ] x

75 () (3) y = log x x = e y, dx = e y dy x s ( log x) t dx = z = (s + )y x s ( log x) t dx = (s + ) t+ y t e (s+)y dy z t e z dz = (4) (3) (5) e x log x = n! ( x log x)k k= (6) (5) Γ (t + ) (s + ) t+. 3.5 () y = x µ y t = σ f(x) dx = ) exp ( y dy π ) = exp ( y dy π = e t dt π t = ( ) Γ =. π () y = x µ σ xf(x) dx = (σy + µ) exp π ) ( y dy = µ.

76 (3) y = x µ y t = σ x f(x) dx = ) (σy + µ) exp ( y dy π = (σ y + σµy + µ ) exp π σ = te t dt + µ π t ( ) = σ 3 Γ + µ = σ + µ. π ( y ) dy 3.6 () x = tan θ ( + x ) n dx = () x = sin θ ( x ) n dx = π/ π/ cos (n ) θ dθ = a (n ). cos n+ θ dθ = a n+. 3.7 () () log b a (3) π log a b (4) log a b 4. () y = x x 6. () x = y y 6. 4

77.8.6.4...4.6.8 6.: 4. ()..4.6.8 -.5 - -.5-6.: 4. ().8.6.4. -.5 -.5.5.5.75 -. -.4 -.6 6.3: 4.

78 4. t = 8 5 4 5 8 7 3 6.3 4.3 () f(, ) = f(r cos θ, r sin θ) = r cos θ + r sin θ = r () x + y =.5.5 6.4 -.5.5-6.4: 4.3 4.4 () f(x, y) y = ±.5 ± ±.5 ± x = ±.5 ± ±.5 ± 6.5 () 6.6 4.5 () Φ(, ) = (, ) Φ(, ) = (, )

79 - - - 4 - - -4 6.5: 4.4 () 4 - - - - 6.6: 4.4 ()

8 () x = t y = t < t u = u(t, t) = t, v = v(t, t) = t u = v u < (3) x = t y = 3t < t u = u(t, 3t) = 4t, v = v(t, 3t) = 3 4t v = 3u < u (4) x = cos θ y = sin θ θ π u = u(cos θ, sin θ) = cos θ, v = v(cos θ, sin θ) = sin θ u + v = v (5) x = cos θ y = sin θ θ < π u = u( cos θ, sin θ) = cos θ sin θ, v = v( cos θ, sin θ) = u + v = 4 4.6 cos nπ d(p n, P ) = n + sin nπ n = n

8 4.7 r > θ < π x = r cos θ y = r sin θ r (x, y) (, ) r < 4 (x ) 9 4 4 (y ) 9 4 f(r cos θ, r sin θ) = r cos θ(r cos θ ) r sin θ(r sin θ ) r cos (r cos θ ) + r sin θ(r sin θ ) ( ) ( ) 9 9 r cos θ r sin θ < 4 4 ( ) ( ) r cos + r sin θ 4 4 lim (x, y) (, ) = 8r cos θ sin θ (r ) f(x, y) = r > θ < π x = + r cos θ y = + r sin θ r (x, y) (, ) r < 4 x 9 4 4 y 9 lim f(x, y) = 4 (x, y) (, ) 4.8 () x = r cos θ y = r sin θ (r >, θ < π) xy log(x + y ) = r cos θ sin θ log(r cos θ + r sin θ) < r log r (r )

8 () x = αy (x, y) (, ) t x = αt y = t (t ) lim (x, y) (, ) xy x + y = lim 4 t αt t α t 4 + t = lim 4 t α α + = α α + 4.9 (α, β) (, ) (α, β) f(x, y) f(α, β) = (x3 + y 3 )(α + β ) (α 3 + β 3 )(x + y ) (x + y )(α + β ) = α x 3 + α y 3 + β x 3 + β y 3 α 3 x β 3 x α 3 y β 3 y (x + y )(α + β ) = α x (x α) + β y (y β) + α y 3 + β x 3 β 3 x α 3 y = α x (x α) + β y (y β) + α y 3 α β 3 + α β 3 + β x 3 β 3 x x 3 y + x 3 y α 3 y = (x α)(α x β 3 (x + α) + y (x + αxα )) + (y β)(β y x 3 (y + β) + α (y βy + β )) lim f(x, y) = f(α, β) (x, y) (α, β)

