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1 0. =. =. (999). 3(983). (980). (985). (966). 3. := :=. A A. A A. := := 4 5

2 A B A B A B. A = B A B A B B A. A B A B, A B, B. AP { A, P } = { : A, P } = { A P }. A = {0, }, A, {0, }, {0}, {}, A {0}, {}. N = { N } = { N}. 6 7 A B... b A b A B A B = { A, B} A B A B. A B = { A B} A B A, B. A B A B Ab A. A B A B A B. 8 9 A \ B = { A, B} AB. A B A \ B B A B B c B (A c ) c = A. n i= A i = A A A n A,A,A n, A B A B n i= A i = A A A n A,A,A n. A \ B B c.3 0

3 X AB = X AB X \ (A B) =(X \ A) (X \ B). () X \ (A B) =(X \ A) (X \ B). () X X A B A B ( ) X \ ( n i= A i) = n i= (X \ A i), (3) X \ ( n i= A i) = n i= (X \ A i). (4) X ( n i= A i) c = n i= Ac i (5) ( n i= A i) c = n i= Ac i (6) = X \ A = X \ B = X \ (A B) , 5,. (7). (0, 5,, ) (7) 4 5 (7) 0 5,.. R n = {(,,, n ):,,, n }, R =, R =. :=. =(,, n ), =. n. 0. ii :=. 6 7

4 =(,, m ), b =(b,,b n ) 4 = (0, 5, ) = 0 5 = 0 5 = (0, 5, ). i = b i, b. i =,,,m, m= n = b b + b =( + b,, n + b n ) 5 =(, ), b =(3, 4) + b =(4, 6). 8 9, V. + b = b +. ( ) V. ( + b)+c = +(b + c). V.3 b, + = b V.3b. b =(b,b,,b n n ). b5c =(5, 6) V. V.3 0. α α =(α,,α n ) α V.4 α( + b) =α + αb( ). V.5 (α + β) = α + β( ). V.6 (αβ) = α(β). V.7 =. b5α =0, β = V.4 V.7 V. V.7 6 X = {(, ) R : + =0} =(, ), b =(b,b ) X ( + b )+( + b )=( + )+(b + b )=0. + b X α + α = α( + )=0 α R. X X V. V.7. X 7 X = {(, ): + =}, b X ( + b )+( + b )=( + )+(b + b )=. + X X. R n 7R n. 3 6 X V. V.7 3

5 , b R n b b + + n b n (8) b. b. b = ( b) ( b) (9) = ( b ) + +( n b n ) (0) (, )(b,b ) ( b ) +( b ). b b b 4 5, b b b. b (, ) (b,b ) b + b =0, b R n (, b) =0 b 8 (, 0) (0, ) (, ) (, ). 6,, n α,,α n α + α + + α n n,, n α,,α n α + α + + α n n = 0 () α,,α n,, n α = = α n =0 (),, n 9 =(, 0, 0), =(0,, 0), z =(0, 0, ) α + β + γz = 0 α = β = γ =0,, z., =(,, 0), =(0, 0, ), z =(, 4, 3) +3 z = 0.,, z.. 7 8

6 ,, n j.,, n () α j 0, j = α j (α + + α j j + α j+ j+ + + α n n ). j = α + + α j j + α j+ j+ + + α n n j α,,α j,,α j+,α n (). 9 =(, ) =(, ). α R; = α, α 0. = =(, ) =(, ),, 30 3 R n,, n,, n. b = O 4,, n,, n 3 3,, n = α + + α n n = β + + β n n. (α β ) + +(α n β n ) n = 0,, n α = β, α = β, α n = β n.,,, n γ + + γ n n = 0 γ,,γ n, =(α + γ ) + +(α n + γ n ) n. R n,, m, m n,,, m (,, m )., ( ) (, ) R 3, ( ), (, ).,, m d, d (,, m ) (,, m ) d

7 3. m, n. m n ij, i =,,,m, j =,,,n m n n n m m mn () ()A A i i in, i =,,,m Ai i := ( i i in ) i j, j, j =,,,n Aj. mj j j := j. j mj A, A =., A =(,,, n ) m Ai j ij A i j (i, j) A A =( ij ) ( ) 3 5 A = A = 3, (, ) =8, (, 3) =5, =(, 3, 5), =(3, 6). m na =( ij ) B =(b ij )(i, j) ij = b ij, i =,,,m, j =,,,n A B A = B. m na B A+B (i, j) ij +b ij A + B =( ij + b ij ). ( ) ( ) ( ) 3 b c + +b 3+c + =

8 A ααa αa =(α ij ). A B B +( )A =[b ij ij ] B A, B A. ( ) ( ) =. b c 3 3b 3c b 4 5 = 3 3 b. 3 c 6 z 3 c z 4 M M7. M. A + B = B + A. ( ) M. (A + B)+C = A +(B + C). ( ) M3. A + X = B X. M4. α(a + B) =αa + αb. () M5. (α + β)a = αa + βa. () M6. (αβ)a = α(βa). M7. A = A m na = ( ik ), n lb = ( b kj ) A B AB nk= k b nk= k k b k nk= nk= k b kl AB := k b nk= k k b k nk= k b kl nk= mk b nk=. k mk b k nk= mk b kl Ai i =( i, i,, in ) B j b j =(b j,b j,,b nj ) i b j = ik b kj k= (3) A, B,, A =., B =(b, b,, b l ),, m 43 AB = b b b l b b b l m b m b m b l A B AB, A B A l m B m n AB l n 3 = = ( ) ( ). ( ) ( ) = = AB BA AB BA. 46

9 , M8 M0. 6 M8. (AB)C = A(BC) ( ) M9. A(B + C) =AB + AC ( ) M0. (B + C)A = BA + CA ( ). m m m 0O. ii,i=,,,m 0 I. A,AI = IA = A. (4) n n nn n n nn ( ) b 7 A = (4). c d 49 A AX = XA = I X AA A A (A ) = A. (5), A B, AB, (AB) =(B A ). (6) 8 (5) (6). 50 m na =( ij ) A (j, i) (i, j) n m A A. 5 ( ) 3 A =, A = A A (A ) = A. (A + B) = A + B, (7) (αa) = αa, α, (8) (AB) = B A (9) A = A A =( ij ) ij = ji. 9. (7) (8) (9) 5 5

10 A B AB f : A B A f B. AB f A B b b f f() b A B f : A B f {f() : A} f Imf f(a). A B Imf f b f b f Imf f f : A B b B b { A : f() =b} f b f (b). A B f (b) f b f b f (b) 6 f : R R, f() = Imf = { R : = }. >0f () ={, } B s B { A f() B s } f B s f (B s ). 7 f : R R; f() = A f (B S ) f B B S f B s f (B s) >0 0 =0 <0 b, B s = { R 0} R f (B s )={ R 0} Imf = BB b, f() =b A f A B A, f( )=f( ) = Imf bf b f A f B b f 8 0f : R R; f() = b R = b f b b 9 f 8, f( )= f 57 58

