f x0
x
x% = .
100
500
30%.
500 · 30% = 500 · 0.3 = 150 ,
500 + 150 = 650 .
500 · (1 + 0.3) = 650 .
40%,
30%, 10 10%.
10%,
30% · (1 + 10%) = 30% · (1.1) = 33%.
300
20%, 30%.
300 · 1.2 · 1.3 = 468
468 - 300
= 0.56 = 56%.
300
1.2 · 1.3 - 1 = 0.56 = 56%.
468 - 300 300 · 1.2 · 1.3 - 300
= = 1.2 · 1.3 - 1.
300 300
p
p% Á := 1 +
100
P p1%
p2%
P · (100 + p1) % · (100 + p2) % - P p1 p2
= 1 + · 1 + - 1
P 100 100
p1 p2
= 1 + · 1 + - 1 · 100% = (Á1 · Á2 - 1) · 100%,
100 100
p1 p2
Á1 = 1 + Á2 = 1 + p1, p2
100 100
K0
r
n
In n
Kn n
In = rK0 · n
Kn = K0 + rK0 · n = K0 (1 + rn) .
rK0.
Kn+1 - Kn = K0 + rK0 · (n + 1) - (K0 + rK0 · n) = rK0,
500
4
 12%
1 = 360
Kn = K0 + rK0 · n.
364·3+365 1457
K0 = 500 r = 0.12, n = = ,
360 360
1457
K = 500 + 0.12 · 500 · = 742.83 .
360
198
n =
360
198
K = 500 + 0.12 · 500 · = 533
360
- - 2500 181 -
3250 - 5750 193 12
- 4200 1550 204 13
- 1900 -350 217 12
1600 - 1250 229 32
- 300 950 261 12
- - 950 273 -
12
I1 = 2500 · 0.12 · = 9.86
365
11
I2 = 5750 · 0.12 · = 20.79
365
13
I3 = 1550 · 0.12 · = 6.62
365
12
I4 = -350 · 0.18 · = -2. 07
365
32
I5 = 1250 · 0.12 · = 13.15
365
12
I6 = 950 · 0.12 · = 3.75
365
9.86 + 20.79 + 6.62 - 2. 07 + 13.15 + 3.75 = 52.10
950 + 52.10 = 1002.10
k
ik
mk
1
k,
k
" k = 2
" k = 4,
" k = 13,
" k = 52,
" k = 365 360
mk
Im = ikK0 · mk,
k
Km = K0 (1 + ik · mk) .
k
 4
1.3%.
k = 12, m12 = 4, i12 = 0.013, K0 = 1200,
K4 = 1200 + 0.013 · 1200 · 4 = 1262
n K0,
n,
n
1
mk
k
mk = nk.
ik ik
1 2
1 1
K0 n
k1 k2
ik ik ,
1 2
ik mk K0 = ik mk K0,
1 1 2 2
mk = nk1, mk = nk2,
1 2
ik nk1K0 = ik nk2K0.
1 2
1
ik1 k1
= ,
1
ik2 k2
ik = ik k2 ,
1 2
k1
1
ik r
k
r = ikk.
 i2 = 18%.
13 2
3
k = 12
2 1
i12 = i2 = 18% · = 3%.
12 6
3 m12 = 12 · 3 = 36
I = i12 · m12 · K0 = 0.03 · 36 · 400 = 432
360
13 k =
13
2 13
i360 = i2 360 = 18% · = 1.3%.
13
180
13
360 1080
3 m360 = · 3 =
13 13
13
1080
I = i360 · m360 · K0 = 0.013 · · 400 = 432
13 13
13
1
2 k = ,
2
2
i1 = i2 1 = 18% · 4 = 72%
2
2
1 3
m1 = · 3 =
2 2
2
3
I = i1 · m1 · K0 = 0.72 · · 400 = 432
2 2
2
360
k =
26 · 7
10000 - 9521.06
ik = = 0.0503 = 5.03%,
9521.06
K = K0 + ikK0.
360
r = ikk = 5.03% = 9.95%.
26 · 7
 K0 n m
n1, n2, ..., nm
m
n = ni.
i=1
i ri i = 1, 2, ..., m.
i
In = rini · K0.
i
n
m m
I = rini · K0 = K0 rini,
i=1 i=1
m m
K = K0 + K0 rini = K0 1 + rini .
i=1 i=1
r n
Å»
m
rnK0 = K0 rini.
Å»
i=1
n
m
1
r = rini.
Å»
n
i=1
r1, r2, ..., rm
r1, r2, ..., rm.
