九州大学学術情報リポジトリ Kyushu University Institutional Repository 物理工科のための数学入門 : 数学の深い理解をめざして 御手洗, 修九州大学応用力学研究所 QUEST : 推進委員 藤本, 邦昭東海大学基盤工学部電気電子情報工学科 : 教授 http://hdl.handle.net/34/500390 出版情報 : バージョン :accepted 権利関係 :
Δ =,, 3, 4, 5, L n = + + 3 + 4 + 5 + + n
S = + + + + + =
S = + + 3+ 4 + 5 + + k + = k = k= S 4 = + 4 + 9 + 6 + 5 + +k + = + + 3 3 + 4 + 5 + +k + = k + k =0 = t = c / a bt y = c + bt = c + b c a = c + c b a t = bt a = bc a bt y = c + c b a + bt = c + c b a + c b a
t 3 = bt a = b bc a a bt 3 y = c + c b a + c b a + bt 3 = c + c b a + c b a + c b a y = bt + c = 0.5t + 3.0 y = at =.5t.5t = 0.5t + 3.0 t = 3.0s y = 4.5 m y = c + c b a + c b a = 3 + 3 + 3 + c b a + 3 3 3 + = 3 + 3 0.5.5 + 3 0.5.5 + = 3 = 9 = 4.5 m 3 3 + 3 0.5.5 3 +
S = 3 + 3 + 3 + 3 3 + 4 3 + 5 3 + 6 3 + 7 3 + 8 3 S = 3 + 3 + 3 3 + 4 3 + 5 3 + 6 3 + 7 3 + 8 3 9 3 S = 3 S = S = 3 + 3 + 3 3 + 3 4 + 3 5 + 3 6 + 3 7 + 3 8 + = 3 + 3 + 3 3 + 3 4 + 3 5 + 3 6 + 3 7 + 3 8 + =
3 n Ln r r r r = + + + + + rl n = r + r + r 3 + + r n + r n+ r=.0 r=0.99 0.98 0.97 0.96 0.95 0.90 0.80 0.70 n+ r n = à L n = r n ( r) L r + 3 n Ln r r r r = + + + + + L = r
S = 7 + 7 + 7 + 3 7 + + + 4 n 7 S = 7 + 7 + 7 + 3 7 + + + 4 n 7 7 S = 7 + 7 + 3 7 + + + 4 n 7 7 S = 7 S = 6 r = 7 L = S = 6 7 S = 7 + 7 + 7 + 3 7 + + 4 7 + = n 6 4 + 4 + 4 + 3 4 + + + = 4 n 4 0 + 0 + 0 + 3 0 + + + = 4 n 0 9 0 + 9 0 A B + A B + 9 0 + A B 3 3 + A B + 9 0 4 4 + + 9 0 + + A B n n + = + = A B < S = + + 3 4 + 5 + + S = + + 3 + + + 4 n S = + 3 + 4 + + S = + n + S = + + 3 4 + 5 + = 4 3 + 3 4 + 5 + = 3
+ + + 3 + 4 + 5 + 6 + 7 + = 8 + 3 + 4 5 + 6 7 + = 8 3 + L = + r r + r 3 + n rl n = r + r r 3 + r 4 + ( + r) L n = + r ( ) n+ r n+ + r L n = + r n+ r n+ n+ r n n+ r n+ + r L = + r = + r ( + r) L = r r + r 3 + ( ) n+ r n + = r = L r ( + r) + + r = + r = + = 4 3
+ 3 4 + 5 + = 3 + 7 7 + 7 3 7 4 + 7 5 0 0 + 0 3 0 4 + = 9 0 9 0 A B A B + 9 0 + A B 3 3 A B 9 0 4 4 + = + = A B < A B < A B + A B A B A B + A B + A B 3 3 + A B A B 4 4 + + A B + = n A B + A + = B B A L n = 0.3333333333 = 0.3 + 0.03 + 0.003 + 0.0003 + 0.00003 + = 3 0.+ 0.0+ 0.00+ 0.000+ 0.0000+ = 3 0 + 0 + 0 + 3 0 + 4 = 3 + 0 + 0 + 0 + 3 0 + 4 3 = 3 0 3 = 3 0.3333333333 3 = = 0.99999999999999 X = 0.99999999999999 0X = 9.9999999999999 0X X = 9 X =
0.666666666 = S = 0.666666666 0S =.666666666 9S =.5 à S = 9.5 = 6 à S = 6 0.4857 4857 4 =? S = 0.4857 4857 4857 0 6 S = 4857.4857 4857 ( 0 6 )S = 4857 à S = 99999 4857 = 7 à S = 7 0.09090909090 = S = 0.09090909090 00S = 9.090909090 à S = 99 9 = à S = 99S = 9 0.08333333333 = 0.076930 76930 76930 = n = k S n = + + 3+ 4 + 5 + + n k = = n n + ( + ( n ) +n) ( + ( n ) + ( n 3) + +3 + +) + ( n +) + ( n +) + ( n +) + + ( n +) + ( n +) + ( n +) + S = + +3 +4 + + n n S n = n + n S n = n + = n( n +) n
S n ( + ) n n = S n = 6 ( 6 + ) = S n = n k= k = n ( n + ) = + + 3+ 4 + 5+ + n n = k S n == + + 3 + 4 + 5 + + n k = ( n + ) = n n + 6
n S n = k = in + 3i(n ) + 5i(n ) + 7i(n 3) + +(k )i n k = { } = (k )i n (k ) k = n { } S n = (k )in (k )i(k ) = (k )in k + (3k ) k = k = k = k = k = k = n k = { } n n n n n = n k n k + 3 k = n n n + nn S n + 3 n n + n 3S n = n n + n + 3 n ( n + ) { = n n + 3n + } = n n + n = n n + n n + 3 n + 3 ( n + ) 3 3 k + k = 3k + 3k + 3 3 = 3i + 3i+ 3 3 3 = 3i + 3i + 3 3 4 3 = 3i3 + 3i 3 + 3 3 5 4 = 3i4 + 3i 4 + 3 3 n n in i n ( ) = 3 + 3 + ( n + ) 3 n n n n(n + ) = 3 k + 3 k + = 3S n + 3 k = k = 3S n = ( n + ) 3 n(n + ) 3 = (n + ) k = { n + 4n + 3n } = n = (n + ) n + + n 3 n (n + ) (n + )n
