limit&derivative
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- きみお てらわ
- 5 years ago
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Transcription
1 - - 7
2 ) e ) ) ) ) si e )
3 ). + + ) + + ) + + ) + ) ) + 5 ) = ) = ) + + ) ) + + ) + + ) + + = + + )5 < + 5 < = 5 5 < + 5 < 5 5 < + 5 < 5 =, + 5 = 5
4 ) + + ) + + ) + + ) + ) ) + 5 ) = ) = + = = ) ) + = + 9., = + ) 9 + ) = + = + + = + ) = ) = 5 ) ) + = ) + + ) 9 + ) + ) ),.)
5 ) 5. si θ = ) θ 0 θ cos θ θ 0 θ = ) P T θ H A, 0) θ, 0 < θ < π ) AP, P H A P T PH < AP < AT si θ < θ < ta θ si θ < θ < si θ cos θ θ cos θ < si θ θ < θ < 0 θ = h θ 0 cos θ =, θ 0 si θ θ =. θ 0 cos θ θ θ 0 cos θ) + cos θ) θ + cos θ) θ 0 si ) θ si θ θ + cos θ) θ 0 θ + cos θ) = ), θ 0 0 ) si θ si θ θ θ cos θ θ cos θ θ cos θ θ. { si θ = si π = = θ = π = si θ θ 0.5 i) θ = π 6 = cos θ = cos π = = = θ = ).596 = ) = cos θ θ 0.97
6 ) 6 θ = π 6 θ ii)θ = π si θ = si π = si5 0 ) = 6 = θ = π =.596 = si θ θ cos θ = cos π = 6+ = = θ = ).596 = ) = cos θ θ θ si θ θ cos θ θ = = si π π π π = cos = = si, = = cos = = = cos 0, ) = cos.) si ) 0 si si si ) ) 0 cos 5) 0 ta ) 0 cos ) 0 6) 0 cos cos
7 ) 7. = = = e = =.5 A0, ) = a A0, ) A = a a A0, ) a e ) f) = e f fh) f0) e 0) h 0 h h h 0 h = e A0, ) h 0 e h h = e ) = e f ) h 0 e +h e h h 0 e h h e = e Ph, e h ) e ) = e A0, ) h e h AP eh h g) = a h 0 A g ) h 0 a +h a h h 0 a h h a a = e log e a = e log a ) = h, ) = h h 0 h h 0 h h 0 a h h h 0 e h loge a h h 0 e h loge a h log e a log e a = log e a a ) = a log a. e e ) e = e.
8 ) 8 e ) 0 e = e =t log+t)= t 0 log + t) t = t 0 + t) t = e ± + ) = e 0 e e +. log + ) log + ) = +, = = e, = log + ) 0, ), 0, 0) = log + ), = = log, = = = log, 0) = e = log = = = = e = + = = log + ) = = log = log = e = log log = = log = e a e a. a = XX > 0) = log X e a = e e a e a = log ) = > 0) = log. a e a e ) 0 5 ) 0 log + ) ) 0 e si ) 0 log + ) 5) 0 logcos ) ) = t log + t) t 0 t,, ) = log + t) t = = log e + t) t = e t 0 t 0 + ) = e e A
9 ) 9.. e ) e ; e + ).) a = ) + =,,, ). a = + =, a = + ) = 9 =.5, a = + ) = 6 7 =.70707, a = + ) = =.065, a 00% ) )? ) + ) = + ) : + ) =. = + 8 : + ) = + ) ). + : + ) + ) = + )., + ) = a ) = + a a 0., 0.0, ), a + ) = e ) + ) + ) + ) e e ) ) 5 5 ), )
10 ) 0 P, + ), P, + ) ), P, + ) ) k,, P k, + ) ) k,, P k k =,,, ) = ) + { ) = + } f ) = { ) + }, { f ) + ) } = e ) = f ) = e ), f ) + = e f ) ) + = e, f ) ) + = e. 00% e e e!) P, + ) A0, ) AP P = +.) P = e Q, e ) Q A, = + A = e ) + = + ) e ) + = e = + ) = + = + P P P P A0, ) A0, ) P k P k+ ) + k+ + ) k = + ) + ) k = + ) k = P k )