83 (α, β) = (, ) x = r cos θ y = r sin θ (r >, θ < π) f(r cos θ, r sin θ) = r 3 cos 3 θ + r 3 sin 3 θ r cos θ + r sin θ = r cos 3 θ + sin 3 θ < r (r ) 4. () () f x (x, y) = 3 + xy 3 + xy + y. f y (x, y) = 6x y + x + xy +. f(x, y) = x y y x = x y e x log y ( y ) f x (x, y) = yx y e x log y + x y log ye x log y = x y y x x + log y x y f y (x, y) = x y y x ( x y + log x ) (3) f x (x, y) = e x sin y. f x (x, y) = e x cos y.

84 4. (x, y) (, ) f x (x, y) = x4 + 3x y + xy 3 (x + y ) f y (x, y) = y4 + 3x y + x 3 y (x + y ) (x, y) = (, ) 4.8 f x (, ) = f y (, ) = (x, y) (, ) f x f y y = x t 4 + 3t t + tt 3 lim f x (t, t) = lim = 3 t t (t + t ) f x(, ) lim f y (t, t) = lim t4 + 3t t + t 3 t = 3 t t (t + t ) f y(, ) 4. 4.8 f x (, ) = f y (, ) = h = t cos θ, f(h, k) f(, ) = h k + ε(h, k) h + k k = t sin θ lim ε(h, k) = (h, k) (, ) ( < θ < π, θ π ) 4, π, 5π 4, 3π t 3 (cos 3 θ sin 3 θ) t = t(cos θ sin θ) + tε(t cos θ, t sin θ)

85 cos 3 θ sin 3 θ = cos θ sin θ + ε(t cos θ, sin θ) t ε(t cos θ, sin θ) cos 3 θ sin 3 θ = cos θ sin θ cos θ sin θ = cos θ + cos θ sin θ + sin θ = + cos θ sin θ cos θ sin θ = cos θ = sin θ = θ 4.3 (a, b) f(a + t cos θ, b + t sin θ) f(a, b) = f x (a, b)t cos θ + f y (a, b)t sin θ + ε(t cos θ, t sin θ)t lim ε(t cos θ, t sin θ) = t f(a + t cos θ, b + t sin θ) f(a, b) lim t t = f x (a, b) cos θ + f y (a, b) sin θ 4.4 () () df = f x dx + f y dy = xdx + ydy + x + y df = f x dx + f y dy = ( + x)xe x cos y dx x e x sin y dy

86 4.5 () x u y u () x u y u (x, y) (r, θ) x ( v a y = c v = ad bc x ( v cos a y = sin a v ) b d ) sin a cos a (x, y) (r, θ) = cos a + sin a = (3) x x ( ) u v v u y y = v u u v (x, y) = uv u( v) = u (r, θ) 4.6 () (u, v) (x, y) = u x v x u y v = y = y y =

87 () (u, v) = Φ(x, ) = (, ) (, ) (3) u = y xy v = xy y = u + v u + v x = v u + v = y = u + v v = Φ {(u, v) u + v } {(, )} (4) u + v () x = v u + v y = u + v (5) (u, v ) u + v = R < r < R r (u, v ) (4) 4.7 Φ (u, v) = ( u + v, ) v u + v () f xx = y 3 x 3 f x = f (x y ) 3 y = (x y ) 3 3xy 3 f (x y ) 5 xy = f yx = 3x y 3x 3 y f (x y ) 5 yy = (x y ) 5