n S n = k = + + 3 + 4 + 5 + + n k= = n ( n + )( n +) 6
,, 3, 4, 5, 6,, n, = L = L 3 = L 3 4 = L 4 H = + + 3 + 4 + 5 + + n '%'()*' %'()* *,%'()*, +%'()*+ =
H = + + 3 + 4 % + 5 + 6 + 7 + 8 % + 9 + 0 + + + 3 + 4 + 5 + 6% + '''' > + + 4 + 4 % + 8 + 8 + 8 + 8% + 6 + 6 + 6 + 6 + 6 + 6 + 6 + 6% + '''' 4 8 = + + + + + '''' = ( ) ) ( %' ( ' M = M
%., -, '( )*+, -, %., -, )*+, -, m ( ) = m. % '(,/. % )*+, -, = + % '(), - % * % '() ) +) m ( 3 ) = m 3 ( 3 ) = 3 3 = 3 + + 3%
L n = + + 3 + 4 + 5 + + % n ' L = + + 3 + 4 =.08 > L = 0.9 + 0.9 + 0.9 3 + 0.9 4 + 0.9 + 5 L = 0.9 + + 3 + 4 + 5% =.055 >
'( % H = + + 3+ 4 + + n H = + + 0.9 3 + 0.9 4 + + + 0.9 0.9 0.9 5 n
H = + 9 0% + 9 0% + 9 0% 3 + 9 0% 4 + '' + 9 0% n '''' H = + +. 3 +. 4 + + +... 5 n
H = + s + 3 s + 4 s + 5 s + + n s + H = + 9.9 0 % + 9.9 0 % + 9.9 0 % 3 + 9.9 0 % a = a = + 9.9 0 % a 3 = + 9.9 0 % a n = + 9.9 0 % + 9.9 0 % + 9.9 0 % 3 4 + 9.9 0 % + '' + 9.9 0 % n '''' + 9.9 0 % a 4 = + 9.9 0 % 4 + '' + 9.9 0 % n + 9.9 0 % + 9.9 3 0 % a n a a n a < lim n a n = a H = + 9.9 0 % + 9.9 0 % + 9.9 0 % 3 + 9.9 0 % 4 + '' + 9.9 0 % n '''' a = / r = / ( 0.99) = 00 ε a n a < ε a n a n+ < a n -a < a n -a n+ < 95.45-00.0 < 4.855050 95.45-98.9 < 3.07794 98.9-00.0 <.7770 98.9-99.3495 <.663 99.3495-00.0 < 0.650479 99.3495-99.769 < 0.438
99.769-00.0 < 0.38097 99.769-99.98 < 0.50945 99.98-00.0 < 0.087509 99.98-99.968 < 0.055508 99.968-00.0 < 0.03900 99.968-99.9883 < 0.0036 99.9883-00.0 < 0.06765 99.9883-99.9957 < 0.0074048 99.9957-00.0 < 0.0047396 99.9957-99.9994 < 0.0070955 99.9994-00.0 < 0.0005764 a n -a n+0n < 3.60306659-5.9056569 <.30585093 5.9056569-8.083678 <.30585093 8.083678-30.50887 <.30585093 30.50887-3.8340697 <.30585093 3.8340697-35.59906 <.30585093 35.59906-37.485775 <.30585093 37.485775-39.765 <.30585093 H = + +. 3 +. 4 + + +... 5 n a n -a n+0n < 9.58445-9.790 < 0.0567 9.790-9.95349 < 0.6337 9.95349-0.0833 < 0.97700 0.0833-0.863 < 0.030800 0.863-0.68 < 0.088794 0.68-0.3333 < 0.065039 0.3333-0.3849 < 0.05664 0.3849-0.460 < 0.040369
π
9
f () u : : f () u tan θ 30
= lim h>0 f ( ) f + h h = lim >0 f ( ) f + y = = df = f 3
y = sin = 0 = = y = cos 3
33
dr = i + j = i + % j = i + f '() j = ( i + f '() j) dr = i + y j = a + + b ( a + b) = a = a = a = a { = lim a( + h ) + b} ( a + b) h >0 h y = + 4 ah = lim h> 0 h = a 34
+ 4 ( + 4) = + = y = 0.6 y = 4 5 5 = d = ( + ) = d( ) = ( = + + () ) = = d = = 35
= lim h >0 ( + h) = lim h h >0 ( + h + h ) h = y = 4 + 4 = 4( + ) + 4 4 + 4 8 = 8 = y = 8 y = 5 y = 3 3 = d = ( + )3 3 ( = 3 + 3 + 3() + () 3 ) 3 = 3 = 3 % %' %' %' ( % % %' 36
= d 3 = lim h >0 ( + h) 3 3 = lim h h >0 ( 3 + 3 h + 3h + h 3 ) 3 h = 3 d 3 = 3 d( 3 ) = 3 y = 4 3 + 4 = y = y = 5 3 y = 3 3 + + ( + h) = + h ( + h) = ( + h) ( + h) = + h + h 37
( + h) 3 = ( + h) ( + h) = 3 + h + h + h + h + h 3 = 3 + 3 h + 3h + h 3 ( + h) 4 = ( + h) 3 ( + h) = 4 + 3 3 h + 3 h + h 3 + 3 h + 3 h + 3h 3 + h 4 = d 4 = d ( n ) ( + h ) 4 4 = lim h > 0 h = lim h > 0 d n = 4 + 4 3 h + 6 h + 4h 3 + h 4 = lim h > 0 ( 4 + 4 3 h + 6 h + 4 h 3 + h 4 ) 4 = 4 3 h n ( + h ) n n n + n n h +... + nh n + h n = lim h h > 0 h = n n = n n (+h) n 0-5 =0h= ( + h) n n=5 = 3 = 33 4 = 464 5 = 605 ( a + b) 6 ( a + b) n = a n + n C a n b + n C a n b + n C 3 a n 3 b 3 +... + n C k a n k b k + + n C n b n 38