11 ) P k f ) = { ) + }, f ) = f ) f ) = e e ) = e ) ) 5 + = e. ) + ). ) + ) ) + ) ) ) ) + 00% ) =.) = + ) 8 8 = + ) ) P 8 ) + P ) + = + = + ) + P ) + 8 P + P + 8 P A0, ) A0, ) ) f ) = f) f ) = f ) f ) t) e ) = e,
12 ). f) = a Q h) = f) g) a f) fa) a = f a) = f) A P a f) fa) f a) a) a = 0 = g) a a g) = f a) a) + fa) = a! ) a f) g) a = 0 f) g) h) = f) g) h) a a) = a = g) = f) a f) = a = a f) fa) + a)f a) f) h) = f) {fa) + a)f a)} a ). ) f) = si, f ) = cos 0 si = 0 si si = 0 si ) a α, β 0 β a α = 0 β α β α ). β a α β α = k k ) = 0
13 ) a) 0m 5m,.5m.) =, =. h),, ) i) f) = a, a h) = f) g) < = f) = g).. Q = f) = g) Q Q P A P P a ii) f) = a f) = a = a Aa, 0) = 0 a+0 f) fa) a =, a 0 f) fa) a =. f f) fa) a) a a f) = a = a = a = 0 = a = = 0 = a Q = a Q = a Q Q Q Aa, 0) P P P Q Aa, 0) P P Aa, 0) P = f a) a) + fa) = a = f)
14 ). ).. : f) = + = 0 f 0) = 0 = f) = 0 = 0, ) 0, ) = f) a + a f) a, f) = 0 = 0 0 f) 0 0 ) = f) f) + g) = + f) g) f) h) = f) g) 0 h) 0 f) ) 0 0 = 0 h) =.) 0 = + = + = + A0, ) A0, ) = + = + = + A A0, ) = = + = +
15 ) 5 = ) 0, ) = 0 a + b + c + d = c + d = b + c + d f) = + =, ) ) =, ) : = a ) f) a ) +, a f) )., f) ) { ) ) = + 0 ) 0 ). = f) ) ) ), f) ± = + ) = = + = ) 0,) = + f) = f) ) f),, ), ) f) ) = ) + ) ) + ) ) + ) = f) ) f) ) f) + ) ) = f) =, ) f) = a ) α). ) + = A ) + B ) + C ) + D, + = ) + ) + ) + = f) ) + ) +
16 ) 6 a) a) f) : f f a) a) + fa) a) ± = a A a f) {f a) a) + fa)} a) f) {f a) a) + fa)} a) A f) fa) + f a) a) + A a) = a A a f) {fa) + f a) a)} a) ) A = a = f) = fa) + f a) a) + A a) f) fa) + f a) a) + A a) f) = a ) a f) { fa) + f a) a) + A a) } a) = 0 a) a = fa) + f a) a) = f) fa) f a) a) = A a) = f) = + =, 0,,, 0.5 ) = 5 ) + 8 = 8 = f) = f) = 5 + = = + ) = ) + = = +
17 ) 7.. f) = + 0, ) f ) = + f 0) =. 0, ) = + ) = 0 ) 0 f) + ) + + ) 0 ) + + ) ) ) + ) + ) ) ) = + = + = + + = + + = + + = + = = = 0 = = ) ) 5 + = + 8 ) 5 = 5 = +, = { 5 + { 5 ) =.5 ) ) } =.75 ) log e π. 6 7 f) = + = ) 8 [ ] f) = f) = 0, =
18 ) 8. )?) ) f), g) = a fa) = ga) = 0, = a g )=\ 0 f ) a g ) a f) g) a f ) g ) g a)=\ 0 a f) g) a f) fa) a g) ga) a = f a) g a) = a g ) = 0 = a 0 0 ).) ) log cos ) 0 ) si ) f) =, g) = log f) = g) = 0. g ) =, g )=\ 0. log f) f) g) g) = f ) g ) = = f ) =, g ) = ) f) = cos, g) = f0) = g0) = 0 g ) = g 0) = 0. g ) = 0 = 0, cos 0 cos ) si 0 ) 0 = 0 si = ) f) = si, g) = f0) = g0) = 0 g ) = g ) = 0 = 0 si si ) cos 0 ) 0 cos ) si 0 ) 0 6 = 6 9.) e + e ) 0 cos ) ) 0 ) 0 0 ) ) =. 0 )