88 () f x = ye xy f y = xe xy f xx = y e xy f xy = f yx = ( + xy)e xy f yy = x e xy (3) f x = x cos(x + y ) f y = y cos(x + y ) f xx = cos(x + y ) 4x sin(x + y ) f yy = cos(x + y ) 4y sin(x + y ) f xy = f yx = 4xy sin(x + y ) 4.8 () f x = x f x + y y = y x + y f xx = (x + y ) x x = x + y (x + y ) (x + y ) f yy = (x + y ) y y (x + y ) = x y (x + y ) f xx + f yy = () f x = e x (x cos y y sin y) + e x cos y = e x {(x + ) cos y y sin y} f xx = e x {(x+) cos y y sin y}+e x cos y = e x {(x+) cos y y sin y} f y = e x ( x sin y sin y y cos y) = e x {y cos y + (x + ) sin y} f yy = e x {cos y y sin y+(x+) cos y} = e x {(x+) cos y y sin y} f xx + f yy =

89 4.9 () f x (x, y) = 4x + y f y (x, y) = x 6y + 5 f xx (x, y) = 4 f xy (x, y) = f yx (x, y) = f yy (x, y) = 6 n f = (n 3, k n) x k yn k f(x, y) = f(a, b) + f x (a, b)(x a) + f y (a, b)(y b) + {f xx(a, b)(x a) + f yy (a, b)(y b) } + f xy (a, b)(x a)(y b) = a + ab 3b a + 5b + + (4a + b )(x a) + (a 6b + 5)(y b) + (x a) + (x a)(y b) 3(y b) () f x (x, y) = e x+y n= n f = x k ex+y yn k f y (x, y) = e x+y { f(x, y) = (x a) n! x + (y b) } r f(a, b) y = e a+b {(x a) + (y b)} n n! n=

9 4. () f(x, y) = x 3 y 3 + y x f x = 3x y 3 f y = 3x 3 y + f y dy dx = y 3 3x 3x 3 y + () f(x, y) = x y y x f x = yx y log yy x f y = log xx y xy x 4. f y dy dx = yxy y x log y xy x x y log x () f x = 3x 3y f y = 3y 3x f x = f y = x = y = x 4 (, ) (, ) f xx = 6x f yy = 6y f xy = 3 H(, ) = f xx (, )f yy (, ) {f xy (, )} = 9 < (, ) H(, ) = f xx (, )f yy (, ) {f xy (, )} = 6 6 9 > f xx (, ) = 6 > (, ) f(, ) =

9 () f x = x f + y y = x y y ( + y ) f x = f y = (, ) (, ) f xx = f + y yy = ( 3y )x + y(3 y ) ( + y ) 3 f xy = 4xy ( + y ) H(, ) = f xx (, )f yy (, ) {f xy (, )} = >, f xx (, ) = > (, ) f(, ) = H(, ) = f xx (, )f yy (, ) {f xy (, )} = ( ) < (3) f x = { x(x + y)}e x y f y = { y(x + y)}e x y f x = f y = x = y x = y = ( ) ( 4,, ) f xx = (4x 3 + 4yx 6x y)e x y f yy = (4y 3 + 4xy 6y x)e x y f xy = (x + y)(xy )e x y ( ) ( ) ( ) { ( )} H, = f xx, f yy, f xy, ( ) ( ) ( ) = 3e 3e e >,

9 ( ) f xx, < e ( H, ( ), ( ) f, ) ( = f xx, ) ( f yy, ) = 3e 3e (e ) >, ( f xx, ) > ( f, ) = e ( {f xy, )} (, ) 4. z = a x y < x < a < y < a 3 < x + y < a 3 xy K x 3 y z = f(x, y) = x3 y (a x y) 3 f x = x y (3a 4x 6y) f y = x3 y(a x 3y) 3 3 ( a f x = f y = K (x, y) =, a ) 6 = f xx = xy (3a 4x 6y) 4x y 3 f yy = x3 (a x 3y) 6x 3 y 3