n C = n C = n n n = n ( n ) ( n ) C = n n n n 3 = 3 ( n 3)3 C = n( n ) ( n )... n (k ) n k k = n ( n k)k n C k 5 = 5 4 3 = 0 ( n ) ( n ) ( n 3) ( + h) n = n + n n h + n n n h + n n n 3 h 3 + n n 3 4 + n( n ) ( n ) h n 4 4 + n( n ) h n 3 3 + nh n + h n + h n 3 n 4 h 4 + n C r a + b ( a + b) 4 = ( a + b) ( a + b) ( a + b) ( a + b) = ( aa + ba + ab + bb) ( a + b) ( a + b) = aaaa + (baaa + abaa + aaba + aaab) + (bbaa + baba + abba + baab + abab + aabb) + (bbba + bbab + babb + abbb) + bbbb 39 4 a 3 b 4= 43 = 4 3= 3 = 6 4/ 3= 43 / 3 = 4 4 C = 4/ (4 ) = 4 / = 4 4C 3 = 4/ (4 3)(3) = 4 3 / 3 = 4 a b 4= 43 = 4 = = 4 4C = 4/ = 4 3 / = 6 y = n = nn = n n
d ( 0 ) = 0 0 = 0 y = 0 = 0.5 = 8 = 8 7 6 5 4 3 = 56 6 0 = 56 70 = 4030 5 y = a y = b 0.6 = a 5 3 = 5a = 0.6b0.4 y = a + b + c y = a 3 + b + c + d y = y = a b 4 y = m y = 6 5 + 3 3 y = a n b c + d 0. 40
y = f ( ) = f ( + ) f () f ( ) + df ( ) f () = df ( ) Δy = f ( + Δ) f () f () + Δf ( ) f () = Δf ( ) Δ Δy Δ = Δf Δ = df y = f ( )g( ) = f ( + )g + f ( )dg( ) + g( )df f ()g() f ( ) + df ( ) + df ( )dg( ) ( g( ) + dg( ) ) f ()g() small = f dg( ) + g df ( ) y = f ( )g = f ( )g y = ( 3 + ) y = ( 3 +) + ( 3) = 9 + + g( ) f ( )
y = ( 4 4 + ) ( 3 + ) y = f ( ) / g( ) = f + g( + ) f () g() f g = df ( )g() f ()dg ( ) g( ) + dg( )g( ) + df ( ) + dg( ) f () g() = f ( )g() + df ( )g() f ()g g( ) + dg( ) g f ()dg( ) g() > dg() g() > g()dg() df ( )g() f ()dg g y = f ( ) g( ) = f ( ) g() f ()g() g y = 3 + 3 + y = 3 + 3 = 6 + 3 = 3 + ( 3 + ) ( 3 + ) y = 44 + 3 +
y = / g y = f y = g( ) g y = + y = / g( ) ( + ) y = f ( )g( ) à y = f ( )g( ) + g( ) f ( ) y = f g à y = f ( ) g() f ()g() g y = g( ) à y = g( ) g dz = d( y ) = ( + ) ( y + ) y = + y dz = d y = dz dt = dt d y = + y + y dt = + y = dt + y dt
d y = y + + y y + + = y = + y d y = d y = yd + = y + y = d = + = + + = + = d = z = y dz ( y) dt = d dt = dt y + dt = + y d y d y = y d = y = f () z = g(y) z = g( f ())
y=f() / =f()' y z=g(f()) z=g(y) dz/ =g(y)' z z / = dz / = 3 dz = dz = 3 dz/=(dz/)(/) y z /=0.5 dz/=3 y = g(u) u = f () à y = g( f ()) = du du y = ( + 3) 5 u = ( + 3) y = u 5 du = 5u du 4 = = du du = 5u 4 = 0( + 3) 4 y = 5( + 3) 4 ( )
y = ( 3 + + ) 4 y = 4 y = ( + ) 3 y = + 4 = ( t ) u = t = du u dt = t = u du = d du u = u 3 dt = du du dt = u t = 3 4t ( t ) 3 y = f () = g(y) y = f () ( ) = y f = f (y) =g(y) y=f() y =g(y)=f-(y) y y = = y y = ± y = = = y
= y à = y = y = = = = y y = y = ± dz = dz = y = 3 = y 3 y = y = 4 = y 4 y = 4 θ θ y = sinθ θ θ θ θ θ θ θ
dθ = d sinθ = dθ dθ cosθ dθ = cosθ θ sin A sin B = cos A + B sin A B sin( θ) dθ = lim sin θ + dθ dθ 0 dθ sin dθ lim dθ 0 dθ = = lim dθ 0 cos θ + dθ dθ sin dθ = lim cos θ + dθ dθ 0 θ θ θ θ sin dθ dθ = cosθ sin = 3 3 + 5 5 sin = 3 + 5 4 lim 0 sin = π sin θ θ θ dθ sinθ dθ = d ( cosθ) dθ sinθ = = sinθ dθ dθ cos A cos B = sin A + B sin A B
dθ = lim cos θ + dθ dθ 0 dθ cos( θ) = lim ( )sin θ + dθ dθ 0 sin dθ dθ sin θ + dθ = lim dθ 0 dθ = sinθ sin dθ θ = sinθ cosθ d tanθ = cosθ dθ dθ = d sinθ dθ = cos θ + sin θ cos θ = r sinθ d ( cosθ) dθ = cos θ cos θ r sinθ ( sinθ ) cosθ cosθ cos θ