19 ) 9 ) 5 0 ) Y cos 0 XY Pt, cos t) t t = 0 P C =\ 0 A, cos ), A t = c A, cos ) Q C cos = A ) X C A Q Q C A ) Q t = c 0 < c < < c < 0), Q C f c) g c) Y 0 A ) cos = f c) g c) 0 < c < < c < 0) c 0 ) 0 c 0. c 0 cos 0 f c) g c) f c) c 0 g c) si c c 0 c t 0 si t t = t = c Q A, cos ) C X t = 0 g t) = 0 C : f c) g c) t = 0 g t) = 0 6 a log Y { X = log t C : Y = t C :ft) = t, gt) = log t t > 0) g), f)) = 0, 0) Ag), f)) = log, ) t = log = A ) A Q X C A Q Q C A Q t = c 0 < c < < c < 0), Q C f c) g c) A ) log = f c) g c) c c log c c c t = t = c < c < < c < ) = f c) g c) fa) = ga) = 0 ga), fa))
20 ) e ) e = 0 ) ) e = 0.) ) f) = e f ) = e = 0 = f) ) = + ) f) a) A 0 0 A 0 e + ) A 0 e + ) {e + )} e 0 ) 0 = 0 e = = 0 e + + = 0 e e + ) a = e = + = e + ) = = e + ) = = e + ) A a f) {fa) + f a) a)} a) F ), G), F a) = Ga) = 0 G ) = a) = a 0 f a) a f ) f a) a A a f) {fa) + f a) a)} a) a f) {fa) + f a) a)}) a) ) = 0 f) a f ) f a) a) = f a) f) fa) + f a) a) + f a) a) f a) a f ) f a) a
21 ) f a), = a f) a). f) fa) + f a) a) + f a) a) 8 f) = e f ) = f ) = e, e = 0 0 e f0) + f 0) + f 0) = + + log + ) = 0,.. f) = e = 0 ) 9 e e = 0..) e = 0 = +. = + + e A +) 0 0 a + b) ± 0 a + b + c) ± 5) ) f) + + B ) e + + B 0 0 {e + )} e 0 ) 0 6 { )} e + + e + ) ) 0 = 6 0 e = 0 ) e + + = 6 0 ) e = 0 ) = 0 e g) = f) g) 6 ) 0, 5), 0 f) +) = f) + ) 0 0 f) + ) 0 = { +0 ) 0 ) = 0
22 ) = e = + + ) = e + + ) = e + + = 6 e : + + a = e + + ) = 6 > 0 e > + + < 0 e < + + e + + ) = 0 = 6 = e = = + + = + = e e = 0,, e f) a) f) B a { fa) + f a) a) + f a) a) } a)
23 ) 0 0, { } f) fa) + f a) a) + f a) a) B a a) 0 0 { } f) fa) f a) a) f a) a) a { a) } a f ) f a) f a) a) a) 0 0 a {f ) f a) f a) a)} { a) } a f ) f a) 6 a) = f a) 6 a { } f) fa) + f a) a) + f a) a) a) f a) 6 f) fa) + f a) a) + f a) a) + f a) 6 f a) a f ) f a) a a) f a) f) = fa) + f a) a) + f a)! a). a) + f a)! a) 0 f) = e f ) = f ) = f ) = e, e = 0 e f0) + f 0) + f 0) + f 0) = + +!! = 0.. si f) = si = 0. = 0 ) = ) A 0 f) A 0 si si ) cos 0 ) 0 0 cos ) ) 0 si = 0
24 ) = 0 si = + 0 = ) si + 0 f) B 0 si B 0 si si ) cos 0 ) 0 cos ) si 0 ) 0 6 = 6 si 6 ) = 0 si f ) = cos. f ) = si. f ) = cos f) f0) + f 0) + f 0)! + f 0)! = 6 = = si = si = si = 6 si : + 0 a = = si = 6 si = 0,,
25 ) 5.. f) = a, 6) f ) a) = a f) = fa) + f a) a) + f a)! a) + f a)! a) + + f ) a)! a). = 0 f) = f0) + f 0) + f 0)! + f 0)! + + f ) 0)! = 0 = 0 a) e + +! si! cos! +! + 5 5! +! + +! 7 7! 6 6! log + ) ) α + α! + αα )! + + ) )! + + ) )! + + ) + αα )α )! + + αα ) α ))! α. α α = ) ) + )! 7) 6) f ) a) f ) a) = f a), f ) a) = f a), f ) a) = f a). 7) + )α α =, = ) = = + +, 0 + = 0.
