i

i 3 4 4 7 5 6 3

( ).. () 3 () (3) (4) /. 3. 4/3 7. /e 8. a > a, a = /, > a >. () a >, a =, > a > () a > b, a = b, a < b. c c n a n + b n + c n 3c n..... () /3 () + (3) / (4) /4 (5) m > n, a b >, m > n, a b <, m = n a /b, m < n (6). a = /, 3/8.3. () () (3). () b/a () (3) 9/ (3) 3. () () (3) / 4. () /3 () (3) 5. 6. () e= e (7 7 9 ) x = + h () e a (3) e (4) e (5) e a 7. () θ = sin x. () lim x a f(x) = f(a). (.3.4) (.3.3) 3. 4. 9. a f(nπ) = ( ) (a+)n a

( ).. 5. 5. f () =. x. () ( + x ) () (cos x + sin x)ex (3) x cos(x ) (4) x log x cos x 3. () + sin x () sin x x 4. () x + A () x a (3) x + A (4) a x 5. () p(p ) (p n + )a n (ax + b) p n () ( 3 n cos (3x + π ) n + cos (x + π )) n n k= nc k n k cos (x + π ) n cos (x + π ) n n!(n + x) (3).. (3) ( x) n+ ( 6. () x sin x + nπ ) ( ) ( ) (n )π (n )π + nx sin x + + n(n ) sin x + ( () n/ e x sin x + nπ ) A.3. sin x + cos x = sin(x + π/4) 4 (. () a a + a 4 x 4) ( x 3 () cos x log x + sin x ) e sin x (3) (4) a 4 x 4 x sin x [ (5) log(ax + b x ) x + (log a)ax + (log b)b x ] x(a x + b x (a x + b x ) /x ) 3. dy d = tan t, dx 4. dy dx = cosh t sinh t,.3 y dx = a cos 4 t sin t d y dx = a(sinh t) 3. () x = x = () x = /e (3) x = (4) x = 3/ 3 3/4 a b (a + b cos x) 3. () x. sin(/x) / f(c + h) f(c) f(c + h) f(c). lim. h h h 3. () < x π x > π () π/4 < x 5π/4 4. (4) a n r <

3 5. () f (k) (x) = P k (x)(x ) n k, P k :. f(x) = (x + ) n (x ) n 6. : f(x) = A(x a )(x a ) (x a n ). : f(a i ) f (a i ) f (a i+ ) a i a i+ f(x) = x 7. () n =, n = k, k n = k + (4): ()(3) H n+ (a i ) = H n(a i ). (5): H n (x) = a < a < < a n ()(3) 6 (, a ), (a, a ), (a n, a n ), (a n, ) H n (x) H n (a n ) >.4. () /3 () /6 (3) e (4). () e/ () (3) ab (4) /3 (5)..4.4 x = /y. γ = min{α, β}.5. () + x + x + + x n + R n () + x + x 4 + + x (n ) + R n n..3 3. () + x! + x n (n )! + R n, R n = cosh(θx) x n (n : ), R n = sinh(θx) x n (n : ) (n)! (n)! n: + x! + x n (n )! + R n R n = sinh(θx) x n (n)! n: + x! + x n (n )! + R n R n = cosh(θx) x n (n)! (7 7 3 ) () cosh x, sinh x e x (3)..4 lim n n! = 4. () + ) ( + (x) + + (x)n! (n )! + R n, R n = cos (n) (θx) (x) n cos x = ( + cos x)/ (n)!. f(a ± h) = f(a) ± f (a)h + (/)f (a + θ ± h). a = /3, b = /6 log( + x) = x (/)x + (/3)x 3 + O(x 4 ), ( + bx)/( + ax) = ( + bx)( ax + a x + O(x 3 )) = + (b a)x + a(a b)x + O(x 3 ) a n.6..5..3

4 ( 3 ) 3.. E = {(x, y) x + y < }, E = {(x, y) x + y > }, E = {(x, y) x + y = }, E. () (4) ()(3) 3. () () (3) (4) (5). () () (3). () () (3) 3.. () f x = 5x 4 y + 9x y, f y = x 5 y + 3x 3 + 3y () f x = y (x y), f x y = (x y) x (3) f x = x + y, f y = y (4) f x = y x + y x + y, f y = x. () f x = x + y e x +y y, f y = x + y e x +y x x + y y () f x = x (x y ), f x y = x (x y ) y (3) f x = (x + y ) log y x x y x(x + y), f x y = (x + y ) log y x + x y y(x + y) (4) f x = y(y x ) (x + y ), f y = x(x y ) (x + y ) f x = f y = 3. () z = x + y () z = x + 5y 6 4. () f xx = y e xy, f xy = f yx = ( + xy)e xy, f yy = x e xy () f xx = (y x ) (x + y ), f xy = 4xy (x + y ), f yy = (x y ) (x + y ) (3) f xx = log y x (log x) 3 (log x + ), f xy = xy(log x), f yy = y log x log x y = log y log x 5. () a m b n e ax+by () α(α ) (α m n + )( + x + y) α (m+n) (3) a m b n e ax cos (by + π ) n (4) ( ) n sin (x y + π ) (m + n) 7. () dz = (cos y y cos x)dx (x sin y + sin x)dy x () dz = x y dx y x y dy (3) dz = xy cosh(x y)dx + x cosh(x y)dy. x C x f y =, f x (x, y) = x sin(/x) cos(/x) (x ), f x (, y) =.

