A. A. A-- [ ] f(x) x = f 00 (x) f 0 () =0 f 00 () > 0= f(x) x = f 00 () < 0= f(x) x = A--2 [ ] f(x) D f 00 (x) > 0= y = f(x) f 00 (x) < 0= y = f(x) P (, f()) f 00 () =0 A--3 [ ] y = f(x) [, b] x = f (y) f (y) f(x) f (y) f(x) f (y) A--4 [ ] y = f(x) y = f (x) y = x A--5 [ ] x = f (y) dx dx = dx = f 0 (x) A--6 [ ] (sin ) 0 =, x 2 (tn x) 0 = +x 2 y = f(x) dx A--7 [ ] x = f(t), y = g(t) f 0 (t) > 0 f 0 (t) < 0 y x dx = dt dx dt = g0 (t) f 0 (t)
A--8 [ ] C x = f(t), y = g(t) t = t 0 P 0 (x 0, y 0 ), x 0 = f(t 0 ), y 0 = g(t 0 ) C P 0 x x 0 f 0 (t 0 ) = y y 0 g 0 (t 0 ) A--9 [ ] C x = f(t), y = g(t) ( dx dt, dt )=(f 0 (t), g 0 (t)) C P (t) A--0 [ ] P (x, y) (r, θ) ( ( p x = r cos θ r = x 2 + y 2 () (2) y = r sin θ tn θ = y x A 2. [ ] A-2- [ ] f(x) [, b] (, b) f() =f(b) f 0 (c) =0 c (, b) A-2-2 [ ] f(x) [, b] (, b) f(b) f() = f 0 (c) b c (, b) A-2-3 [ ] f(x) D D, + h f( + h) =f()+hf 0 ( + θh) θ (0 < θ < ) A-2-4 [ ] f(x) D () f 0 (x) > 0 = f(x) D (2) f 0 (x) < 0 = f(x) D (3) f 0 (x) =0 = f(x) D A-2-5 [ ] f(x), g(x) D f 0 (x) =g 0 (x) C g(x) =f(x)+c
A-2-6 [ ] f(t), g(t) [, b] (, b) f 0 (t) 6= 0 g(b) g() f(b) f() = g0 (c) f 0 (c) c (, b) A-2-7 [ ] f(x), g(x) (, b) f 0 (x) 6= 0 x g 0 (x) f(x) 0, g(x) 0 lim x f 0 (x) lim x g(x) f(x) =lim x g 0 (x) f 0 (x) A 3. A-3- [ ] 0,, 2, n, c X () n x n = 0 + x + 2 x 2 + + n x n + (2) n=0 X n (x c) n = 0 + (x c)+ 2 (x c) 2 + + n (x c) n + n=0 (2) c A-3-2 [ ] X n x n x = r x < r x n=0 A-3-3 [ ] X n x n r r = 0 n=0 r r = lim n, r = p n n n+ lim n n A-3-4 [ ] y = f(x) f 0 (x) x ; f(x) ; f()+f 0 ()(x ) y = f(x) x = f 00 (x) x ; f(x) ; f()+f 0 ()(x )+ 2 f 00 ()(x ) 2 y = f(x) x =
A-3-5 [ ] f(x) D n + D, b f(b) =f()+f 0 ()(b )+ f 00 () (b ) 2 + 2! + f (n) () (b ) n + f (n+) (c) (b )n+ n! (n +)! c b A-3-6 [ ] f(x) 0 D n + D x f(x) =f(0) + f 0 (0)x + f 00 (0) x 2 + + f (n) (0) x n + f (n+) (c) 2! n! (n +)! xn+ c 0 x A-3-7 [ ] f(x) x R n+ (x) R n+ (x) = f (n+) (c) (n +)! xn+ 0, (n ) f(x) f(x) =f(0) + f 0 (0)x + f 00 (0) 2! x 2 + + f (n) (0) x n + n! A-3-8 [ ] () e x =+x + x2 2! + x3 xn + + + (r = ) 3! n! (2) sin x = x x3 3! + x5 x2n+ + +( )n + (r = ) 5! (2n +)! (3) cos x = x2 2! + x4 x2n + +( )n + (r = ) 4! (2n)! (4) p r à p r! = ( + x) p = p(p )(p 2) (p r +) r! à p! à x + p 2! à x 2 + + p n! x n + (r = ) (5) log( + x) =x x2 2 + x3 xn +( )n 3 n ( ) + (r =)