11 f : A B b B f() =b A. bf b A f B. f f. b b 0 8 f f B A f() =b f (b) =. A f B f b f 59 f : A B, g : B C 4 f f : A B b B f (b). f (b) A A f f() B, g(f()) C. Ag(f()) C. f g f g. g f() =g(f()) 60 6 A B C g f f b g c g f f : R R, f() =+, g: R R, g() = (g f)() =( +), (f g)() =( )+. U, V U V f f( + ) = f( )+f( ), f(α) = αf() f. f : R R, f() =,,, Rλ f( + ) =( + ) = + = f()+f() f(λ) =(λ) =λ() =λf() f. 6 63

12 3 A m nf A : R n R m f A () =A, R n λ f A ( + ) = A( + ) =A + A = f A ()+f A (), f A (λ) = A(λ) =λ(a) =λf A () f A. f A A 5 nr n mr m f m na. f : R n R m f() =A m na. A f R n {e, e,, e n } R m {e, e,, e m} =(,,, n ) R n = e + e + + n e n, f f f() = f(e )+ f(e )+ + n f(e n ) (0). f(e ),f(e ),,f(e n ) R m f(e j )= j e + je + + mje m, j =,,,n () j, j,, mj, j =,,,n. 64 A =( ij ) (0) () f() = ( e + e + + me m ) + ( e + e + + me m) + + n ( n e + ne + + mne m ) = n m m = (Ae )+ (Ae )+ + n (Ae n ) = A( e + e + + n e n ) = A. n n. mn 5 5f : R n R m f A (f(e ),f(e ),,f(e n )) = (e, e,, e m)a A R n f. 4 R {e, e } R 3 {e, e, e 3 } f : R R 3 f f(e ) = 3e e + e 3, f(e ) = 4e +5e 6e (,,,n)i, j, i j (,,,n) σ =(i,i,,i n )(,,,n).. n n A A = sgn(i,i,,i n ) i i inn () σ. sgn(i,i,,i n ) σ = (i,i,,i n ) () (,,,n) 65 66

13 5 A 6 A = A A = (3) (,, 3) 3! (,, 3) (,) (,, 3) (,3) (, 3, ) (,3) (3,, ) (,) (3,, ) (,3) (, 3, ) ()(3) 67 (,,n)(i,i,,i n ) k(i,i,,i n )k (,,n) A = σ = σ = A. sgn(i,i,,i n ) i i inn sgn(i,i,,i n ) i i nin 68 D. αα α i n = α i n. D.,. i j n = j i n. D.3, i i n =0. 69 D.4. i + b i n = i n + b i n. D.5. n = nn n nn = nn. 0 D.D D.D = , = = / = ( 3 ( 3/))=

14 7 n A, B AB = A B = BA. A =( ij )=( n ) BA =(B B B n ) BA = B B B n. B =(b b b n ) B = b b b n = Be Be Be n. (4) j = n i= ij e i, D. D.4, BA = B n i= i e i B n i= i e i B n i= in e i = n i= i Be ni= i i Be i ni= in Be i = ν ν νnn Be ν Be ν Be νn. D.3ν, ν,,ν n 0 (,,,n). D. BA = sgn(ν, ν,,ν n ) ν ν νnn (ν,ν,,ν n) Be Be Be n () (4) BA = A B AB = A B A = 0 A i,i=,,,n, A =[,, n ] i, i =,,,n, α,,α n α + + α n n = 0. α n 0 D. D.3 D.4 A = α n,, n,α n n = α n,, n,α + + α n n = α n,, n, 0 =0. 75 A A = 0. i,i=,,,n, A =[,, n ] i, i =,,,n, j, j =,,n, e j j + + nj n = e j j := ( j,, nj ) 0. X := [,, n ] AX = I. 7 D.5 A 0. A X = AX = I = 76 8 n,, n A =[,, n ] A =

15 , = = λ m n A = λ... 0 () A ij, () A iλ, () A j =( j +λ i ), A () ij, A () iλ, A () i =( i +λ j ) A = b c z A = b : A. 0 0 c z λ 0 A = λb λ : Aλ. 0 0 c z 0 0 λ 0 A = λ + b λ+ : A + λ. 0 0 c z ( b c 8 B = z ) ( B b c = z B 0 0 ( 0 λ 0 λb c = λ z 0 0 B 0 0 λ 0 = 0 0 ) : A. ) : Aλ. ( ) + λb b c : A + λ. + λ z 83 84

16 , =, 87 0 A A. A r A rnka = r mn n n = b n n = b (5). m + m + + mn n = b m A = n n m m mn, =. n, b = b b. b m A = b (6) (6)(5) A (5) n b (A b) = n b m m mn b m (5) (5) rnk(a b) = rnka

17 (A b) c j d c j d.... c rjr d r 0 d r+ (5) c j j + +c n n = d. () c rjr jr + + c rn n = d r (r) 0 = d r+ (r +) (7), (7). 9 d r+ 0(r+),,, n., d r+ =0 rnk(a b) = rnka. rnk(a b) = rnka, d r+ =0 (7),,, n. r j,, jr, ( n) (rnka), (r) jr (r ) jr, () j jr, jr,, j. 9, rnk(a b) = rnka ( n) (rnka) = = =. (A b) = 3 4, 9 3 5, rnk(a b) = rnka = 94 { = =. ( ) =( ) rnka =4 =. 3 = α() 4 = β() =3+5α +8β = α β, (α, β ). = α 4 = β (A b) = + 3 = = = 5, , rnk(a b) =3> rnka =. 96

18 = = = 0 (A b) = 3 5, , 3 +, rnk(a b) = rnka = = 5 3 = 3 = 0 =57, =, 3 = 0 3 ( ) =( ) rnka =3 3=0... [] [( ) =0] A = 0 b = 0 A = n n = n n = 0 (8). m + m + + mn n = 0 A = b b 0 (8)rnkA = rnk(a 0). (,,, n )=(0, 0,, 0) (8) 99 A = 0 rnka <( ). 3 A =( n ) m n. n rnka = n. =(,,, n ) A = n n. n A = 0 rnka = n. A() A = sgn(p,p,p 3 ) p p p3 3. (9) (p,p,p 3 ) (9) sgn(,p,p 3 ) p p3 3 (,p,p 3 ) = sgn(p,p 3 ) p p3 3 (p,p 3 ) = A, 0 0 A := ij ij i j A ij A ij A(i, j) A ij, Ai j ij (30). A ij =( ) i+j ij. (30) () A,, n A = A + + n A n. 0