ω y = sin ωt u = ωt y = sinu dt = d ( sinu) du du dt = ( cosu )ω = ω cos ωt ωt ω d ωt ( ) d sin ωt = d ωt = cos( ωt) à y = cos( ωt) ωdt = cos ( ωt ) dt = ω sin ( ωt ) y = ( sin ) y = sin u = sin du = cos y = u = du = u cos = sin cos du y = cos y = sin 3 y = tan 3 y = sin 5 y = sin cos y = cos( 3 +) d sinθ dθ d cosθ dθ d tanθ dθ = = cosθ d( sinθ ) = cosθdθ = sinθ d ( cosθ) = sinθdθ cos θ d ( tanθ ) = dθ cos θ
= r sinθ y = r cosθ θ dθ = r cosθ = r sinθ dθ = dθ r sinθ = r cosθ = tanθ dθ = at bsint y = a bcost dt = a b cost dt = bsint = dt dt = bsint a b cost
y = sin y = sin y = sin = sin y = cos y cos y + sin y = d( sin ) = = cos y = sin y = AB AD = dθ OB OC = y dθ = y = sinθ y θ θ θ θ à y
y = cos = cos y = sin y d cos = = sin y = cos y = AB BD = dθ OB BC = y θ dθ = = cosθ θ θ θ
y = tan = tan y == cos y = + tan y = + d( tan ) = = + + y dθ θ θ θ θ θ θ θ DB AB = CO BO θ θ + y + y dθ = + y y = tanθ dθ = y = tanθ + y à
y = sin a sin y = a cos y = a à = / a cos y = / a sin y = / a = / a a y = tan a tan y = a ( + tan y) = a à = / a ( + ( / a) ) = a + a y = cos a = cos y = a sin y = a / a cos y = / a = / a a y = sin y = cos y = tan sin d sin = d ( cos ) = d ( sin ) = d cos = d ( tan ) = + d ( tan ) = d sin ( / a) = d cos ( / a) = d tan / a + a a a = + a
y = a da a a a a a = lim = lim = a lim = a log a ( + h) h h h 0 h h 0 h h 0 h ----------------------------------------------------------------------------------------------------------------------------------------------------- lim (+ / m) m = e m e lim log m a (+ / m) m = log a e à lim mlog m a (+ / m) = log e e log e a = log e a h = log a (+ / m) / m = a h m = / ( a h ) h lim mh = lim h 0 h 0 a h = log e a h a lim = log h 0 h e a ----------------------------------------------------------------------------------------------------------------------------------------------------- log e a = a = e da = a y = e
de = e y = a log a y = log a a y = = a y = a log a y = a log e y = log e a y = e log e a log e y = log e a y = a = e log e a y = a y = e log e a log e a log e a da = a deloge d log e a = e d ( log a) log e a e d log e a y = a log e a = a log e a u = log e a e u = a da = da du du = deu du d ( log e a ) = e d ( log a) u e = a log e a y = log a = a y = ay log e a = d log a = a y log e a = log e a d ( log e ) = y = log e = e y
= ey = = d log e = da = a log a d ( log ) a = log a de = e d ( log e ) = log e log ln log y = 0 y = 5 y = e sin y = e y = n log y = n log LHS = d ( log y) = d ( log y) = y d n log RHS = = n y = n à = yn = nn y = log y = log
LHS = y d log RHS = log + = log + = + log = y ( + log ) = ( + log ) y = y = e sin y = e f ( ) y y = f ( ) y = f ( )( ) + y = o = 4 y 4 = = 4 y = 4( ) + 4 = 4 4
a + y b = a + y = 0 b à = b y a y y = b ( y a ) y a y a y = b + b à b + y a y = b + a y à a a + y y = a y + b b a b + y y b = = a + y b = a + y b = a + y y = b a y b = a y y b = V = y S = + y
y = S V = S = S 4 3 dv = S 4 3 = 0 à = S 6 y = 6 S S S 6 = 6 S S 6 = S 6 t AP = AP c = a + c
t PB = PB = b + d c t ( ) = a + c c + b + d c t = 0 ( ) = c a + d b + d c sinα = sin β c c sinβ sinα = c c sinα = 0.65999 sin β = 0.599363 I = V r + R V P = RI = R r + R dp dr = V R = V ( r + R) ( r + R) R r + R ( r R) ( r + R) r R = V = V ( r + R) 4 ( r + R) 4 ( r + R) = 0 3 R = r Ω Ω
y = 3 y, f (), y, f (), d y = d y, f (), d 3 y = d d y 3 = d d y (n), f (n) (), d n y n y = sin = d d n y = d d d d n n = cos = sin + π d y = sin = sin + π