93 f xy = x y(3a 4x 6y) 6x y 3 ( a H, a ) ( a = f xx 6, a ) ( a f yy 6, a ) { ( a f xy 6, a )} { 6 = 4 ( a ) ( } { a ( a ) 3 ( a 3 6) 6) } { ( a ) ( a ) } > 6 ( a f xx, a ) ( a < 6, a ) ( 6 a K f, a ) 6 f ( a, a ) = a6 6 6 3 4 4.3 x + y = x = cos θ y = sin θ ( π θ π) F (θ) = f(cos θ, sin θ) F (θ) = 3 cos θ + 3 cos θ sin θ + sin θ = cos θ + 3 sin θ + θ F (θ) = sin θ + 3 cos θ F (θ) = θ = 5 6 π π 3 π 6 3 π θ = 5 ( 6 π π 6 F 5 ) ( π ) 6 π = F = 4 θ = π 6 3 3 π F ( π ) ( ) 3 π = F 3 π =

94 4.4 g(x, y) = (x +y ) (x y ) Φ(x, y, λ) = f(x, y) λg(x, y) g x = 4x(x + y ) g y = 4y(x + y + ) Φ x (x, y, λ) = x{ λ(x + y )} = Φ y (x, y, λ) = y{ λ(x + y + )} = g(x, y) = (x, y) = (, ) (±, ) f(x, y) (x, y) (, ) f(x, y) > (x, y) = (, ) g x (, ) = 4 g y (, ) = (, ) g(x, y) = x y x = r cos θ y = r sin θ g(r cos θ, r sin θ) = r 4 r cos θ = r = cos θ (, ) = ( cos, sin ) θ f(r cos θ, r sin θ) = r g (, ) (, ) 5. D = {(x, y) x, y x } = {(x, y) y, y x y }.

95 5. () 7 6 5.3 () 5 () e (3) 8 5 () (3) (e ) (4) 45 5.4 () () (3) (4) αa a dy dx dy e y/α y/β x+a a y y dy f(x, y) dx + f(x, y) dy+ f(x, y) dx + f(x, y) dx + βa αa a a 4 dx dy dy e a y/β a a y y e dy log y f(x, y) dx f(x, y) dy+ f(x, y) dx f(x, y) dx 3a a a dx f(x, y) dy x a 5.5 D a = (ak, ck), b = (ch, dh) 5.6 6 (3e4 + 5e 4 ) 5.7 () π 3 a3 () π 4 (ea ) (3) a4 4 x = a + r cos θ y = r sin θ 5.8 () D n : x + y n, x, y () log( + ) D n : y x, x n (3) log b a D n : x, a y b, xy n b n e xy y dx dy = dy e xy dx D n a

96 5.9 () e e () πa (3) π 4 a4 () (3) 5. 4π 3 abc 5. < t < min{a, b}, D = {(x, y) t x a, t y b} f(x)g(y) dx dy D 5. u = x 5.3 f(x, y) = e x y f(x, y) dx dy E(a) D(a) f(x, y) dx dy, 6. dy dx = x y 6. f(x) = x, g(y) = y C y dy = x dx + C

97 log y = log x + C, y = e C x, y = ±e C x ±e C C C y = Cx 6. 6. C 3 dy = dx + C, y 3 = x + C 3 y y = (x + C) 3 y = C y = 6.3 y = ± y 6. C dy = ± dx + C y sin y = ±x + C, y = sin(±x + C) = ± sin(x ± C)

98 sin(x ± π) = sin x ±C C y = sin(x + C) y = ± C y = ± 6.4 6. p(x) = a, q(x) = b a = y = b y = bx + C a 6. y = b a + Ce ax C ( 6.5 6. p(x) = x, q(x) = x y = e x ( e x) ) e x xdx + C = e x (e x + C) = + Ce x C = e x x 6.6 6. C, C C C y, y y = C f(x) + g(x), y = C f(x) + g(x) y y = (C C )f(x) (y y ) + p(x)(y y ) = (C C )f (x) + (C C )p(x)f(x) = f (x) + p(x)f(x) =