26 ) 6 si 6 si 6. ) si = 0! = π 6 si π 6 π 6 π ) ) si π 6 = 0.5, π = log a a a 5 ) ) log + ) = ) log ) = log + ) = ) 0 < < + + = = { ) log + ) } = { 5 ) log + ) + ) } 5 = log = e = = + + =.5 e + +! +! +! + +! = = = = = e.78888, +)! = 7 8! = ) + e
27 ) 7..5 e + e ) 0 ) ) + ) 0 e + e ) + e + e 0 e ) 0 e ) e e ) 0 e = + + ) ) ) ) ) ) ) = )e + e + ) 0 e + e = + ) + ) = 0 = 0 ) e ) ) ) e + e ) + = + ) + ) + + ) = 0 = 0 ) = =.K.). = t t +0 ) ) + + t +0 t t t t + t t t +0 t t t 0 + t = + t t) 8 = + t t + t t t + t t t ) + t) t 0, = t t =.K.) 0 0,,, 0 0 MuPAD.)
28 8. e e = 0 e =t t 0 log + t) t = t 0 + t) t = e ± + ) = e ) e 0 = e + e e.)?) a = ) + = ±, ±, ±, ). a = + =, a = + ) = 9 =.5, a = + ) = 6 7 =.707, a = + ) = =.06, + ) ) ) ) = C 0 + C + C + + C = + + )! = + + )! ) ) +! + ) )! ) + +! + + )! a + ) = + + ) + ) ) + + )!!! < + +! +! + +! < ! = > = = + ) < a e ) e + ) = e ) ) a a a e )
29 9 ) + e a = ) ) ) = = = + ) + ) a = ) ) ) = = = + ) + ) ) = + ) = e ) = ) = + ) + ) ) + ) + ) = e + ) = e ± ) ± + = e ± e < + + ) < + ) < + ) ) < + ) < + ) + ) + ) = e e + ) = e ) + ) + = e + ) = e!) ± ) t = t 0 ± + t) t + ) = e t 0 ± e = + ) e?) ) t 0 ± ± + = e.) e h 0 = = a A0, ) a.. e ),
30 0. 7 ). ) f), g) = a D fa) = ga) = 0, = a g )=\ 0 a f ) g ) f) a g) f ) a g ) F a) = F b) F c) = 0 c a b b D a F a) = F b) F ) = µf) λg) µ, λ ) µ, λ F a) = F b) µfa) λga) = µfb) λgb) µ {fa) fb)} = λ {ga) gb)} µ = gb) ga), λ = fb) fa) F ) = {gb) ga)} f) {fb) fa)} g) ) F a) = F b) F c) = {gb) ga)} f c) {fb) fa)} g c) = 0 c a b. a < < b b < < a g )=\ 0 gb) ga)=\ 0 g c)=\ 0. fb) fa) gb) ga) = f c) g c) ) = gt), = ft) C C Aga), fa)), Bgb), fb)) C A B AB Pgc), fc)) fa) = ga) = 0 a f) g) = f c) g c) c a a c a c a f c) g c) Y t = c Bgb), fb)) P C Aga), fa)) X f) a g) f c) c a g c) f ) a g ) Q.E.D.) Y Y t = c Bg), f)) P C a t = c P Bg), f)) C Aga), fa)) X Aga), fa)) X
31 . ) ) )0 ) ) 5) 6) 7) 8) 9) 5) 6) ) + ) ) ) ) + ) = ) ) + ) 7) < + < + < + + < ) ) ) + ) = ) < + + < =, 8) + + = + + ) ) + + ) ) = ) 9) ) + ) + + ) ), ) << ) ) + ) ) ) ) ) = ) ). ), ) + ) ) ) ) ) = ). 7),, + + = +. + = 8), + = + + = ) ) + ) ) ) = ) + ) +. = ) + ) ) ) = ) ) ). ) ) =. + ) = )
32 = log, 0), = S a = T log e =, b S a, T b S = T a b. b a = S a = e b T = log A, 0), P + a, 0), Q + a, log + a )), R + a, a ) = log A = log S = log = = APQ < S < APR a log + a ) < S < a = T b loge b ) < T < b = = log R Q + a, log + a )) S = T, a log + a ) < S = T < b b loge b ) < T = S < a A, 0) a = P + a, 0) = log log + a ) < b a a < loge b ) loge b ) Se b, 0) b e a 0, b 0 log + a ) a t 0 log + t) t =, loge b ) = log e = Commet b a = S T log e = =\ 0 T log = 0 S