y = cos = sin + 3 π y(4) = sin = sin + 4 π π y (n) = sin = sin + n π y = cos z = f (, y) z, z, f (, y), f (, y) z y, z, f (, y), f (, y) y y y z y z r = i + y j dr = i + y j
dr = i + j z = f (, y) r = i + y j + f (, y) k dr = i + ( f (, y) / ) k = i + f k dr y = j + ( f (, y) / y) k = j + f y k
3 0 ( ) % 4( % (,, ( % ) * ' - +. (, 56 76 ( % ) * ' - +. (, /0 /0
Δ Δ Δ Δ % ' (
= b a n Δ S n a b [ ] = + + + + % = n = n b a = (b a) n n y = ( ), ( ), ( ), ( ), ( ), ( ), ( ), S n = ( ) + ( ) + ( ) + + ( ) = n( ) Δ n S ab = lim f ( i ) = f () n i= b % a n lim n i= b % a b ( = = lim % n = lim n n ) * a b a n b ' a n +, - = b ' a 0 % = ( + + + + ) = 0 0 0 i= Δ
Δ y = = = + C y = + C %%% ' = + = + 999 = = y + C ds = ds = s + C dt = dt = t + C d = + C
d adz d ' % d d y ' % d = d = + C adz = a dz = az + C d ' % = + C d d y ' % = d y + C k df () / = = y = = = + C / = d = + C = = + C = = y y = + C d y = d % d % ' = ' = d % ' = + C
b = a [ ] a b = b a b = = a b b 0 a 0 [ ] b 0 [ ] a 0 = b a k d d d a d c 0 z 0 b k = k a [ ] b a = k( b a) d d = c [ ] c d = d c d = 0 [ ] 0 = d = z 0 [ ] 0 z = z % 4 3 Δ S n = ( 0 + + + 3 + 4 + 5 + + n ) = ( 0 + + + 3 + 4 + 5 + + n) n ' = k % ( ) = n n + k =0 b n n + = lim S n = lim n n 0 b = b 0 b ' % n( ) = b lim + ' n % n( ) = b
b = b %0 = b 0 = b 0 0 ( k ) = ( 0 + + + 3 + 4 + 5 + %%% + 0) = 55 0 0 k =0 = = + C 0
b d a b d = % ( a ' b a = b ) a b y tdt a 5 0 Δ S n = ( 0 + + ( ) + ( 3 ) + ( 4 ) + ( 5 ) + + ( n ) ) n ' = ( 0 + + + 3 + 4 + 5 + + n ) 3 = k % ( ) 3 = n n + 6 b n n n + = lim S n = lim n n 6 0 b = b 3 0 i= 3 ( n + ) % b( ' n) * k = 3 = b3 ( n + ) 6 lim % + ( n ' n) * % + ( ' n ) * = b3 Δ 3 3 y= y= 3 /3
= 3 3 + C 0 + y + z dz = 3 0 0 0 y = b a %' ' % % b a y = y 3 3 % ' b a = b3 3 ( a3 3 a z dz t dt a a a 5 0 3 = + C 4 0
y = y = y = 3 = y = y / = y /3 0 y / = 0 y /3 = = + 3 + = m = 0 m + 3 3 4 y = m Δ = b / n ΔΔ Δ S n = ( m + ( ) m + ( 3 ) m + ( 4 ) m + ( 5 ) m + + ( n ) m ) = m + m + 3 m + 4 m + 5 m + + n m n m+ b m+
S n = m + m + 3 m + 4 m + 5 m + + n m nn m 03 + 3 b m+ 0m + m + m + 3 m + 4 m + 5 m + + n m n m + n m + n m + n m + n m + n m + + n m 3 + 3 = 4 + 4 03 + 3 + 3 3 + 3 + 3 = 9 4 = 4 + 8 03 + 3 + 3 + 3 3 3 3 + 3 3 + 3 3 + 3 3 = 4 + lim S = lim 0 m + m + m + 3 m + 4 m + + n m n n n n m + n m + n m + n m + n m + + n m n+ b m+ = bm+ m + n b m+ n 0 a ' '()( * + % = b r S m = ( b br )b m + br br = b m+ r = b m+ r + b m+ r + r m+ = bm+ r r m+ ( br ) m + ( br br 3 )( br ) m + ( br 3 br 4 )( br 3 ) m + + r m+ + b m+ ( r)r ( m+) + b m+ ( r)r 3( m+) + + { + ( r m+ ) + ( r m+ ) 3 + + }
L m = + r + r + r 3 + + r m rl m = r + r + r 3 + + r m + r m+ ( r) L m = r m+ m+ r L m = r ( r) r m+ = L m = + r + r + r 3 + + r m S m = bm+ r b m+ = r m+ + r + r + r 3 + + r m m S = bm+ + m = b( r) b m = 0 m + bm+ b = 0 b 0 n n = n + n+ + C n = = log + C
y = m y = t y = y = y = 0 y = S m = ( br b) b + ( br br ) % br ' + br 3 br + ( r ) + ( r ) + ( r ) + ((( + = r = m r % br ' + br 4 br 3 % br 3 ' + ((( +
d n n % = n ' ( n ' = n n n / n n n / n = n d n / n d n / n = n n / n = n n = n n = f () = f () ' % )% % (% )% % (% )% % % *+,)%(%- ()%(% f () = f () + C %./,*+ = f () y = f () = f () + C f ()