19 , i,, A = i A i + + in A in = j A j + + nj A nj. (3) A A (3)( i,, in ) j, j i, ( j,, jn ) j A i + + jn A in (3) A i j, i A 0 D.3 (3). ( i,, in ) A { j. A i = j = 0 i j. A jn A A A A n A A A n A n A n A nn = A A = A I. (33) (A ji )dja. (33) A8 A 0 A = dja. (34) A 03 4 () n n = b n n = b (35). n + n + + nn n = b n A =( ij ), (35) j = A (b A j + + b n A nj ), j =,,n. (36) =(,, n ), b =(b,,b n ), (35) A = b. (37) A (37)A = A b. (34)(36) ,,, 3, (38) k k 0 n, n (38) n, (38) { n }. 33 () k, k =,, P k P 0. P 0, P, P, P

20 { n } n+ n = d (d ) (39) { n }d (39) 0 = = 0 + d = + d, = + d = +d, 3 = + d = +3d,. d n = + nd. (40) () 33k + P k+ P k. P 0 = P i P k+ P k = i P (4) {P n } P ip. (40) P n =(+ni)p =3 d = n =3+n : {3, 4, 7, 9, } =5 d = 3 n =5 3n : {5,,,, } 3 d. ➀ = 0, d =. ➁ =0, d =. ➂ = 3, d = 3. 0 { n } k+ = r k, r (4) { n } r (4) 0 =, = r 0 = r = r = r 3 = r = r 3 r. n = r n. (43) 36 () 33 P k+ P k = i P k, i (44) P k P k+ =(+i)p k. {P n }+i P 0 = P (43) n P n = P ( + i) n. 37 = r =.5 n = (.5) n : {, 3, 4.5, } = r =0.5 n = (0.5) n : {,, 0.5, } = r =.5 n =(.5) n : {, 3, 4.5, } = r = 0.5 n =( 0.5) n : {,, 0.5, } 4, r 4. ➀ =5, r =. ➁ =, r =3. ➂ =, r = % 0. 3

21 { n }n. k = n. k=0 5. ( k ± b k ) = k ± b k,,(45) k=0 c k k=0 k=0 k=0 = c k, c. (46) k=0 4 ( k ± b k ) = ( 0 ± b 0 )+( ± b )+ +( n ± b n ) k=0 =( n ) ± (b b n ) = k ± b k, k=0 k=0 c k = c 0 + c + + c n k=0 = c( n ) = c k. k=0 5 3 n. n(n +) k =++ + n =. (47) k= ( +) ( ) =,,, n 6 6 ( ) d { n } n. (n + )( + nd) k = ( + kd) =. (48) k=0 k=0 (45) (46) 3 ( + kd) = k=0 + kd k=0 k=0 = (n +) +(++ + n)d n(n +) (n + )( + nd) = (n +) + d = (n +)(6+n) (3 + k) = =(n + )(n +3). k=0 6 3n. 7 ( ) r { n } n. { k = r k = rn+ r = rn+ r, r, (n +)r, r =. k=0 k=0 (49) r k = + r + r + + n k=0 = ( + r + r + + r n ). r =. 8 9

22 , r r = r, r r = (r )r, r 3 r = (r )r, r n+ r n. = (r )r n r n+ =(r )( + r + r + + r n ). +r + r + + r n = rn+ r. = rn+ r 39 ( ) ( ) k n+ ( ( ) ) n+ 3 =3 =6. k=0 7 4n. 0. { 0 =, k+ = r k + d, k =0,,. r d. (50) (50) (50) (50) = r 0 + d = r + d, = r + d = r + d( + r), 3 = r + d = r 3 + d( + r + r ),. (50) n = r n + d( + r + r + + r n ) { r = n + d rn r, r, + nd, r =. (5) 3 40 C, n. i. kp k k + P k + C k +i(p k + C) P 0 =0 {P k } { P 0 = 0, (5) P k+ = (+i)p k +(+i)c, k =0,,,. (5) (50) =0, k = P k, r =+i, d =(+i)c (5) P n =(+i)c ( + i)n. i 4 { n } {b n }, n = b n+ b n b b 0 = 0, b b =, b 3 b =,. b n+ b n = n. k = b n+ b 0. k=0 5

23 4 k=0 k = n(n + )(n +). (53) 6 (k +) 3 k 3 =3k +3k + (n +) = 3 k +3 k + k=0 k=0 k=0 = 3 k +3 k +(n +). k=0 k=0 3 k = 3 (n +)3 k=0 k=0 k (n +) 3 = 3 (n +)3 n(n +) (n +) 3 = n(n + )(n +) k= k(k +) = n n +. (54) k= k k + = k(k +) k(k +) = n + = n n +. 8 ➀ nk=0 k 3 = 4 n (n +). ➁ nk=0 k(k +)= 3n(n + )(n +). ➂ nk=0 k(k + )(k +)= 4n(n + )(n + )(n +3). ➃ nk= k(k+)(k+) = (n+)(n+). 7 8 { n } 0 n n (55), { n }, 0 n n (56). (55) (56) n n l { n } l n n = l n l (n ). l { n } 9 30

24 n n { n } n n = n (n ). n n { n } n n = n (n ). { n } n n 3 43 n n = l l n n = n n = n n n n + =0. (n +)=, n n (n n )=. {( ) n }, {( ) n }. n (n )= ({ n = + nd} ) +, d > 0 ( + nd) =, d < 0 n, d =0. 45 ({ n = r n } ) n rn = +, r >, r = 0, <r< r. 8 { n }, {b n }. n ( n ±b n ) = n n ± n b n,, n (c n )=c n n, c, n ( n b n ) = n n n b n, n n n n bn = n b n, n b n n n 4n + = n 3 n 4+ = 3 n 4. n 3 +3 n n + = n n + 3 n + =+. n ( ) n (n n 4 ) = n 4 n n =. n ( n + n) = =0. n n ++ n 9 { n }. ➀ n = 3n + n +n. ➁ n = n 4n +3 n. ➂ n = n + n

25 { n } (57) () n=0 n n. n S n = k = n k=0 (57)n n S n = S (57) S (57) S (57) n=0 n = S S n=0 n = n=0 n = n=0 n 47 (), r { n }. { (r S n = r k n+ ) = r, r, (n +), r =. k=0 ±, r, 0(), n = r, <r<, n=0, r r. ➀ =, r =. ➁ = 3, r =3. ➂ =, r =. 9 n, b n. ( n ± b n ) = n ± b n,. n=0 c n n=0 n=0 n=0 = c n c. n= ➀ n n n =0. ➁ n n 0 n. ➀ n S n n S n = S ➁. n n = (S n S n n )=S S = n = n ++ n, n =,,, n n = =0. n n ++ n S n = = k = k= k= k ++ k ( k + k)= n +. k= 6. S n = ( n + )=+. n n