4 3 5
K = 3600 1 + 0.06 · + 0.055 · + 0.045 · =
12 12 12
1 11 3 421
3600 1 + + + = 3600 · = 3789
50 800 160 400
21
r = = 0.0525 = 5.25%.
Å»
400
 K0
K D
r
n
K = K0 (1 + rn)
K0 = K (1 + rn)-1
K + Krn - K Krn
D = K - K0 = K - K (1 + rn)-1 = = = Krn (1 + rn)-1 .
1 + rn 1 + rn
K 1000
K0 = = = 892.86 ,
1 + rn 1 + 0.16 · 0.75
1000 1000
K0 = = = 862.07
1 + 0.16 1 + 0.16
F
D
P
d
n
D = dF · n
P = F - D = F (1 - dn) .
P
F = .
1-dn
F - D > 0
dn < 1,
d
1
n < ,
d
n
1
d < .
n
d = 14%.
3
D = 0.14 · 1500 · = 52.50
12
P = F - D = 1500 - 52.50 = 1447.50
" d =
12%
" r = 15%
P = 10000
F :
P 10000
F = = = 10638.30
1
1 - dn - 0.12 ·
1
2
1
I = rP n = 0.15 · 10000 · = 750.00
2
K = 10000 + 0.8 · 750.00 = 10600.00
 r
P
= P + rP n · 0.8
1 - dn
1
= 1 + rn · 0.8
1 - dn
1
rn · 0.8 = - 1
1 - dn
nd
rn · 0.8 =
1 - dn
1.25d 1.25 · 0.12
r = = = 0.159 6 = 15.96%.
1 - dn - 0.121
1
2
d r n,
D = I K0 = P,
dF n = rP n,
P
= rP,
1 - dn
d
r =
1-dn
r
d = .
1+rn
n
1 1
n = - .
d r
d n
 n r n
d.
d n nd < 1
n r.
1
n n < ,
d
26·7
F = 10000 P = 9521.06 n = .
360
D F - P 10000 - 9521.06
d = = = = 0.0947 = 9.47%.
26·7
nF nF · 10000
360
D 10000 - 9521.06
r = = = 0.0995 = 9.95%.
26·7
nP · 9521.06
360
d n r
d n.
D I
= = r.
nP nP
d r
n D I
Å»
1
n (n < )
d
D > I Ô! n > n,
Å»
D < I Ô! n < n.
Å»
 P F
P n
D = dF n,
d 1 - dn
I = rP n = rF (1 - dn)n = F (1 - dn)n = · dF n.
1 - dÅ» 1 - dÅ»
n n
D 1 - dn
Å»
= .
I 1 - dn
1
n <
d
1 - dÅ»
n
D > I Ô! > 1 Ô! n > n
Å»
1 - dn
1 - dÅ»
n
D < I Ô! < 1 Ô! n < n
Å»
1 - dn
n > n, D > I, n < n, D < I.
Å» Å»
n I
n r, D
1
n d (n < ).
d
d r
D > I Ô! r < Ô! d > .
1 - dn 1 + rn
d r
D < I Ô! r > Ô! d < .
1 - dn 1 + rn
P
F =
1 - dn
D dF dP 1 d 1
= = · = · ,
I rP 1 - dn rP 1 - dn r
1
n <
d
d 1 d r
D > I Ô! · > 1 Ô! r < Ô! d >
1 - dn r 1 - dn 1 + rn
d r
D < I Ô! r > Ô! d <
1 - dn 1 + rn
 d
" F
" D
" P = F - D
" n
F,
d n
P = F (1 - dn) .
F = 200,
P = 195,
D = F - P = 5.
276 - 183 = 92
92
n = .
360
D 5 5 9
d = = = = = 9.78%.
92 92
nF 200 · 5 · 92
360 9
 n
9 9
d 9 40 3 10 30
92 92
r = = = · = · = = 10.03%.
9 92 1
1 - dn - · 1 - 92 39 23 13 299
1
92 360 40
10.03%.
10.03%.
d = 16%,
r = 17%.
d 0.16 0.16 16 100 1
= = = · = = 16.67% < r.
90 16 1
1 - dn - 0.16 · 1 - · 100 96 6
1
360 100 4
n,
Å»
1 1 100 100 25 · 17 - 400 25
n = - = - = = = 0.367 647 058 8
Å»
d r 16 17 68 68
0.367 647 058 8 · 360 = 132.352 941 2
 K0 > 0 r,
n
n " N (:= {1, 2, . . .}) .