33 5) ) ) ) 0 si si ) 0 si si ) si si = 6 ) cos ) + cos ) 0 + cos ) 6) cos cos = cos + ) cos ) = ) si si + cos ) = + cos = = cos cos si si ) cos si + si si ) = si si α ± β Commet cos cos 0 0 si si si θ θ, cos θ θ ) 0 0 ) si si = 6) 0 si si ) si ) 6 = 6 cos = ) = 9, cos cos cos 9 ) ) = = ) ) ) e log 5) 0 e log 5) log 5) log 5 = log 5 ) 0 e si log + ) e si 0 si si log + ) = = 5) logcos ) 0 Commet 0 log + cos ) 0 log + cos ) cos 0 e +, log + ) ) 0 5) 0 e si log + ) = logcos ) = log ). cos ) = ) = logcos ) =
34 5 { ) + ) } { = e ) + ) } = e ) = m m = + ) + ) m = e m m ) ) ) + ) ) + ) + = e Commet ) 00% e ) 00% e ) 00% e. ) -00%) e. 6 f ) = + f) = + = = = f) ) = ) = ) + ) f) ) A { f) ) + } + + 5) + 5) ) A ) ) ) ) = Commet = a ) ) ) = f) { ) + } = ) ) f) + ) ) ) f) a 0 + a a) + a a) + = a =. f) = + f.) +. ) = =.05
35 5 7 ). = 9 9 = ) ) = ) ) Commet = + 0 ) = = 6.5 ) = 0 = 0 8 f) = f) = 0, = 0, 0) = 0 f) = = ) + 0 = + f) f) = = = f) = f) ) + ) ) = = ) ) ) = = = Commet f) ) f) ) + = + ) + = + = = f)
36 6 9.) e + e ) 0 cos ) ) 0 ) = 0 e + e 0 e + e ) e e 0 ) 0 ) = 0 ) 0 e e 0 e e ) ) 0 e + e = ) ) = 0 cos ) 0 0 { cos )} si + ) 0 si + ) cos + 0 ) 0 cos + ) si 0 ) 0 = ) 0 log + ) = 0, f) = log + ) f ) = +, = 0 ) = ) = 0 A 0 0 A 0 f) 0 log + ) 0 = 0 A 0 log + ) = 0 0 {log + ) } ) 0 + log + ) = = log + ) ) Commet f ) = +, f ) = +), f 0) =, f 0) =. = 0 f) = log + ) f0) + f 0) + f 0)! =
37 7 + = 0 f) = + = + ) f ) = + ), f ) = + ), f ) = 8 + ) 5 f0) =, f 0) =, f 0) =, f 0) = 8 = 0 f) f) f0) + f 0) + f 0)! + f 0)! = + +! + 8! = )
38 8. A, A P A P Q P A Q Q 0 Q 0 Q 0 A A ) = cos A0, ) Pt, cos t) π < t < π, t=\ 0) Q ) = cos A0,) A P Q ) cos t) = si t P si t. = 0 cos t = t) si t A Q P = cos t t si t A A P Q 0, cos t ) t si t ) ), P A Q Q 0 ) cos t t 0 ) Q 0 0, ) t = si t = P A A P Q Q 0 A = ) Commet A A A P A P Q P A Q Q 0 Q 0 )
39 9 B A Q P A B C A C Aa, fa)) = f) 0 f a) = 0 = f) A Pb, fb)) P = b) + fb) f b) = f) A = a Q P A b a b a b a { b a r = a b) + fb) f b) P { { f b) } a b) + fb) a b) + fb) f b) f a) } f b) f a) b a + fb) } = f a) + fa) f fb) fa) a) b a b a r = fa) = f a) Q A a b f a) = 0 = f a) f) = cos f ) = si, f ) = cos. = f) A0, ) r = f 0) = A 0, r = + {f a)} ) f a) f) = cos A0, ) = A0, ) = A0, ) ), f = f) A C = f)