99 B f(x) = Be R p(x) dx C Cf(x) = CBe R p(x)dx Cf(x) CB C y = Ce R p(x)de + g(x) g (x) + p(x)g(x) = q(x) g(x) 6.7 i, j 3 i, j y i y j = (C i C j )f(x) C C, C i =, y = Cf(x) + g(x), y i (x) = C i f(x) + g(x) y y = C C y y C C (C C )/(C C ) C y y y y = C, y = C(y y ) + y

6.8 ( e R ) R p(x) dx = p(x)e p(x) dx 6.9 y + p(x)y = 6. C y = Ce R p(x) dx C x C(x) y = C(x)e R p(x) dx C(x) y y = {C (x) C(x)p(x)} e R p(x) dx = C (x)e R p(x) dx p(x)y y y C (x) = e R p(x) dx, C(x) = e R p(x) dx q(x) + C C y = C(x)e R p(x) dx C C { } y = e R p(x) dx e R p(x) dx q(x)dx + C

Lagrange( ) 6. 6. y 6. C = C y = C e R p(x) dx + e R p(x) dx e R p(x) dx q(x) dx y ϕ(x) ϕ(x) = e R p(x) dx y y ϕ y + p(x)y = ( ) y(x) y (x) = ϕ(x) C y(x) y (x) = Cϕ(x) y y = (C + C)e R p(x) dx + e R p(x) dx e R p(x) dx q(x) dx { } = e R p(x) dx e R p(x) dx q(x) dx + C + C 6. C C + C C

6. C y = Ce R p(x)dx 6. y, y C, C y = C e R p(x) dx, y = C e R p(x) dx 6. y = y = x p(x) = x e R p(x) dx = e log x = x Cx 6.3 C y = C x + x 6.3 y, y (y y ) + p(x)(y y ) = 6.3 B y y = Be R p(x) dx x = x y (x ) y (x ) = B = y y

3 x = p I y (p) = y (p) y y y (x ) < y (x ) y (p) > y (p) x = q I y (q) = y (q) y (x) y (x) y (x ) < y (x ) x I y (x) < y (x) 6.4 y, y (y y ) + p(x)(y y ) = B y y = Be R p(x) dx y y 6.8 B y z = y y z z + p(x)z = C z = y y = Ce R p(x) dx y y y y = C B C B C y = C(y y ) + y

4 6.5 y + x y = x y 3 Bernoulli p(x) = x, q(x) = x 6.3 a = 3 C ( ) y = e R x dx e R x dx x dx + C ) = x ( x x dx + C = x ( x + C) = x 3 + Cx (x C)x y = 6.6 x y + x y = y x log x a =, p(x) = x, q(x) = x log x Bernoulli e R x dx = e R x dx = e log x = x x log x dx = x log x x dx = x log x + x 6.4 y = x(x + x log x + C) = + log x + Cx C y = + log x + Cx

5 6.7 6.4 x ( a)y a y = ( a)p(x)y a + e R ( a)p(x) dx e R ( a)p(x) dx ( a)q(x) ( a) y a Bernoulli 6.8 y = z + x y z z Bernoulli z + x z = x z 6.4 a =, p(x) = q(x) = x C 6.4 z = e R x ( dx e R x dx ) x dx + C ( ) = x x dx + C = + Cx y = x + + Cx = x + + Cx + Cx 6.9 6.5 y Riccati y C y = y + Cg (x) + g (x) = Cy g (x) + (y g (x) + ) Cg (x) + g (x) +