d f () % = f () = f () y = 3 y = 3 = 3 = 3 = 3 y = 3 = 3 + + + C = 3 + C n = n + n+ + C y = ( + + ) = + + = 3 3 + + + C { f () + g() } = f () + g()
y=f()+g() y=f() y=g() ( 6 5 + 3 ) y = y = 4 + 3 % ( ' y = 3 + % ' ( + S = ( ) ( )( ) ( )( )( 3) n n n n n n n n n n 3 4 n + = + n + + + + + 3 4 ( ) ( )( ) ( )( )( 3) 3 4 3 4 + = + + + + +
= ( ) = + % ' + ( ) % ' % ' + ( ) 3 % ' % ' 3 % ' + ( ) 4 % ' % ' 3 % ' 4 % ' + 3 4 5 % ' % ' 3 % ' 4 % ' 5 % ' + ( ) 6 % ' % ' 3 % ' 4 % ' 5 % ' 6 % ' + ( ) 7 + 6 7 = 8 4 6 6 5 8 8 7 56 0 04 33 048 4 ((( 5 y = y = 8 4 y = 8 4 6 6 S = 4 dt = 4 8 4 6 6 5 8 8 7 56 0 04 33 ' 048 4 % ( ) 0 0 * = 4 3 3 85 5 5 7 33 -, 67 7 89 9 56 043 3 0485 5 / +. π S = = 4 6 40 5 5 7 86 33 % 040 ' ) = 3.4593 ( π 0
= ( ) ( = + ()( ) + )(() ) ( () + )(() )(() ) ( () 3 + )(() )(() ) (() 3) () 4 3 4 ( + )(() )(() ) (() 3) (() 4) ( () 5 + )(() )(() ) (() 3) (() 4) (() 5) () 6 5 6 ( + )(() )(() ) (() 3) (() 4) (() 5) (() 6) () 7 + 7 = + + + 3 + 4 + 5 + 6 + 7 + + n + (n =,,3,) y=+ + + 3 + 4 + 5 + 6 y=+ + + 3 + 4 y=+ + y=+ y=+ + + 3 y=+ + + 3 + 4 + 5 y=+ + + 3 + 4 + 5 + 6 + 7 = + + + 3 + 4 + 5 + 6 + 7 + 0 = log 0 0 = + + 3 3 + 4 4 + 5 5 + 6 6 + 7 7 + 8 n + + + (n =,,3,) 8 n
y = + / + 3 / 3+ 4 / 4 + 5 / 5 + 6 / 6 + 7 / 7 + 8 / 8 y = + / + 3 / 3+ 4 / 4 + 5 / 5 + 6 / 6 y = + / + 3 / 3+ 4 / 4 y = + / y = + / + 3 / 3 y = + / + 3 / 3+ 4 / 4 + 5 / 5 + 6 / 6 + 7 / 7 y = + / + 3 / 3+ 4 / 4 + 5 / 5 + = + + + + + + + + 3 4 5 6 7 8 = + + ( ( ) + )(() ) ( ( ) + )(() )(() ) ( ( ) 3 + )(() )(() ) (() 3) ( ) 4 3 4 (() )(() ) (() 3) (() 4) ( ( ) 5 + )(() )(() ) (() 3) (() 4) (() 5) ( ) 6 5 6 (() )(() ) (() 3) (() 4) (() 5) (() 6) ( ) 7 + = + + + = + 4 6 + 8 0 + 4 + + 7 n n + (n =,,3,) y = + 4 y = + 4 6 + 8 y = + 4 6 + 8 0 + y = y = + 4 6 y = + 4 6 + 8 0 + 4 y = + 4 6 + 8 0
dt = + t 3 0 3 + 5 5 7 7 + 9 9 + 3 3 5 n n + + () 5 n + (n =,,3,) dt = tan + t 0 y = 3 / 3 y = 3 / 3 + 5 / 5 7 / 7 y = 3 / 3 + 5 / 5 y = 3 / 3 + 5 / 5 7 / 7 + 9 / 9 y = 3 / 3 + 5 / 5 7 / 7 + 9 / 9 /+ 3 /3 y = 3 / 3 + 5 / 5 7 / 7 + 9 / 9 / y = 3 / 3 + 5 / 5 7 / 7 + 9 / 9 /+ 3 /3 5 /5 tan / 4 = 4 = tan = 3 + 5 7 + 9 + 3 5 () n + + + (n =,,3) n π π = / 3 = 0.577 tan / 3 6 = 3 3 % 3' ( + 5 % 3' ( % 7 % 3' ( 3 + 9 % 3' ( π 4 % 3' ( 5 + 3 % 3' ( 6 = / 6 5 % 3' ( 7 + ))( ' π π (
= = + + + + % ' ( ) + % ' % ' 3 % ' 4 % ' % ' % ' 5 % ' % ' 3 % ' 4 % ' 5 % ' 6 % ' = + + (3 (4 4 + (3(5 ( 4(6 6 + (3(5(7 (4(6 (8 8 + 7 3 ( ) 5 + ( ) 3 + % ' % ' 3 % ' 4 % ' % ' 3 % ' 4 % ' 5 % ' (3(5(7 (9 ( 4 (6 (8(0 0 + ( ) 7 + ((( = + + 3 8 4 + 5 6 6 + 35 8 8 + 63 56 0 + 3 04 + 49 048 4 + ((( 6 (3(5(7 (9( ( 4 (6 (8(0( + ( ) 4 ( ) 6 (3(5(7 (9 ((3 ( 4(6 (8(0((4 4 + ((( y = + ( / ) + ( 3 / 8) 4 + ( 5 / 6) 6 + ( 35 /8) 8 + ( 63/ 56) 0 + ( 3 /04) + 49 / 048 y = + ( / ) + 3 / 8 4 + 5 / 6 0 + ( 3 /04) + 63 / 56 6 + ( 35 /8) 8 y = + ( / ) + ( 3 / 8) 4 + ( 5 / 6) 6 + ( 35 / 8) 8 + ( 63/ 56) 0 y = + ( / ) + ( 3 / 8) 4 + ( 5 / 6) 6 + ( 35 / 8) 8 y = + ( / ) + ( 3 / 8) 4 + ( 5 / 6) 6 y = + ( / ) + ( 3 / 8) 4 4 y = + ( / ) dt = + 0 t 3 3 + 3 8 5 5 + 5 6 7 7 + 35 63 3 49 8 9 9 + 56 + 04 3 3 + 048 5 5 dt = sin 0 t