K1 = K0 + rK0 = K0 (1 + r)
K2 = K1 + rK1 = K1 (1 + r) = K0 (1 + r)2
n K0 (1 + r)n
n Kn
Kn = K0 (1 + r)n ,
n
In = Kn - K0 = K0 ((1 + r)n - 1) .
 (1 + r) .
Kn+1 = Kn (1 + r)n , n " N *" {0} .
K0 Kn (Kn > K0) n
Kn
n
r = - 1,
K0
K0 Kn (Kn > K0) r
n
Kn
Kn ln K0
n = log1+r = .
K0 ln (1 + r)
K0, Kn r
Kn
ln
K0
ln(1+r)
K0 = 10000
r = 12%
r = 10%
Ü
K5 = K0 (1 + rn) = 10000 (1 + 0.12 · 5) = 16000.00 ,
Ü
K5 = K0 (1 + r)n = 10000 (1 + 0.1)5 = 16100.00
Ü
r
Å»
r
Ü
Ü
K5
16100
- 1
- 1
K0
10000
r = = = 12.20%,
Å»
n 5
r
Ć
r
K5 16000
5
5
r = - 1 = - 1 H" 9.856%.
Ć
K0 10000
k
mk mk " N
ik
Km
k
mk mk - tego K0
k
Km = K0 (1 + ik)m ,
k
mk
k
Im = K0 ((1 + ik)m - 1) .
k
K0 = 1000
k = 4
i4 = 6%. mk = 8
k
K8 = K0 (1 + ik)m = 1000 (1 + 0.06)8 = 1593.85
k
rk ik rk
rk := k · ik.
mk
K0
rk mk
Km = K0 1 + ,
k
k
n
rk nk
Kn = K0 1 + .
k
rk nk+k
K0 1 +
Kn+1 rk k
k
= = 1 + .
Kn K0 1 + rk nk k
k
Ák n
rc.
k
rc n
Kn K0
k
rc nk rc nrc
rc
c
Kn = lim K0 1 + = lim K0 1 + = K0er n.
k" k"
k k
n " N n
t
t e" 0 K(t)
rc
c
K(t) = K(0)er t.
t = 0
t = t0 t e" t0
c
K(t) = K(t0)er (t-t0).
c
r = er - 1
0
K(t) = K(t0)(1 + r)(t-t ).
t = t0 +1 r
r%
 "t "t
t K (t) t + "t
K (t + "t) = K (t) + K (t) rc"t
K (t + "t) - K (t)
= rcK (t) .
"t
K (t + "t) - K (t)
lim = rcK (t)
"t0
"t
K (t) = rcK (t) .
c
K (t) = cer t,
c t = 0
K0
K0 = c,
c
K (t) = K0er t.
t
t
ik
1
ik n K0
2
n
Kn.
1
k1 ik
1
k1
k2 ik
2
1
K0 n ik
k2 1
ik
2
1 2
K0 (1 + ik )nk = K0 (1 + ik )nk
1 2
1 2
(1 + ik )k = (1 + ik )k .
1 2
rk rk
1 2
ik ik
1 2
rk1 k1 rk2 k2
1 + = 1 + .
k1 k2
Ák Ák
1 2
rk rk
1
Ák = Ák .
1 2
ik ik ik ik
1 2 1 2
k1 k2, rk rk
1 2
Ák Ák ik , ik
1 2 2 2
ik ik
1 2
1 2
(1 + ik )k = (1 + ik )k ,
1 2
rk1 k1 rk2 k2
1 + = 1 + ,
k1 k2
Ák = Ák .
1 2
ik
1
k1 ik
2
k2
k1
k2
ik = (1 + ik ) - 1.
2 1
ik k
ref
ref = (1 + ik)k - 1 = Ák - 1,
Ák ik
 ik k
ref
1
k
ik = (1 + ref) - 1.
rk
1
k1 rk
2
k2
k1
k2
rk1
rk = 1 + - 1 k1.
2
k1
rc
k ik,
c
er = (1 + ik)k .
rc
c
ref = er - 1 = Ác - 1.
ik
k, r
n
(1 + ik)nk = 1 + rn.
n
K0 n
r(i) i = 1, 2, ..., n.
K1 = K0 1 + r(1) ,
K2 = K0 1 + r(1) 1 + r(2) ,
...
n
Kn = K0 j 1 + r(i) ,
i=1
n
In = K0 j 1 + r(i) - 1 .
i=1
 r
Å»
n Kn,
n
K0 (1 + r)n = K0 1 + r(i) ,
Å»
i=1
n
n
r = (1 + r(i)) - 1.