B. B 4. [ ] B-4- [ ] () x p dx = p + xp+ + C (p 6= ) (2) dx =log x + C (3) e x dx = e x + C x (4) sin xdx=cosx + C (5) cos xdx=sinx + C (6) (8) cos 2 x dx =tnx + C (7) x 2 2 dx = 2 log x + C ( 6= 0) x + (9) x 2 + 2 dx = x tn + C ( 6= 0) (0) 2 x dx x 2 =sin + C ( >0) () x 2 + A dx =log x + p x 2 + A + C (A 6= 0) B-4-2 [ ] {f(x)+g(x)} dx = kf(x) dx = k f(x) dx + f(x) dx ( k ) g(x) dx sin 2 x dx = tn x + C B-4-3 [ ] ϕ(x) =t f(ϕ(x))ϕ 0 (x) dx = f(t) dt B-4-4 [ ] f(x)g 0 (x) dx = f(x)g(x) f 0 (x)g(x) dx B 5. [ ] B-5- [ ] f(x) [, b] [, b] n = x 0,x,x 2, x n,x n = b b nx f(x) dx = lim f(x i ) x ( x = b n n ) B-5-2 [ ] i= f(x) D D, b, c b c b f(x) dx = f(x) dx + f(x) dx c
B-5-3 [ ] f(x), g(x) [, b] b b f(x) > = g(x) = f(x) dx > = g(x) dx f(x) =g(x) B-5-4 [ ] f(x) [, b] b f(x) dx = f(c)(b ) c b B-5-5 [ ] f(x) D F (x) = F (x) D F 0 (x) =f(x) x f(t) dt F (x) f(x) B-5-6 [ ] f(x) [, b] f(x) F (x) b f(x) dx =[F (x)] b = F (b) F () B-5-7 [ ] f(x) [0, ] n nx lim f( k n n n )= f(x) dx k= 0 B-5-8 [sin n x, cos n x ] n π 2 0 sin n xdx= = π 2 0 cos n xdx n n n 3 3 n 2 4 2 π 2 n n n 3 4 n 2 5 2 3 (n ) (n ) B-5-9 [ ] r = f(θ) ( < = θ < = β) θ =, θ = β S S = 2 β {f(θ)} 2 dθ = 2 β r 2 dθ
B-5-0 [ ] x = f(t), y = g(t) ( < = t < = β) dx dt, dt s r β s = ( dx dt )2 +( dt )2 dt B-5- [ ] f(x) [, b] f 0 (x) y = f(x) ( < = x < = b) s r b s = +( b p dx )2 dx = +{f 0 (x)} 2 dx B-5-2 [ ] r = f(θ) ( < = θ < = β) s β p r β s = {f(θ)} 2 + {f 0 (θ)} 2 dθ = r 2 +( dr dθ )2 dθ B-5-3 [ ] f(x) [, b) c c f(x) dx f(x) [, b] lim c b b f(x) dx =lim c b c f(x) dx (, b] c f(x) [, ) lim f(x) dx c f(x) [, ) f(x) dx = lim f(x) dx c c (, ] C. C 6. [ ] C-6- [ ] x, y, z (x, y) z z x, y f(x, y), f(x, y, z),,f(x, x 2,, x n ) (x, y) D
C-6-2 [ ] P (x, y) A(, b) f(x, y) C f(x, y) C C f(x, y) f(x, y) A f(, b) C P A P A P A PA 0 (x, y) (, b) lim f(x, y) =C lim (x, y) (, b) f(p )=C P A f(x, y) C ((x, y) (, b)) f(p ) C (P A) C-6-3 [ ] f(x, y) D A(, b) lim (x, y) (, b) f(x, y) lim f(x, y) =f(, b) (x, y) (, b) f(x, y) A(, b) D D C-6-4 [ ] lim h 0 f( + h, b) f(, b) h A(, b) f(x, y) x f x (, b) f(, b + k) f(, b) lim k 0 k A(, b) f(x, y) y f y (, b) A(, b) f x (, b), f y (, b) f(x, y) A(, b) f(x, y) P (x, y) f x (x, y), f y (x, y) x, y f(x, y) x y C-6-5 [ ] z = f(x, y) f x (x, y), f y (x, y) f xx, f xy, f yx, f yy n C-6-6 [ ] z = f(x, y) f xy,f yx f xy = f yx C-6-7 [ ] z = f(x, y) f x,f y x, y t z = f(x(t), y(t)) t dz dt = f dx x dt + f y dt = z dx x dt + z y dt