26 R. D R R f f DR = f() f : D R D f R. f() D f f { = f(), D} = f() f :. f O D { = f(), D} f = f() f() [, b] ={ b}, (, b) ={ <<b}, [, b) ={ <b}, (, b] ={ < b}, [, ) ={ }, (, ) ={ <}, (,b]={ b}, (,b)={ <b}, (, ) ={ << } = R. [, b], (, b) [, b] [, b) (, b] (, b) f(), f(). b b b b i,b i R, i =,,n, f() = n n f() = n n b 0 + b + b + + b n n 0. f {(, ) = f(), D}. 46, ➀ = + +. ➁ = ++. ➂ =( )( ). ➃ = ( )( ). I, < f( ) <f( ) I f, > f( ) >f( ) I f 50 = n, n N :=, n (, ), n (, 0][0, )

27 = 4 = 3 f : D V D = f().. f, f. O f : V D, = f (). (58) (58) = f (). = f () = f() =. 5 n = n = n. 50 f()f() α f() α f() =α,, f() α( ). α f() cf() =c f(), (f() ± g()) = f() ± g(), f()g() = f() g(), f() g() = f() g(), g() 0. c,, ± n n = ( )=. = = +, > 0. = (n + n + + n )=n n. O = n n O = = n + n + + n O 53

28 . ➀ ➁ ➂ 4 4. ( 3 + ). 0, +0 + f() α β f() =α, f() =β, + α β f() =α =, O 0+ =. 3. ➀. ➁ +. ➂ n ( []). ➃ n+ ( []). n[] n <n+ n. 56 f() f(), f() =f() (59) f() = I f()f() I [, b], b f() =f(), + f() =f(b) b (59). 57 = f() = f() = f() () f()[, b] f() <f(b)f() <k<f(b)k f(c) =k, <c<b c. f() >f(b) f(+) f() f() O f() O f( ) =f(+) O 3 () [, b]f() f(b) = f() = f() k f() O c b O b 58 59

29 54 f() = [, ], 40 f() = (0, ) = = >0, = (, ) (0, ) (, ) O O = > 0 <<. = = log, (0, ) (, ) (0, ) 60 6 = = log () O 4 ➀ = +. ➁ = /. 5 ➀ = log. ➁ = log. ➂ = log =, log =0, =, log =, + =, log = log + log, =, log = log log, ( ) =, log = log. 6, b, c ➀ log b log b =. ➁ log = log b log b. ➂ log b log b c log c =. e ( n = + n) n (, b n = + n) n+, n =,, { n }, {b n }. n n b n < < :, b >b > : { n }, {b n } 63 64

30 (60) n N e. ( e := + n ( = + n n) n+. (60) n n) e e =.788. ( + =e R, (6) ± ) = h (6). h 0 ( + h) h =e. (6), e e e log ln ➀ = e ln. ➁ log = ln ln. 5 (6) log( + h) =, h 0 h (63) e h =. h 0 h (64) log( + h) = log( + h) h = log e =. h 0 h h 0 k := e h h 0 k 0 e h k = h 0 h k 0 log( + k) = k 0 k log( + k) = e +h e h 0 h log( + h) log h 0 h k := h = e e h = e, h 0 h ( = h 0 h log + h ) = k 0 k log( + k) =, 8. ➀ e 0 3. ➁ e e. log ➂ e e. 69 XOP O OPOX O XOP OP. OXOP P XOP O X 70

31 OOX A, OP QQ OP A. OP, AQ α α XOP OP OX α OP θ. θ = α +nπ, n =0, ±, ±, π π π 0 π π π 9. 0, 65, 00, O P(, ) OP θ θ P, P. θ. sin θ =, cos θ =, tn θ =. θ. π π sin θ O π : sinθ π θ 7.5π π.5π -0.5π cos θ O - : cos θ tn θ π 0.5π O 0.5π : tnθ 0.5π.5π θ π.5π π θ sin θ, cos θ (, ) tn θ ( π + nπ, π + nπ), n =0, ±, ±,. sin θ, cos θ θπ tn θ θ π sin θ, cos θ π tn θ π 30 ➀ = sin. ➁ = cos. ➂ = sin. ➃ = tn tn θ = sin θ cos θ. sin θ + cos θ =. sin( θ) = sin θ, cos( θ) = cos θ, tn( θ) = tn θ. ( sin θ ± π ) ( = ± cos θ, cos θ ± π ) = sin θ,. sin(θ +nπ) = sin θ, cos(θ +nπ) = cos θ, tn(θ + nπ) = tn θ, n =0, ±, ±,. sin( ± b) = sin cos b ± cos sin b, cos( ± b) = cos cos b sin sin b. 3 ➀ sin θ = sin θ cos θ. ➁ cos θ = cos θ sin θ = cos θ = sin θ. ➂ tn θ = tn θ tn θ

32 7 sin θ cos θ =, =0. (65) θ 0 θ θ 0 θ OPQ < OPQ < OPT sin θ< θ< θ tn θ < sin θ < cos θ > sin θ > cos θ. θ cos θ (θ 0) (65). Q T O θ H P (65) cos θ θ = cos θ θ(cos θ +) = sin θ θ(cos θ +) = sin θ sin θ θ cos θ +, sin θ θ 0 cos θ + = 0 =0. 3 ➀ sin 3θ θ 0 θ. tn( θ) ➁ θ 0 θ. ➂ cos θ θ 0 θ [ π, π ] = sin. = rcsin. ( π, π ) = tn. = rctn. = rcsin = sin,, π π, = rctn = tn, <<, π <<π. 33. ➀ rcsin( ). ( ➁ rcsin 3 ). ➂ rctn( ). ➃ rctn( 3). 0.5π O 0.5π = rctn 8 8

33 7. f() 0 f() f() ( ) n = 0 (66) f() ( ) n 0, f() n f(). f() =o( ) n. (67) n =0( ) 0 = f() =o() ( ) 0 f()., n n n f() =0 ( ) f() =o n. 56 cos cos 0 = 0 (cos +) ( ) sin = 0 cos + =. cos = + o( ). 34 ➀ n = n + n n ( )+o( ). ➁ e =+ + o(). ➂ log( + ) = + o() = f(). f(). (, ) = f() (, f()). = f()+m( ) m. f() =f()+m( )+o( ). f() f() m( ) ( ) = f() f() m =0 f() f() = m. (68) 87 (68). f() = f (), = f() f f() f() () =. f() f() =f()+f ()( )+o( ) f() =f(). f() =. 88