. () () f x (, ) = f y (, ) = 3. () () 5 3.3 sin 4t. () () 5 + 3 cos 4t sin v cos u. () z u = sin u + cos v, z sin u cos v v = sin u + cos v v ( () z u = cos (uv) sin (uv) ), (uv) u ( z v = cos (uv) sin (uv) ) (uv) (3) z u =. u η =. () u ξη = u u + v, z v = v u + v 3.4. () (3, ) () (, ) (3, 3) (3) (, ) (4) (, ) (/3, /3). () + x + 6 x3 xy + 4 x y + 4! y4 + R 5 () + ax + b +! (ax + by) + 3! (ax + by)3 + 4! (ax + by)4 + R 5 ( (3) x log + y y + y 3 ) ( 3 3 x3 log + y ) + R 5 3! log( + y) = log ( + y/). () (, ), (, ±), (±, ), (±/, ±/)( ) (±/, /)( ) ( /, ±/)( ) () (, ) H = (3) (, ), (, ), (, ) (, ) H =. AOB= x, BOC= y 3. 3.5. () z x = c x a z, z y = c y 3. () dy dx = x z y z, dy dx = y x y z () dy dx = x z y z, b z () z x = x y z dy dx = x + y y z, z y = y x z

6. () u x = x v u v, u y = y v u v, v x = u x u v, v y = u y u v () u x = x a u 3. () y = x + y x y, y = 5(x + y ) 4. ap + bq + cr + d a + b + c, u y = y a u, v x = a v, v y = a v (x y) 3 () y = b x a y, y = b a y 3

( 4 ) 7 4. + C. () ( (x + ) 3/ x 3/) () x sin x + cos x (3) 3 3 x cos 3x + 9 x sin 3x + cos 3x 7 (4) (x x + )e x. () a(α + ) (ax + b)α+ (α ), () (/)(x ) 5 (3) log(e x + e x ) (4) (/x)(log x + ) 3. () 7 (3x ) 3x + () x4 4 [ log x ] 4 log ax + b (α = ) a 4. () log sin x () x sin x + x (3) (log x) (4) 3 8 (x )4/3 (4x + 3) (5) tan e x (3) 3 (log x)3 (4) log log x 5. () x tan x log(x + ) () x log(x + ) x + tan x (3) (4) 5 (x + 3)3/ (x ) 6. () log x 6 log x + + 3 log x () log x + x + x + (3) 4 x 4 x + x + 4 log x x + tan x ( x tan x x + tan x ) (7 6 ) t = x x + (4) log x x 7. () x + x + log x () x + log x + + x + log(x + ) + tan x () x + log x + + tan x (7 6 ) x + (3) [ tan ( x ) tan ( ] x + )) (4) (x + ) + 4 log x + 3 x + (4) (x + ) + 4 log x + 3 x + (7 6 ). () (( + tan(x/)) () x + ) tan x t = tan(x/) tan(x/) / tan(x/) = / tan x t = tan x (3) log sin x + sin x 3. () tan(x/) () log tan(x/) + (3) x tan x/(= x sin x + cos x ) = x(tan(x/)) + tan(x/) 4. () tan x x t = tan(x/) 8 () ( log tan x + tan x = ) log cos x + sin x cos x sin x (3) log x + + x + 5. () x(sin x) + x sin x x t = sin x.

8 () x + sin x + 4 x (3) x ( tan 3 tan x x 6. () tan x + x (x )(x ) t = x = t dx dt = tx x(t)dt dt x (t) (7 6 ) ( ) () x3 + + 3 3 log x3 + x t = x3 + + x = t dx dt = tx x(t)dt dt x (t) (7 6 ) 7. () x ( ) 4 tan x 3 x (x () + 4) 3 tan 3 x (3) 3 log x 6 log(x + x + ) + tan ( x + ) 3 3 ) 4.. () log () log (3) ( + log(/3)) (4) π. () 3 8 π () log log 3 (3) log 3 (4) 7 8 e 4 + 3 8 3. () /3 () log (3) /6 (4) / log (5) π/4 4. () 8/3 () log(5/3) (3) (π/4)( / 3) (4) (π/4) 5. A.3. 6. () () I(, m + n). () π/4 / () π/(3 3) (3) π/ log = x. () tan x = tan x tan x () 4 tan π/8 tan () π/8 (3) π 4 tan () tan(π/8) = 3. () < < () ( x) /3 < < 5. (3) π (x π/)f(sin x) dx = 9. f(a) = a [, ] 4.3 a xf(x) dx + a ( tan x ) xf(x) dx. () π/ () π/ (3) 4 /4 (7 ) (4). () 4 /4 () (3) / (4) π a 3. () π/ () (3) (4) ab(a + b) a 4..4 (5) log + b 4. x n n. 5. (log x) n n. () () (3) (4) x