40 0 = f) = cos A0, ) : + + ) ) = = + 6 ) = ) A + + A A C : = + 6 = ) 6 ) = cos A = cos ) = A = f), = g) fα) = gα), f α) = g α), f α) = g α). = f) = g) α, fα) A = f) A fα) = gα), f α) = g α) = ) = ), 0) = ) = ), 0) ) = f) = g) = ) f) = f ) =, f ) = A0, 0), r 0 = f 0) = r r r ) A = = r < r > ) = B0, b) P, ) d d = + b) = + b) A)
41 ± ) a ± ) + a ) a + a + = 0, + a. =, + a a ) ) ) ) ) ) ) = = = ) = ) ± ) ) ) ± { ) ) = = ) < ) ) = ) + = ) a + b. ) = t t
42 = = = ) ) = 0. = { ) = )} = 0 { ) = )} = 0 =, = + ± ). = = = = + ) ) ) + )
43 a a, fa)) a ) ) + + ) ) ) = + = + + = + ) + + = + = + 0, ) ) ) = + = + + ) ) + ) ) ) ) ) ) ) = = + + ) ) ) ) + = ), = 6 ). 0 + ) = t, t ) 6 ) = = + 5 t 0 + t) + 5 t 9 + t t 0 t ) t 9 + t = 9 + t 9 + t ) = + t 9 + t 9. t ) + t t =
44 ) ) = ) 9 ) = = ) = ) = ) = ) 9 6 ) = = ) = + + ) = + ) + = Commet ) ) ) + ) + + = +
45 5 5 S S = a k= k= a k a + a + a + + a }{{} S ) a k S k= S a =\ 0 S ) ) k= ) ) k= ) k= ) k= + ) + + ){ }, S = S S = ) ){ ) }, S S = { ) } ) = 0 k= ) { ) } = =
46 5 6 ) kk+) = k k+, a k = fk) fk + ) S = = k= kk + ) = ) + = + k= k ) k + ) + ) + + ) + k= + ) ) = + ) k+ k+ = S = = k k+ k+ k+) k k+) = k k + ). a k = fk) fk + ) k k + ) k= { ) + ) + ) + + } + ) = + + k= ) = S {a } ) S ) a =\ 0 ) + ) + k= ) k= ) k= ) a, r r=\ ) S = a r ), 0 < r < r a < r < r ar = a + ar + ar + + ar + k= a r ) r = a r )
47 )a k = fk) fk + ) S a = =\ 0 ) S S = + + }{{} + ) ) ) }{{}}{{}}{{} > ) ) ) }{{} = = + ) k= k= k= =. + ) ) a k = k < k= kk ) k = = ) = + ) + ) + = S kk ) ) + + k =,,, ) ) a =\ 0 a = 0 ) ) ) ) a = 0 a =\ 0. ) a = S S. S = S ) a {S S } = S S = 0. a < M M {a }, a > M M {a }, k = π 6 k=
48 5 8 a > 0 =,,, ) = [ k k= a k a k k= ] [ = k k= ] k= k = + = k= + k > [ ] + d = log = log + ) k= k log + ) = k= k k= k = k= k < + [ d = + ] = < S = = a > 0 Commet k= k. ) k=. k k= k k ) = k= Zs) =. k. < + k= k+ k+ kk+), k, kk+) k+ k+ k= k k= k+ k+ kk+) kk + ) < k, > 5.) k + k + k + [. d = + 0 ] 0. = 0 0 = 0. 0 <. = <.. s = s + s + s + + s > Zs) 0 < s s
49 5 9 = = + + S = k= k + k + >, ) = [ ] k + k + k= k= k + k + = + + = =, = a ) S = k= ) k = + ) ) v k = ) ) k k ) 5 8 ) ) + + ) ) 5) k= k + + k k= k ) ) ) = = = ) a k = a a + a a + + k= {a } a + < a ) a k ) a = 0, ) = log
50 5 50 ) ) = 5 ) ) a ) S S = k = { } k k + k= k= = { ) + ) = ) +. 5) S = k= = = k + + k = k= k ) k + = k= + ) S = ) + + =, )} + k + k k + + k ) k + k ) = k= k k + { ) + 5) + 5 7) + + } + ) + = k= k + + k = ) ) ) ) = {,, { ) } { }. + { ) + + < + < + = } + + { } } { } + k= k k = =, k= k + k + )) + <, + > = = = + = = = + =
51 5 6 ) ) f) = f) = ) + ) + = + + = f) = + = = + ). = + ) = +, = {f) + )} ± ± = 0 = +, = + = + ). f) = ) = f ) =, 0 = 0 ). ) = ) ) = ).