34 = f()+f ()( )P(, f()) = f() = f() P Q(, f())q P f() f() (PQ ) = = f()+f ()(b ) f(b) f() O = f()+f ()( ) P b Q = f() 57 f() =e f () =e = e (0, ) = f(0) + f (0)( 0)=+. f() = log f () = = log (, 0) = f() + f ()( ) =. 35. ➀ = 3, (, ). ➁ = e, (,e). ➂ = log, (e, ) = f() I f() I, I. = f(), f (), d d,. d f(), Df() d f () = h 0 f( + h) f() h f () f() = f () f () =, f () f () =. f() f () c 0, c n n n n e e log sin cos = sin ( + π ) cos sin = cos ( + π ) tn cos, > 0, rcsin << rctn + ( ) h h, = f()f(+ ) f().,, f (). f () = 0. = f () + o( ). 0, f (). = f() d df(). d =df() =f (). (69) 93 f() = f () = d =. (70). (69) (70) 58 d = f ()d d d = f (). d 3 =3 d, de = e d. 94

35 36 ➀ d n ➁ d log ➂ d cos 8. ➀ (f() ± g()) = f () ± g () () ➁ (cf()) = cf () c ➂ (f()g()) = f ()g()+f()g () ( ) ➃ f() g() = f ()g() f()g () g() = m + b n = m m + nb n, = e sin = (e ) sin + e (sin ) = e (sin + cos ), = tn = ( ) sin cos = (sin ) cos sin (cos ) cos = cos +sin cos = cos. 37. ➀ = ➁ = e. ➂ = +b c+d () = f(u), u = g() = f(g()). d d = d du du d. u, u, = u u. 0 u 0 0 = u 0 u u 0. d d = d du du d u, = u n = d du du d = nun u, = log u = d du du d = u u. = log = 0 6 = log u u = du d = {, > 0, < 0 =. d d = d du du d = u = = = =. >0. log = log., =. = = =. 38. ➀ =( +3 +4) 5. ➁ = log( + +). ➂ = e. ➃ =. ➄ = +. ➅ = log( + +) ( ) = f() = f (). d d =. d d =

36 63 ➀ = rcsin =. ➁ = rctn = +. ➀ = rcsin = sin. d d = d = cos. d π <<π cos >0 cos = sin = ➁ = rctn = tn.. d d = = = d d cos + tn = >0 ➀ = rcsin. ➁ = rctn. 0 0 = f()f (). f(), f (), d d d, d f(), D f().,, n. n (n), f (n) d n d n (), d n, d nf(), Dn f(). f (n) (), f() n nf (n) (), f()n =0 f (0) f(). 03 f() f (n) () ( ) ( n +) n e e log ( ) n (n )! n sin sin ( + nπ ) cos cos ( + nπ ) 40 ➀ f() = sin f (n ) (0)=( ) n, f (n) (0)=0. ➁ f() = cos f (n ) (0) = 0, f (n) (0)=( ) n. ( ) ➂ f() =(+) f (n) (0) = n!. n 04,. [, b] [, b]i. () f()., f() <f()f() = f() >f()f() = f() O 05 06

37 3 f() = f () =0 =, f() <f(). > < f() f() < 0, f() f() > 0, f() = f () 0, f () 0 f () =0 = ((Rolle) ) f() =f(b) c. f (c) =0, f(), b <c<b 3 f()[, b] <c<bc = c f(), 3 f (c) =0. f() =f(b) <c<b c f (c) =0 08 O c 4 f () =0 = f() 4 c ➀ f() = 3, 0 ➁ f() = 3,. b () f(), b c. f(b) f() = f (c), <c<b. b (, f()), (b, f(b)) = F () =f() f(b) f() ( b)+f(b). b f(b) f() ( b) f(b) b 0 F () =F (b) =0 F (c) =0 c, <c<b. F (c) =f f(b) f() (c) b. f(b) = f() 43 c. ➀ f() = log, e. ➁ f() =, b. f() O c b

38 34 ((Cuch) ) f(), g() [, b] [, b]g () 0c. f(b) f() g(b) g() = f (c) g (c), <c<b. g(b) =g()g ( 0 )=0 0, < 0 < b. g() g(b) f(b) f() F () =f() g(b) g() g() F (b) =F () F (c) =f f(b) f() (c) g(b) g() g (c) =0 c, <c<b, f(b) f() g(b) g() = f (c) g (c), <c<b. c. g() = c. f() = 4, g() =, 0 <<b ➀ I f () =0 f I ➁ I f () > 0 f I ➂ I f () < 0 f I I,, < f( ) f( ) = f (c) c, <c<. ➀ f (c) =0 f( )=f( ) f(). ➁ f (c) > 0 < f( ) < f( ) f(). ➂ f (c) < 0 < f( ) > f( ) f() >0. ➀ f() = log( + ). ➁ f() =e. 36 ➀ I I f () > 0 f() f()+f ()( ), I (7) =. ➁ I f () < 0 f() f()+f ()( ), I (7) =. 6 ➀ g() =f() f() f ()( ) g() =0 g () =f () f (), g () =0. f () > 0 f (). < g () <g () =0, > g () >g () =0. g() = g() 0 =.. ➁. (7), (7) ➀, > 0. ➁ log e. I, 0 <α< α f(α +( α) ) <αf( )+( α)f( ) f I f(α +( α) ) >αf( )+( α)f( ) f I 8

39 f( ) αf( )+( α)f( ) f( ) f(α +( α) ) O α +( α) O f(α +( α) ) f( ) αf( )+( α)f( ) f( ) O α +( α) O = f() = f()+f ()( ) = f()+f ()( ) = f() 9 37 ➀ I f () > 0 f I. ➁ I f () < 0 f I. ➀ I, α, 0 <α< = α +( α) (7) f( ) > f( )+f ( )( ), f( ) > f( )+f ( )( ). αf( )+( α)f( ) >f( ) =f(α +( α) ). f. ➁ ((L Hospitl) ) f () f() =g() =0, g () = α f() g() = α. f () g (), g () 0,>,, f() f() f() = g() g() g() = f (c) g (c), <c<. + c + f() + g() = f (c) c + g (c) = f () g ().. 0 0,, 0,, 00,, 0.. log log = 0+0 = 0+0 = 0+0 ( ) = ➀ n e, n. ➁ 0+0. ➂ sin cos 0 sin.,. f g f() =g(), f () =g (),, f (n) () =g (n) () (73) f() g() f () g () ( ) n = = n( ) n f (n) () g (n) () = n! = 0 f() =g()+o( ) n. (74) 3 (73) g()g f. g., g g() =α + β( )+γ( ) f() =g(), f () =g (), f () =g () α, β, γ (74) g () =β +γ( ), g () =γ. α = f(), β = f (), γ = f (). f() =f()+f ()( )+ f () ( ) + o( ). 4