6 6.5 g (x) > y g (x)g (x) g (x)(y g (x) + ) = g (x) > C C = f 4(x)y + f (x) f 3 (x)y f (x) x y (f f 4 f f 3 )y + (f f f f ) + (f f 4 f f 3)y + (f 3f 4 f 3 f 4)y = f (x)f 4 (x) f (x)f 3 (x) Riccati 6. y i y j = (C i C j )(f f 4 f f 3 ) (f i f 3 + f 4 )(C j f 3 + f 4 ) y y 3 = (C C 3 )(f f 4 f f 3 ) (C f 3 + f 4 )(C 3 f 3 + f 4 ), y y 3 = (C C 3 )(f f 4 f f 3 ) (C f 3 + f 4 )(C 3 f 3 + f 4 ) y y 3 = (C C 3 )(f f 4 f f 3 ) (C f 3 + f 4 )(C 3 f 3 + f 4 ), y y 4 = (C C 4 )(f f 4 f f 3 ) (C f 3 + f 4 )(C 4 f 3 + f 4 )

7 6. y Riccati 6.9 y 6. y i 6. / y y y y3 = C C / C C3 y y y y 3 C C C C 3 y C 3 y, y, y 3 Riccati 6. y, y Riccati (y y ) + {q(x) + r(x)y (x)}(y y ) + r(x)(y y ) = (y y ) ( ) {q(x) + r(x)y (x)} = r(x) y y y y y = z + y y z z Bernoulli z + {q(x) + r(x)y (x)}z = r(x)z z ( ) {q(x) + r(x)y (x)} z z = r(x) z = y y ( ) {q(x) + r(x)y (x)} = r(x) y y y y

8 y y ( ) ( ) {q(x) + r(x)y (x)} = y y y y y y y y 6. C = Ce R {q(x)+r(x)y (x)} dx y y y y Riccati y, y y 6.3 x y = r (x) z r(x) z + z z r(x) z y y Riccati r (x) r(x) + z z z + p(x) + q(x) z r(x) z r(x) z + z r(x) z = r(x)z 6.4 W [f, f ] x x < f (x) f (x) x = W = f (x) f (x) = x x = f (x) f (x) x < = W = f (x) f (x) = x x =

9 W [f, f ] f, f C, C g(x) = C f (x) + C f (x) f (x) f (x) g() = C f () + C f() = C + = g( ) = C f( ) + C f( ) = + C = C = C = 6.5 λ = a, λ = b (D λ ) y = (D λ )(Dy λ y) = D(y λ y) λ (y λ y) = y λ y λ y + λ y = y λ y + λ y = y + ay + by = L(y) 6.6 (D λ )(D λ )y = {D (λ + λ )D + λ λ }y (D λ )(D λ )y = {D (λ + λ )D + λ λ }y

(D λ )e λ x = (e λ x ) λ e λ x = λ e λ x λ e λ x = 3 6.7 6.7 (D λ ) e λ x y = (D λ ){(D λ )e λ x y} = (D λ ){e λ x Dy} = e λ x D y 6.8 6.8 (D λ ) y = (D λ ) e λ x e λ x y = e λ x D e λ x y 3 6.9 λ + 6λ + 9 = (λ + 3) = 6.9 λ = 3 C, C y = (C + C x)e 3x

6.3 λ 5λ + 6 = (λ )(λ 3) = 6.9 λ =, λ = 3 C, C y = C e x + C e 3x 6.3 Wronskian W e 3x xe 3x W = 3e 3x e 3x 3xe 3x = x e 6x 3 3x = e 6x > e 3x, xe 3x 6.3 Wronskian W e x e 3x W = e x 3e 3x = e5x 3 = e5x > e x, e 3x 6.33 6. p = a, q = 4b a p q + ap + b = p + a = L(e px cos qx) = e px (p q + ap + b) cos qx e px q(p + a) sin qx = L(e px sin qx) = e px q(p + a) cos qx + e px (p q + ap + b) sin qx =