y = +/ ( 3) 3 + 3/ (8 5) 5 + 5 / (6 7) 7 + 35 / (8 9) 9 + 63 / (56 ) + 3 / (04 3) 3 + 49 / (048 5) 5 y = +/ ( 3) 3 + 3/ (8 5) 5 + 5 / (6 7) 7 +35 / (8 9) 9 + 63 / (56 ) + 3/ (04 3) 3 y = +/ ( 3) 3 + 3/ (8 5) 5 + 5 / (6 7) 7 + 35 / (8 9) 9 + 63 / (56 ) y = +/ ( 3) 3 + 3/ (8 5) 5 + 5 / (6 7) 7 + 35 / (8 9) 9 y = +/ ( 3) 3 + 3/ (8 5) 5 + 5 / (6 7) 7 y = +/ ( 3) 3 + 3/ (8 5) 5 y = +/ ( 3) 3 sin / / 6 = 3 = sin = + ' 3% ( ) + 3 ' 85 % ( ) 4 + 5 ' 6 7 % ( ) 6 + 35 ' 89 % ( ) 8 63 ' + 56 % ( ) 0 3 ' + 04 3% ( ) 49 ' + 0485% ( ) π + θ cos d sin d ( sin ) = cos d( sin ) = cosd d cos d = d sin = sin + C Δ 4 +
cosd = sin 0 [ ] 0 = sin = /60 θ θ θ sin d d ( cos ) = sin d ( cos ) = sind d sin d = d cos = cos + C sind = cos 0 [ ] 0 = cos +
θ θ θ) cos( a ) d u = a a cos( a ) d = cos a cos (t) dt = sin ( t ) + C d( a ) = a a θ) sin( a ) d a sin( a ) a d a d = sin( a ) = a sin (t) dt = cos ( t ) + C θ) θ d ( tan ) = d cos ( a ) d( a ) = a sin ( a ) d a = cos d ( tan ) = d cos a sin( a ) + C cos( a ) + C cos d = d ( tan ) = tan + C cos sin tan y = cos = + cos = + sin + C 4
y = sin = cos = sin 4 + C y = tan = % cos ' ( = tan + C y = cos( ) y = 3sin( ) y = sin cos d = sin + C cos t dt = sin t sin d = cos + C sin t dt = cos t cos d = tan + C + C + C y = sin d ( sin ) = = d sin = sin + C = sin sin 0 = sin 0 = / 0 y =
y = cos d ( cos ) = d( cos ) = = cos + C = cos cos 0 = cos 0 cos ( / ) = 0 cos 0 = = / = / 0 y = y = tan d( tan ) = + d ( tan ) = + tan = + C +
= tan tan 0 = tan + 0 = 0 / 50 y = + = sin + C = d / a a = sin / a % a' ( + C = cos + C = d / a a = cos / a % a' ( + C = tan + C + a + a = d / a + / a = tan % a' ( + C 9 +6 = sin 9 % 3' ( + C + 6 = 4 d / 4 + / 4 = 4 tan % 4 ' ( + C
f ()g() f () g() [ f ()g()] = f () g() + f ()g() [ f ()g()] = f () g() + f ()g() f ()g() = f () g() + f ()g() f () g() = f ()g() f ()g() f () g() = f ()g() f ()g() cos cos cos = ( sin ) = sin sin + C = sin cos + C cos = ( sin ) = sin ( )sin + C = sin + ( cos ) + C = sin + % cos cos ' + C = ( )sin + cos + C sin sin y = g(u)
u = f () y = g( f ()) = du du y = = du = du du du ( / du)du y = ( 3 + ) 8 y = sin 4 cos u = 3 + du = 3 = du / 3 y = du u 8 = 3 7 u9 + C = ( 7 3 + )9 + C y = ( 3 + ) 8 d( 3 + ) = ( 3 7 3 + )9 + C u = sin du = cos y = y = u 4 du = 5 u5 + C = 5 sin5 + C sin 4 d ( sin ) = 5 sin5 + C + a sin ( a + b ) + a a + b 4
b = a n + b a + + n n n 0 0 0 b b 0 0 = = ( b a ) = a a '(')*% +, - % S a ~ ar n ( ar a) / a ( ar 3 ar ) / ar S a ~ ar n = ra % a / ar = r ar ar = r = r a + ( ar % ar) ar + ( ar 3 % ar ) + ( r % ) + ( r % ) + ( r % ) + + ar + ar 4 % ar 3 = r % = n r % n ar 3 + + ar n % ar n% ar n% S a ~ ar m = m ( r % ) S ar m ~ ar n = S a ~ ar n = n r % % S a ~ ar m % m( r %) = ( n % m) ( r %) 95
S ab ~ abr m = m r % = S a ~ ar m c = ar m S c a % = S bc ba % S c / a ( * * ) * * + S a ~ ar n = S ar n % a S ab ~ abr m = S r n = n ( r ' ), = S abr m % = S r m ab = m ( r ' ) S a ~ ar m+n = S a ~ ar m + S ar m ~ ar m+n = m r % 30 { } r % + ( m + n) % m = ( m + n) ( r %) S r m+ n = S r m + S r n = r m y = r n S y = S + S y log 0 ( 0 0 3 )= + 3 3 S y % log y = log + log y 96
log y = log + log y log / y = log % log y r = ( b / a) n S b / a = n r % S b a % = n ( * ) b / a n ' n ' + -, = b / a / n S b a = % b ' a ( = b / a) n * lim n() / n = log e ( b / a) h a lim = loge a h 0 h h = / n lim n a n ' % n( ) = log e a 647 640 7 b b = a b = log a e % b a = log e b log e a = log e a 97
e e = log e % e = log e e log e = = log e + C a= b=5 =50 S[5]=log e 5=.70805 5 S 5 = % 5 / n n ' =.7873 3 98