40 f() = f()+f ()( )+ f () ( )! + + f () ( ) n + o( ) n (75) n!. (75) f() = (Tlor) =0 f() =f(0)+f (0)+ f (0) + + f (0) n +o( n )! n! (76). (76)f()(Mclurin) 48. ➀ ( + ) =+ + ( ) + o( ). ➁ ➂ ➃ ➄ e = o(3 ). log( + ) = o(3 ). sin = o(6 ). cos = o(5 ) D, = (, ) D D, = f(), = f(, ) n,,, n f(,,, n ). 7 8 =(, ) =(, ) (, ) (, ) := ( ) +( ) 0. =(, ) =(, ) f(, ) l, f(, )=l, f(, )=l (, ) (, ) f(, ) l 65 (, ) (,) ( +4 )=+8= (, ) (0,0) + ➀ (, ) =0(0, 0) ➁ +4 4 (, ) (0,0) + = 0 = 4=4. 0 (, ) = m +4 (, ) (0,0) + = +4m 0 + m = +4m +m.! 49 (, ) (0,0)

41 39 () (, ) (, )f(, ) g(, ). cf(, )=c f(, ), {f(, ) ± g(, )} = f(, ) ± g(, ), {f(, ) g(, )} = f(, ) g(, ), f(, ) g(, ) = f(, ) g(, ) 0. g(, ) 50. ➀ (, ) (0,0). ➁ (, ) (,). 3 f(, )=f(, ) (, ) (, ) f(, ) (, ) f(, ) D f(, ) D 40 () f(, ) g(, )(, ) f(, ) ± g(, ), f(, )g(, ), f(, ) g(, ), g(, ) 0, (, ) D. 3 67,,, ( ) f(, ) = φ(u,v ) = ψ(u,v ) f(φ(u,v ),ψ(u,v )). 68, f(, ) (, ),. 69 { f(, )= (, ) (0, 0) + 0 (, )=(0, 0) 0 f(, ) +. (, ) = m f( m, ) = (, ) (0,0) 0 + m = +m. 34 m(, ) (0, 0). f(, ). =0 f( 0, 0) = 0 0 0). +0=0=f(0,, f(, ),, f(0, )=f(0, 0) 0. f() 0 f() f() n =0 f() n f() =o ( n ) f() = c, c 0, c ± n f() =c n + o ( n ). (77) 35 5 (77) 36

42 z = f(, ) = f(, ) = f(, ) f(, ) f( = + h, ) f(, ) h 0 h =(, )f(, ) z f (, ), f(, ), =., f(, + h) f(, ) h 0 h z f (, ), f(, ), = = = f(, ) = f(, ). (,,f(, ))f (, ) = = f(, ) = f(, ) (,,f(, ))f (, ) 0 (,,f(, )) = f(, ) (, ) 0 (,,f(, )) = f(, ) (, ) f (, ),f (, ) = f(, ) A R = f(, ) A f(, )= (, 3). (, )=(, 3)f(, )= f (, 3)=3 6 ( 3)=48. f (, 3) = 3. ➀ f(, )= +3 (, 3). ➁ f(, )=e (cos + sin ) ( 0, π ). 40 = f(, ) D D (, ) f (, ) f(, ) f (, ),, f,. f(, ) f (, ),, f, f (, )=3 6, 53 ➀ =. ➁ = ln( + ). f (, )=

43 f(, )(, ) f f (, ) f. f, f (, ) f. f (, ), f (, ),, k, k =3, 4,, f =6 6, f = 6, f =30, f = 6, f =6, f = 6, f = 6, f =30, f = ➀ = 3. ➁ = () f(, ), f = f 4 nn. f(, ) = l( )+m( )+f(, ) +o( ) ( ) (78) l, m f(, ) =(, ) = l( )+m( )+f(, ) (,,f(, )) = f(, ) A R f A = l( )+m( )+f(, ) 0 (, ) 3 47 = = (78) f( +, + ) f(, ) = l + m + o( ( ) +( ) ) ((, ) (0, 0)) (79) (79) (, )(0, 0) =0 (79) f( +, ) f(, )=l + o( )( 0). { } f( +, ) f(, ) l =0. 0 f( l = +, ) f(, ) = f (, ). 0 48

44 m = f (, )(79) f( +, + ) f(, ) = f (, ) + f (, ) (80) +o( ( ) +( ) ) ((, ) (0, 0)) (80) = f(, ) f (, ) + f (, ) = f(, ) (, ) = f(, ) f + f (8) d df 49, = f =, f =0(8) d = = d =. (8) d = f d + f d (8) f = f( ) (8) = d d, =0 d = d d d = f ( )d = f(, )=. d = f d + f d = d + d. 55. ➀ = ➁ = e +. ➂ =. 43 () = f(, ) = φ(t), = ψ(t) = f(φ(t),ψ(t)). d dt = d dt + d dt. 5 5 = f(φ(t),ψ(t)) t d dt = f(φ(t + t),ψ(t + t )) f(φ(t),ψ(t)). (83) t 0 t t t, φ(t + t) = +, ψ(t + t )= +. t ( ) +( ) 0. ) o ( ( ) +( ) = o ( t). (80) f( +, + ) f(, ) = f (, ) + f (, ) + o( t) ( t 0) { f( +, + ) f(, ) t 0 t f t f t (83). } =

45 74. d = e, = t, = cos t d dt d dt = d dt + d dt = e t e sin t = e ( t sin t). 56 = ln( + ), = t + t, = t(t ) d dt. 44 () = f(, ) = φ(u,u ) = ψ(u,u ). = +, u u u = +. u u u u u 43. u, u u = +. u u u u u. 75 = u, u. +, = u + u, = u u = + = u u u u = u + u = u. + + u = 3 e, = u + u, = u u u, u ( (Tlor) ) f(, ) m. f( + h, + h ) ( ) = f(, )+ h + h f(, ) + ( ) h + h f(, )! ( ) m + + h + h f(, ) (m )! +R m, (84) ( ) R m := m! h + h m f( + θh, + θh ), 0 <θ<, ( h + h ( ) h + h f. ( ) h + h m f ) f := h f + h f, := h f +h h f + h f := h m m f m + m C h m h m f m + + h m m f m