4.4 9. m, k, g m k mx = kx mg. v = x v =.. m, m b log a + bv a mv = av bv. m bv + a log = t + C, t = v = v a v C = m a log bv + a. v 3. ( + )t kv + mg. mg S, a, k, x Sx = ak x 4. y = a log a + a x a x x y a x = x 4.5. πa. 3π S = 3. 8 4. (3/)π 5. S = 4 6. π/4 7. π/4. π π π/4 y dx dt dt r dθ. 3/ { 3. β β + α ( α β + )} β + + log + α + α +

( 5 ) 5.. {(x, y) y x y, y /4} {(x, y) y x /, /4 y /} 3. {(x, y) y x +, x } {(x, y) x y, x } 4. {(x, y) x /4 y x /4, x } {(x, y) y x y, x } 5.. () 4 () log(6/5) (3) /3 / (7 ) (4) (/)(e ). () (/)π + (/)π + (7 ) () /3 (3) (4) + (3/8)π 3. () /8 () / (3) 4/5 (4) 3/8 4. () (/)e 3/ () 6 log 3/ (3) (/6)( log ) (4) /π. () 4/3 () (4/5)( + ) (3) e 4 (4) 7/4 43/4 (7 ). x y 3., x 5.3. () D = {(x, y) y x, x }, ϕ (x, y), ϕ (x, y) = x () D = {(x, y) x y }, ϕ (x, y), ϕ (x, y) = x + y (3) D = {(x, y) x + y }, ϕ (x, y), ϕ (x, y) = x y. () D = {(y, z) y z, z }, ψ, ψ = (y + z) () D = {(y, z) y + z }, ψ = + y + z, ψ = ( y + z ) 3. () D = {(x, y) x y, x }, ϕ, ϕ = x + y () D = {(x, y) y x, x }, ϕ, ϕ = x y (3) D = {(x, y) x x y x x, x }, ϕ, ϕ = x y 4. () /4 () 3/8 (3). () (/)(log 5/8) () / (3) /5. : x dx n xn n + x dx n f(t) dt = n! x (x t) n f(t)dt

5.4. () π log () π(a log a a + ) (3) π ( e a) (4) π sin a. () 3 () π 3 log 3 (3) 5 π (4) 3 4 log 3. () 3 9 () 3 3 π (3) π 6 9 (4) 7. () 3 64 () (3) e (4). () π () π 4 (3) π 5π (4) 6 64 5.5. () log () (3) (e )/. () π/4 + (/) log () π/ (3) π (4) π 3. () r > () r <. () 3/ () / 3π x + xy + y = (x + y/) + (3/4)y, u = x + y/, v = ( 3/)y. () log( + ) () (/)( cos ) (3) (/4)(e /e) (4) log( + ) 3. () / () / 5.6. 4 3 πabc. π 3 3. 3 V = D (x 3 + y 3 ) dxdy D = {(x, y) x + y, x 3 + y 3 } = {(x, y) x + y, x + y } 4. 4πa 5. π( + 3) 6. 3 π( ) 7. 6 3 3 = ( 3). π{a a + + log(a a + )}. 8 3 π 3. 4 7 4. 4a 5. π

5.7. I = (x + y ) dx, I = (x + y ) dy C C () I = 3, I = 3 () I =, I = (3) I = 4 3, I = 4 3 (4) I = 3, I = 3. /6. b < a, b > a π. C C C C ε C C ε C D C ε

( 6 ) 3 6.. () () (3) (4). () () α > α (3) (4). () (4) (4): 6..6 (ii). () (3) ()(3): < x < π/ (/π)x < sin x < x 4. (3) x > () x < () < a k < 6.. () lim f n(x) = n (x ) (x = ) () 3. () lim n f n(x) = x =, x =, < x < () f n (x). () lim f n(x) = () n. () lim f n(x) = () n 3. () () 4. () sin x (4) (3) 6.3. () () x = + x + x + 5x + 6x = 3 3x x = (3 n+ n+ )x n /3. () () (3) 3. () /e () /e n=. () x n+ = x x, x = t () t = x (3) < x < x >. () () 3. () () () n x = y (n+) () = n y (n) (). y () () =, y () = y (n) () 6.3.3 sin x x