52 6 5 ) f) = = = ) = = = f) = ) = = 0 0) = f) = 0 ) f ) = + = + ) = + ) + ) = =. ) = + ) = + ) = e +e ) = + 5) = + ) ) [ ] = + =, 0
53 6 5 ) f) = + = + = + = + = f) 0, f) = ± ± + = ± + = 0 0, + 0 ) = = f) = = = + = + = + = + f ) = + ) + ) = + + ) = ) + ) + ) f 0) = =, = ± f) = = + +.) - ) f) = 0. = ± 0 0 = ± f) = = f).)
54 6 5 = = = f) = = = f) = 0 ± f) = + ) ) = + ) + ) = +) = f) = = ). ) = +). ) f) ) + = 8 ) = +) = ) = f) = ) 8 = ). = ). f ) = ) ) = ) < 0 f ) ) = +. ) =
55 f) = = ) + ) = + + ) f) = + ) ) + ) f) = + + ) ) ) f) ) f) +) a g) f) = ) {g) f)} = 0 a ) ) a g) f) f) = 0 g) f) f) f) ) ) f) ) ) f) ) )+) ) + =, ) { f) } = ) 8 ) f) )+) + = = f) )
56 6 56 f) ) 0 8.) = )
57 6 57 ) = = ) = + f ) = + = = ± = = + = = = ) = = ) = + f ) = + = = = + = = = = = = + ) = e = e = e e f ) = e +e > 0 = e = e = e +e = e = e ) = = < 0,, = = +
58 6 58 = + = = = Commet = ± = ±. < < f) = )+ = ) = + = 0 =. =, =, =. 5) = : + =, = = + = + = = Commet f ) = = f ) = =. = 0 = = ± = 0 = =.
59 6 59 ) = + ) ) = ), = + = + = f) = + = + = + = + = f) ). = 0 = + = + = = + = + = + ), = =, = = + = + + ) ) = = ). = 0 0 = f) = = = = = = =
60 6 60 Commet f) = = + = + ) + ) f) + = f) + ) + = + = = +, = f)
61 ) +? ) log + ) ) = = log ) + a = b a, b a, b { + + a + b)} = 0 5 { } + + a + b) = c a, b, c )
62 ) > 0 i) ii)!! < si <! < cos <! + 5 5! +! ) 0 si.
63 k =,,, + + ) +k < < = + Commet = + + +k = ? ) log + ) ) = = log ) ) S S = k= log + ) = k k= log k + k = ) = = {log k logk + )} k= = {log log ) + log log ) + log log ) + + log log + )} = log + ) log + ) =, S ) S = k= = log k ) = { log log = log + Commet 0 = log+) log log k= ) + k + )k ) = k log log ) + log k= S log + = = log + ). k+ k k k log log 5 = k= ) + + ) log = log log ) { log log k k log k + } k log + ) ) )}
64 7 6 S + a = b a, b ) 0 ) 0 { } + a = a = 0. a = + b t 0 t t t + + ) t 0 t + t 0 t t + + = t + ) t + + ) t 0 t t + + ) Commet = + a + a b ) = b + a + b ) = 0 = + ) = +, ) a =, b = a, b = ) + ) = { + + a + b)} = 0 + = a + b = ) = + ) 5 + = +. a =, b = t =
65 7 65 t =, t +0 { } { + + a + b)} t +0 t + t + a t b t +0 ) 0 ) 0 t +0 + t + t a bt t { + t + t a bt} = a = 0. t 0 a =. t +0 + t + t bt t + t + t bt) + t + t + + bt) t +0 t + t + t + + bt) t 0 b )t + b)t t + t + t + + bt) t 0 b )t + b) + t + t + + bt = b 0 b = 0. b = 5 { } + + a + b) = c a, b, c ) { } 0 ) a =, b = t =, t +0 { } { } c + + a + b) t +0 t t + t + a t b t +0 ) 0 ) 0 t +0 + t + t a bt t { + t + t a bt} = a = 0. t 0 a =. + t + t bt t +0 t + t + t bt) + t + t + + bt) t +0 t + t + t + + bt) t 0 b )t + b)t t + t + t + + bt) t 0 b )t + b) t + t + t + + bt) ) t +0 ) 0 ) 0 b = 0. b =. ) c t 0 b )t + b) t + t + t + + bt) t 0 b ) + t + t + + bt = b = 5 8.