46 F (t) =. F (m) (t) = F (t) =f( + h t, + h t) ( h + h ) f( + h t, + h t), ( ) m h ++h f( + h t, + h t). F (0) = f(, ( ), F() = f( ) + h, + h ), F (0) = h + h f(, ), (. ) F (m) (0) = h + h m f(, ). F (t) F () = F (0) + F (0)! (84). + + F (m ) (0) + R m, (m )! R m = F (m) (θ), 0 <θ< m! f(, )=e + (0, 0). f i = e +, i =,, k f k = e +, k =,,, i m f k = e +, k =,,,m, m=,,, i m k i e + = + +! R m + ( + )! + + ( + ) m (m )! = ( + ) m e θ( + ), m! + R m, 0 <θ< m =3. ➀ +. ➁ ln( + + b ) (),. f.. u = f(, ) u i, i =,.. 65 u = u = f(, )=u (85) (, ) u = du = d d = u d + u d =0. u u. (86) (86), 66

47 f(, )=u f d f d 9. O 59, u =. (, )=(, ) (, )=(, ) f(,,, n )(,, n ), (,, n ) (,,, n ) f(,,, n ) <f(,, n ) f(,,, n ) (,, n ) f(,, n ), f(,,, n ) >f(,, n ) f(,,, n ) (,, n ) f(,, n ) f(,,, n ) (,, n ), f(,, n )=0, i =,,,n.(87) i g( i )=f(,, i, i, i+,, n ), i =,,,n g( i ) i = i, 3 dg( i) d i =0. i f(,, n )= f(,,, n ) (87) (,,, n ), (,,, n ) f(,,, n ) (87) (87) (,,, n ) f(,,, n ) f(,,, n ) (87). A ij = i j f(,, n ), i,j =,,,n A ij = A ji n[a ij ]. 7 na =[ ij ] q() = ij i j i,j= = A, =(,,, n ) q() 0 q() > 0 q(), q() 0 q(), q() < 0 q(), q() 0 q() 7

48 47 f(,,, n ), : A ij ξ i ξ j 0. i,j= f(,,, n ), A ij ξ i ξ j 0. i,j= (45) f( + h, + h ) f(, ) = h i f(, ) i= i + h i h j f( + θh, + θh ) i,j= i j = h i h j f( + θh, + θh ), i,j= i j <θ<., i j f( + θh, + θh ) = i j f(, )+ɛ ij = A ij + ɛ ij f( + h, + ( h ) f(, ) = i,j= A ij h i h j + ) i,j= ɛ ij h i h j. r = h + h n ξ i = h i r f( + rξ, + ( rξ ) f(, ) i,j= = r A ij ξ i ξ j + ) i,j= ɛ ij ξ i ξ j. 75 f i r 0 ɛ j ij 0 ɛ ij ξ i ξ j i,j= ɛ ij ξ i ξ j ɛ ij 0(r 0). i,j= i,j= i,j= A ij ξ i ξ j (ξ 0,ξ0 ) (0, 0) r (88) f( + rξ 0, + rξ 0 ) f(, ) < 0 f(, ). f(, ) i,j= A ij ξ i ξ j 0. f(, ) i,j= A ij ξ i ξ : A ij ξ i ξ j > 0 f(), i,j= A ij ξ i ξ j < 0 f(). i,j= i,j= A ij ξ i ξ j > 0 ξ + ξ n = (ξ,ξ ) i,j= A ij ξ i ξ j m>0 m (88)r i,j= A ij ξ i ξ j + i,j= ɛ ij ξ i ξ j > 0. i,j= f( + rξ, + rξ ) f(, ) > 0. A ij ξ i ξ j < 0 f(, ) 77 78

49 f(,,, n ) ( ) n i,j= A ij ξ i ξ j ( )0. ni,j= A ij ξ i ξ j > (<)0 f(,,, n ) ( ). k k = k......, k =,,,n k k kk A k n A =[ ij ] q(,, n )= ij i j q(,, n ) k > 0, q(,, n ) ( ) k k > 0, q(,, n ) n =0 n =0 0 r r 0 > 0, > 0,, r > 0, r+ = = n =0, q(,, n ) n =0 n =0 0 r r 0 < 0, > 0, 3 < 0,, ( ) r r > 0, r+ = = n = A ij := i j f(,, n ), i,j =,,n, A A A k H k := A A A k......, k =,,,n A k A k A kk k (Hessin) (Hesse) 49 f(,, n ) f(,, n )=0,, f(,, n )=0 n ➀ f(,, n ) H k 0, k =,,,n H k > 0 f(,, n ) ➁ f(,, n ) ( ) k H k 0, k =,,,n ( ) k H k > 0 f(,, n ) f(, )= ( + ) f =3, f =3, f =6, f =0, f =6. f = f =0 ( ) ( ) (, ) = 3,, 3 3,, 3 ( ) ( ) ,, 3 3,. 3 ( (, )= 6 3, 6 3 ) f = 6 > 0, f f f f = =4> ( (, )= 6 3, 6 3 ) f = 6 < 0, f f f f = =4> 0.. ( (, )= 6 3, 6 3 ) f f f f = = 4 < 0. (. (, )= 6 3, 6 3 ) f f f f = = 4 <

50 ( ) 6 6 (, ) = 3, = ( ) 6 6 (, ) = 3, = ➀ ( + ). ➁ ( + ) ( ). 50 g(,,, n ) = 0 (88) f(,,, n ) (,,, n )=(,,, n ) (,,, n ),,, n,λ f + λ g =0. f n + λ g (89) n =0 g(,,, n )= (88) = φ( ) (90). (88) g φ + g =0. φ = g/ g. (90) f(, ) f 0 = f + f φ = f g g/ f. (9) 87 (9) (9) λ = g/ f (9) 0= f + g λ. (93) 0= f + g λ. (94) f(, ),, λ(93) (94)(89). 88 (89)λ (Lgrnge) F (,,, n,λ)=f(,,, n )+λg(,,, n ) (95) (89) F = F = = F = F n λ =0. (95) ( ) q,q u(q,q ). p, p = p q + p q u(q,q ). 50 λ u q λp =0 u q λp =0 p q p q =0. (96) (96) u/ q p = u/ q p = λ. (97) 90