C-6-8 [ ] x, y u, v x = ϕ(u, v), y= ψ(u, v) z = f(x(u, v), y(u, v)) u, v z u = z x x u + z y y u, C-6-9 [ ] z v = z x x v + z y y v A(, b) f(x, y) f( + h, b + k) =f(, b)+hf x ( + θh, b + θk)+kf y ( + θh, b + θk) θ (0 < θ < ) C-6-0 [ ] df = dz = f x dx + f y f(x, y) C 7. [ ] C-7- [ ] f(x, y) A(, b) P (x, y), P 6= A f(, b) >f(x, y) f(x, y) A f(, b) f(, b) <f(x, y) A f(, b) C-7-2 [ ] f(x, y) A(, b) f x (, b) =0,f y (, b) =0 C-7-3 [ ] f(x, y) A(, b) f x (, b) =0,f y (, b) =0 H(x, y) =f xx (x, y)f yy (x, y) {f xy (x, y)} 2 () H(, b) > 0 f xx (, b) > 0 = f(x, y) A f xx (, b) < 0 = f(x, y) A (2) H(, b) < 0 f(x, y) A ( ) H(, b) =0 C-7-4 [ ] F (x, y) A(, b) A(, b) F (, b) =0, F y (, b) 6= 0 A F (x, f(x)) = 0 f() =b y = f(x) f(x) x = dx = F x(x, y) F y (x, y)
C-7-5 [ ] C : F (x, y) =0 A(, b) F x (, b)(x )+F y (, b)(y b) =0 F y (, b)(x ) =F y (, b)(y b) C-7-6 [ ] g(x, y) =0 f(x, y) A(, b) g x (, b) 6= 0 g y (, b) 6= 0 λ ( f x (, b) λg x (, b) =0 f y (, b) λg y (, b) =0 C 8. [ ] C-8- [ ] xy D x [, b] y = f(x), y = g(x), (f(x) > = g(x)) x =, x = b F (x, y) D b f(x) F (x, y) dx = { F (x, y) } dx D. [ ] D C-8-2 [ ] (C-8-) D y [c, d] x = h(y), x= k(y), (h(y) > = k(y)) y = c, y = d (C-8-) d h(y) F (x, y) dx = { F (x, y) dx} D c g(x) k(y) C-8-3 [ ] θ =, θ = β r = f(θ), r= g(θ), f(θ) > = g(θ) > = 0 D F (P ) β f(θ) F (P ) ds = { F (r cos θ, rsin θ)r dr} dθ D D 9. [ ] D-9- [ ] = f(x)g(y) dx g(y) = f(x) dx + C (C ) g(θ) D-9-2 [ ] dx = f( y x ) y x = u y = xu
D-9-3 [ ] + P (x)y = Q(x) dx y = e R P dx ( e R P dx Qdx+ C) (C ) D-9-4 [ ] dx + P (x)y = Q(x) y y + P (x) =0 dx y = y + Ce R P dx D-9-5 [ ] (C ) P (x, y) dx + Q(x, y) =0 P y = Q x D-9-6 [ ] P (x, y) dx + Q(x, y) =0 f(x, y) = Pdx+ (Q Pdx) = C y D 0. [ ] D-0- [ ] d2 y dx 2 = ky (k ) A, B () k =0 y = Ax + B (2) k>0 k = c 2 y = Ae cx + Be cx (3) k<0 k = c 2 y = A sin cx + B cos cx D-0-2 [ ] y 00 + y 0 + by =0 s 2 + s + b =0 A, B () = y = e x (Ax + B) (2), β = y = Ae x + Be βx (3) λ ± iµ = y = e λx (A sin µx + B cos µx) D-0-3 [ ] y 00 + y 0 + by = R(x) y y 00 + y 0 + by =0 u y = y + u