66 7 66 Commet c t +0 + t + t a bt t t +0 + t + t a bt ct t = 0 + t + t t = 0 a =, b =, c = 5 8 t 0, + t + t + t 5 8 t 6 ) > 0 ) 0 si. i) ii)!! < si <! < cos <! + 5 5! +! cos, si = 0 cos + si + 5!!! 5!. ) F ) = + 5! 5! ) si F ) = +!! F ) = cos! ) cos, ) F ) = si,! ), F ) ) = si > 0 > si F ) ) > 0. > 0 F ) > 0 F ) > F 0) = 0 > 0 F ) > 0 F ) > 0 F ) > F 0) = 0 > 0 F ) > 0 F ) > 0 F ) > F 0) = 0 > 0 F ) > 0 F ) > 0 F ) > F 0) = 0 F ) > 0 F ) > 0 i) F ) > 0 F ) > 0 ii) Q.E.D.
67 7 67 ) > 0! 5 5! < si <!! 5! +0 5! 0 0, = si 0 < si <! si = +0 6 ) si ) +0 ) 0 si = 6 si = +0 6
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sekibun.dvi
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高校生の就職への数学II
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9 5 ( α+ ) = (α + ) α (log ) = α d = α C d = log + C C 5. () d = 4 d = C = C = 3 + C 3 () d = d = C = C = 3 + C 3 =
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(1) 3 A B E e AE = e AB OE = OA + e AB = (1 35 e ) e OE z 1 1 e E xy e = 0 e = 5 OE = ( 2 0 0) E ( 2 0 0) (2) 3 E P Q k EQ = k EP E y 0
(1) 3 A B E e AE = e AB OE = OA + e AB = (1 35 e 0 1 15 ) e OE z 1 1 e E xy 5 1 1 5 e = 0 e = 5 OE = ( 2 0 0) E ( 2 0 0) (2) 3 E P Q k EQ = k EP E y 0 Q y P y k 2 M N M( 1 0 0) N(1 0 0) 4 P Q M N C EP
x () g(x) = f(t) dt f(x), F (x) 3x () g(x) g (x) f(x), F (x) (3) h(x) = x 3x tf(t) dt.9 = {(x, y) ; x, y, x + y } f(x, y) = xy( x y). h (x) f(x), F (x
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Chap9.dvi
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ac b 0 r = r a 0 b 0 y 0 cy 0 ac b 0 f(, y) = a + by + cy ac b = 0 1 ac b = 0 z = f(, y) f(, y) 1 a, b, c 0 a 0 f(, y) = a ( ( + b ) ) a y ac b + a y
01 4 17 1.. y f(, y) = a + by + cy + p + qy + r a, b, c 0 y b b 1 z = f(, y) z = a + by + cy z = p + qy + r (, y) z = p + qy + r 1 y = + + 1 y = y = + 1 6 + + 1 ( = + 1 ) + 7 4 16 y y y + = O O O y = y
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0 9 (1990 1999 ) 10 (2000 ) 1900 1994 1995 1999 2 SAT ACT 1 1990 IMO 1990/1/15 1:00-4:00 1 N 1990 9 N N 1, N 1 N 2, N 2 N 3 N 3 2 x 2 + 25x + 52 = 3 x 2 + 25x + 80 3 2, 3 0 4 A, B, C 3,, A B, C 2,,,, 7,
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名古屋工業大の数学 2000 年 ~2015 年 大学入試数学動画解説サイト
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