51 (97) q, q du d = u dq q d + u ( ) dq q d = λ dq p d + p dq. d dq p d + p dq d = du d = λ. λ u(q,q )=q q = 00 p =0 p =0., (96) q λ0=0 q λ0=0 (98) 00 0q 0q =0 (98)q =0 q =5 6 4, q, q u = q ( + q ) g(,,, n ) = 0 (99) (,,, n ) f(,,, n ) g i = g( i,,, n ), f i = f( i,,, n ), A ij = i f( j,,, n ), i,j =,,,n, 0 g g g k g A A A k η k = g A A A k, k =, 3,,n g k A k A k A kk ( ) g g rnk g n = (00) f f fn η k < 0, j =, 3,,n f(,,, n )., (00) ( ) k η k > 0, k =, 3,,n f(,,, n ) (99)(, ) g d + g d =0. (0) f = f =0f(, )(, ) (0) (0) ( g g f f d + f d =0. (0) )( ) d = f d ( 0 0 ). (03) (03)(d, d ) (0, 0) (, ) (00) 48 (d, d ) (0, 0) i,j= A ij d i d j (04) f(, ) (, ). (00) g i, i =, 0, g 0 (0) d = g g d. (05) 95 96

52 (05) (04) A ij d i d j i,j= = A d d + A d d + A d d + A d d ( = A g ) ( d + A g ) d g g +A ( g g ) d + A d = B d. B := g g A g A g A + A g g 97 (04) A > 0., η 0 g g η = g A A g A A = 0 g 0 g A 0 g A B = 0 g g A B = g B. (00)η < 0 f(, ) η k k 8 80(98) 5. (00) ( ) ( ) p p rnk 0 0 = =. q q p p η = p 0 p 0 =p p = 00 > [, b] = 0 < < < < n = b n [ 0, ], [, ],, [ n, n ] [, b],. i δ i δ i. δ = 0, δ =,, δ n = n n, = m{δ,δ,,δ n }. f()[, b], [ i, i ] i. f( i )δ i = f( )δ +f( )δ + +f( n )δ n. (06) i= 0(06) I, f() [, b] b I = f()d f() b b 30 30

53 f() 0f( i )δ i δ i f( i ) (06). 0 (06) = = b = f() ( ) O b = f() [, b]... b >b f()d = f()d, b b = b f()d = f()d =0. 63 [, b]f() k b f()d = k(b ) ➀ b (f() ± g())d = b f()d ± b g()d (). b kf()d = k b f()d (k ). ➁ b c b f()d = f()d + f()d. c ➂ b b [, b] f() g() f()d g()d. f() =g() ( ) b c (, b); f()d = f(c). b [, b]f() M m m<m53 b b b md f()d Md. m b f()d M. b () m = M c [, b] [, b] S() = f()d d S() =f(). d S( + h) S() = +h f(t)dt = f(c) h h c [, + h] h 0 c f(c) =f() d S( + h) S() S() = = f() d h 0 h S() = f(t)dt b F () =f() f()d = F (b) F (). (07) 308

54 f() F () =f() F () f() ( 8 3 ) =3 3 3 C, ( 3 + C ) =3 3 + C 3 f() F () C (F ()+C) = f(). F ()+C f(). f()g() (G() F ()) = G () F () = f() f() =0. G() F () =C, C. G() =F ()+C., f() F () +C, C f()d f() f()d, f()d b c f(). f()d f()d F () f() f()d = F ()+C, C (08). f() f()(08) f() C. f() ( ) ➀ d = + +,. d ➁ = log. e d = e. ➂ sin d = cos. cos d = sin. ➃ d α = rcsin, > 0. ➄ d + = rctn () (f() ± g()) d = f()d ± g()d, kf()d = k f()d, k ➀ ➁ ➂ ➃ ➄ (α + β) d = (α+β)+. α(+) d α+β = log α + β, e α+β d = α eα+β. sin(α + β)d = cos(α + β). cos(α + β)d = sin(α + β). d + = rctn. 34

55 83 ➀ ( ) d = ( 3 ) d = 3 d ( ) ( d ) = = ➁ [ 0 (3 ) 5 d = 3 6 (3 )6] 0 = 8 ( 6 ( ) 6) = 7. ➂ π [ 0 (sin θ + cos 3θ)dθ = cos θ + ]π 3 sin 3θ ) 0 ( cos π + 3 sin 3 π = = ) ( cos 0 + ) 3 sin 0 ( + 3 ) ( +0 = ➀ ( )d. ➁ (e +) d. ➂ d. 4 ➃ 3 d. ➄ π0 ( sin cos 3 ) d. ➅ 30 d ()g() =t t f(g())g ()d = f(t)dt, b g(b) f(g())g ()d = f(t)dt. g() d F (t) =f(t) d dt d F (g()) = f(g())g (). f(g())g ()d = F (g()) = F (t) = f(t)dt, b f(g())g ()d = F (g(b)) F (g()) = g(b) f(t)dt. g() ➀ ( +)e + d. + = t ( +)d =( +)d = dt. ( +)e + d = e t dt = et = e +. ➁ 0 d. = sin θ θ 0 π 0. = cos θ, d = cos θdθ 0 d = π 0 cos θdθ = π 0 ( + cos θ)dθ = [ θ + sin θ ]π 0 = π d. 5 = t d = dt =4 t = =8 t = d = dt t = 49 t dt 49 = t = [ t] 49 = 49 =6. 69 ➀ ➁ ➂ ➃ ➄ ➅. 8 ( 3 +) d. sin cos d. (e +) 3 d. e 3 d. π 3 0 tn d. 0 d

56 59 ( ) f()g ()d = f()g() f ()g()d. b b f()g ()d = [f()g()] b f ()g()d. 8, (f()g()) = f ()g()+f()g (). f()g() = f ()g()d + f()g ()d f()g ()d = f()g() f ()g()d ➀ cos d. = f(), cos = g () f () =, g() = sin. cos d = sin sin d = sin cos. ➁ e log d. e e log d = log d e = [ log ] e d = (e 0) [] e = e (e )= ➀ sin d. ➁ log d. ➂ rctn d. ➃ 0 e d. ➄ π sin d. ➅ 0 rcsin d. f()[, b],. ➀ b f()d = b+ɛ ɛ 0+0 f()d. ➁ b b f()d = b ɛ ɛ 0+0 f()d. ➂ b = f()d = b b f()d. ➃ = b f()d = b f()d. ➀ ➃, ➀ ➁ d d = 0 ɛ 0+0 ɛ = [ ] ɛ = ( ɛ)=. ɛ 0+0 ɛ 0+0 d = b = b b d [ ] b = ( b ) + =. b ➂ d. = =0 [, 0) (0, ] 0 d = ɛ ɛ 0+0 d = ɛ 0+0 [log ] ɛ = ɛ 0+0 log ɛ =, 0 d = d ɛ 0+0 ɛ = ɛ 0+0 [log ] ɛ = ɛ 0+0( log ɛ) =. d ➂ d = [log ] = log log =

57 = = = 7 ➀ 40 d. 4 O O O ➁ ➂ ➃ ➄ 0 log d. 0 d. 0 e d. d