e lim( + / n) n = e lim(+ / n) n = e n n n lim log a ( + / n) = loga e lim n n a e lim = log 0 e a lim = 0 lim( + / n) n = e n=/ lim n 0 e 3 4 3 4 = + + + + + nlog a (+ / n) = log e a (+ ) / = e log lim log 0 a (+ ) / = log a e lim a (+ ) = log(+ ) lim = 0 log a 0 de 3 4 3 4 = 0 + + + + + + = e e = C + + + 3 3 + 4 4 + = e 0 a y=e y=e y=e a = a / loga + C de = e e = e + C 99
da = a loga da = a log a a = log a a a = a log a e e = e / loge + C = e + C = d log e = = d log e log e + C f () = log f () + C f () f () df () = f () = log f () + C f () f () = t f () = dt dt = log t = log f () + C t a = a + C log a e = e + C f () df () = f () = log f () + C f () tan + + ( + 3) 300
tan = sin = d cos cos cos = + = + = log cos + C + d + = log + + C ( + ) ( + 3) = + % + 3' ( = log + log + 3 + C = log + + 3 + C 3 3 3 cos + sin e log e I = e a sinb I = e a cosb e = e e = e e log log = = log = log + C I = e a sinb = % a ea '( sinb = a ea sinb ) a ea bcosb = a ea sinb ) b a I I = e a cosb = % a ea '( cosb = a ea cosb + a ea bsinb = a ea cosb + b a I 30
I = a ea sinb b a I I = a ea cosb + b a I I = a ea sinb b a a ea cosb + b a I % ' I + b a % = a ea sinb ' b a ea cosb I = e a a + b ( asinb bcosb) I = a ea cosb + b e a a a + b % (bcosb)' = e a a + b [ bsinb + acosb] e 3 e = a a % + a ' ( = log a + a = + a a Tan % a' ( a = a = n n n ( a ) n ( a ) = ( a ) = % % ( ) a = % % a ( + '( 30 ( + '( a % + a ' ( = % % a a ( '( + log a + a
a = n n + ( ) n = = + = log I n = ( ) + ( + a ) n I n = ( + a ) = + a n a ( + a ) = n a % % + a = ) % + I a n * % n,+ = % I a n % n = 0 % + a % n = a n ( + a ) n n ( + a ) + n n 3 4 5 I n n % ( + a n + n 3 % ) n ( + a ) n I n ( + a ) ( ( ' ( + a ) ( n ( ' I n ( ( ' n - +.( ( /+ ' ( ( ' I n = ( + a ) n I = = + a a Tan % a' I = ( + a ) = %I a + % = % a % + a ( ) a Tan * + a ( + a ) ( = % '( a % + a, -. ( '( I ( '( 303
( ) + + 3 y = I = = + ( ) log + = t I = t )t t dt = t ( ) = t t = t t I = ( ) t t % t ' ( t )(t) = ( )t dt ( t ) ( t ) dt = % t = ( )' t + log t + t t u = t ( * ) tdt = du = t u u t = t dt ( t ) * = ( ) % t t + t +. ', / - + t 0 dt ( * ) ( t) + ( + t) % ' t = = 304
I = ( ) % + log + ' ) ) ) ) () = + ( ) log + y = ' % ( ) I = = ( ) ( ) + ( )Tan = t / ( 3) / ( 3) αβ y = y = a + b + c a + = a + + a log + a + a + = t % ' 305
a + = t t + = t a t a + = t t a t = t + a t = ( t a ) / t = t a dt = t + a t t t t dt a + = t + a t + a t 4 + a t + a dt = t t 4 dt = 4t 3 = 8 t + a logt ( a4 8t 4 t + a t + a4 % 4t 3 ' dt = ) 8 t ( a4, * + t -. + a logt = a + + a log + a + % ' 8 t a4 % t ' = 8 t + a % t ' t a % t ' = t + a % t a % t ' t ' = a + y = a + + y = a y = ± y = + t y = + 0. y = + t y = 3 + 3 + y = 3 + t t=0 t= t= 306 t=0 t= t=
a + = log + a + y = a + = t a + = t + a t = t + a dt t a + = t t + a dt = dt t + a t = log t = log + a + t a + = log + a + + tan = t sin tan = t / cos = dt = cos dt = dt + tan = dt + t a + = a + + a log + a + sin = t + t % ' 307
= sin dt + t = t t dt = logt = log tan + C + t tan t t t = t cos = sin = tan = + t + t t + y = y = t + ( + t ) + t + t = 0 %'( { + t } ( + ) = 0 + t *+ ) = t + t y = t t + t + % ' = t + t = cos = t t y = sin = + t + t θ 308
θ θ θ θ θ y = a sin + b cos tan = t = dt cos cos dt dt y = = a sin + b cos = a t + b ab = b ab Tan % a t ' ( = b ab Tan % a tan ' ( + sin + cos ( b / a)dt t + b / a a + = a tan a = asin 309