L(y) = L(C e px cos qx + C e px sin qx) = C L(e px cos qx) + C L(e px sin qx) = C + C = 6.34 λ λ + 5 = 6. a =, b = 5 p =, q = C, C y = (C cos x + C sin x)e x 6.35 Wronskian W q e px cos qx e px sin qx W = pe px cos qx qe px sin qx pe px sin qx + q cos qx cos qx sin qx = e px q sin qx q cos qx = qepx 6.36 L(y) = sin x c, c y = c cos x + c sin x y = c sin x + c cos x, y = y

3 L(y) = 4y + y = (c 4c ) cos x + (4c + c ) sin x = sin x sin x cos x c 4c =, 4c + c = c, c L(y) = sin x y c =, c =, y = ( cos x + sin x) L(y) = λ 4λ + 3 = (λ )(λ 3) = L(y) = y = C e x + C e 3x C, C y = y + y y = C e x + C e 3x + ( cos x + sin x) 6.37 L(y) = x c, c, c 3 y = c + c x + c 3 x ( 3c c + c 3 ) (3c + 4c 3 )x 3c 3 x = x 3c c + c 3 =, 3c + 4c 3 =, 3c 3 = c = 4 7, c = 4 9, c 3 = 3

4 L(y) = λ λ 3 = (λ + )(λ 3) = y = C e x + C e 3x 4 7 + 4 9 x 3 x 6.38 L(y) = e x c y = cxe x y = ce x, y = c( + x)e x, y = 4c( + x)e x ce x = e x c = y = xe x L(y) = λ 3λ + = (λ )(λ ) = C, C y = C e x + C e x + xe x 6.39 y 3 = C (x)y + C (x)y L(y 3 ) = r(x) r(x)y (x) r(x)y (x) C (x) = dx, C (x) = dx W (x) W (x)

5 y 3 = C (x)y + C (x)y, y 3 = C (x)y + C (x)y + r(x) L(y ) = L(y ) = L(y 3 ) = C (x)l(y ) + C (x)l(y ) + r(x) = C (x) + C (x) + r(x) = r(x) 6.4 x = x L(y) = y = x, y = e x x Wronskian x e x W (x) = W [y, y ] = e x = (x )ex L(y) = r(x) r(x) = x 6. (x )e x (x )x y 3 = x dx + ex (x )ex (x )e dx x = x dx + e x xe x dx = (x + x + ) L(y) = r(x) C, C 6. y = C x + C e x (x + x + ) = (C )x + C e x (x + )

6 C C y = C x + C e x (x + ) 6.4 6. y = ax + bx + c L(y) = x a, b, c y = ax + bx + c, y = ax + b, y = a L(y) = x 4ax (5a b)x + (a 5b + 4c) x 4a =, 5a b =, a 5b + 4c = 3 a = 4, b = 5 8, c = 3 L(y) = x y = 4 x + 5 8 x + 3 L(y) = C, C y = C e x + C e 4x L(y) = x y = C e x + C e 4x + 4 x + 5 8 x + 3

7 6. L(y) = y = e x, y = e 4x Wronskian W (x) = W [y, y ] = W [e x, e 4x e x e 4x ] = = 3e5x > e x 4e 4x r(x) = x, W (x) = 3x 5, y = e x, y = e 4x 6. L(y) = x r(x)y (x) y 3 = y W (x) = 3 ex r(x)y (x) dx + y dx W (x) x e x dx + 3 e4x x e 4x dx = 3 ex ( x x )e x + ( 3 e4x 4 x 8 x 3 = 4 x + 5 8 x + 3 L(y) = x y = C e x + C e 4x + 4 x + 5 8 x + 3 C, C ) e 4x 6.4 y = x y = x L(y) = Wronskian x W (x) = W [y, y ] = x x x = x >

8 y = x, y = x, r(x) =, W (x) = x 6. r(x)y (x) r(x)y (x) y 3 = y dx + y (x) dx W (x) W (x) = x dx + x x dx = x + x log x = (log x )x L(y) = C, C y = C x + C x + (log x )x = C x + (C )x + x log x C C y = C x + C x + x log x