Å»
j=1
Á
Å»
n
n
Á = r + 1 = (1 + r(i)),
Å» Å»
j=1
n
i(j), j = 1, 2, ..., m
K0 m
Km = K0 m 1 + i(j) ,
j=1
m m- %2Å‚
m
m
%2Å‚ = (1 + i(j)) - 1.
j=1
m- Ám
m
m
Á = 1 + i(j) ,
j=1
m- r
m
m
r = Ám - 1 = 1 + i(j) - 1
j=1
K0 n
r(j), j =
c
1, 2, ..., n. Kn n
(1) (2) (n) (j)
n
rc
c c
j=1
Kn = K0er er ...rc = K0e ,
rc
(j)
n
rc
c
j=1
er n = e ,
n
1
rc = r(j),
c
j=1
n
r(j), j = 1, 2, ..., n.
c
 Kn,
K0 n
r.
Kn = K0 (1 + r)n
Kn
K0 = .
(1 + r)n
rc.
c
Kn = K0er n,
c
K0 = e-r nKn.
K0.
1
c
e-r
1+r
½
Kn = ½Kn+1,
d
Kn+1 Kn
Kn+1 - Kn
d = = 1 - ½.
Kn+1
r
1 r
d = 1 - = ,
1 + r 1 + r
rc
c
dc = 1 - e-r .
K0, n
Kn
K0 = ½nKn,
d :
K0 = (1 - d)n Kn.
 m
i(j) j = 1, 2, ..., m,
inf
finf m-
m
%2Å‚inf m
m 1 + iinf
m
1 + finf = 1 + i(j) ,
j=1 inf
m
m
m
m
%2Å‚inf = 1 + finf - 1 = 1 + i(j) - 1.
j=1 inf
K0
t
inom t
Knom = K0 (1 + inom) .
Kreal
iinf
Knom 1 + inom
Kreal = = K0 .
1 + iinf 1 + iinf
t inom
Knom := K0 (1 + inom) .
 t iinf
Knom
Kreal := .
1+iinf
1 + inom
ireal := - 1.
1 + iinf
Knom K0 (1 + inom)
Kreal = = = K0 (1 + ireal) ,
1 + iinf 1 + iinf
ireal
t
1 + inom = (1 + ireal) (1 + iinf) .
inom-iinf
ireal =
1+iinf
inom-ireal
iinf = .
1+ireal
iinf > 0, ireal < inom - iinf.
iinf < 0, ireal > inom - iinf = inom + |iinf|
ireal > 0 Ô! iinf < inom
|iinf| 1)
22%.
1 + rnom 1.22
1 + rreal = = H" 1.0796,
1 + rinf 1.13
rreal = 7.96%
 5% 6500
6% 9%.
rnom
8000 = 6500 (1 + rnom)2 ,
8000
rnom = - 1 H" 10.94%.
6500
rnom - rinf 0.1094 - 0.05
rreal = = H" 5.66%.
1 + rinf 1 + 0.05
rinf = (1 + 0.06) (1 + 0.09) - 1 H" 7.49%.
Å»
rnom - rinf 0.1094 - 0.0749
= H" 3.21%.
1 + rinf 1 + 0.0749
Knom 1 + iinf - iinf iinf
Kreal = = Knom = Knom 1 - .
1 + iinf 1 + iinf 1 + iinf
iinf
dinf := .
1 + iinf
100000 · 1.06 + 40000 = 1460000.
R t K (t) t
K (t0) t0.
 r > 0.
t e" t0
0
K (t) = K (t0) (1 + r)t-t ,
t < t0
0-t
K (t0) = K (t) (1 + r)t ,
0
K (t) = K (t0) (1 + r)t-t .
0
K (t) = K (t0) (1 + r)t-t , t " R.
t
t
c
rc 1 + r = er
c
K (t) = K (t0) er (t-t0).
t0 t0
1-t0
t1. K (t1) = K (t0) (1 + r)t
0 1-t0
1 1
K (t) = K (t0) (1 + r)t-t +t1-t1 = K (t0) (1 + r)t (1 + r)t-t = K (t1) (1 + r)t-t .
K
K1, . . . Km,
m
K (t) = Kj (t) ,
j=1
Kj
m m
0 0 0
K (t) = K (t0) (1 + r)t-t = (1 + r)t-t Kj (t0) = Kj (t0) (1 + r)t-t .
j=1 j=1
t c (t)
"t
Å» Å»
t t + "t
Å»
c (t) "t.
t0 = 0 t = T
T
c (t) dt.
0
 t = T. c
t = T, t < T
T
r > 0
c (t) (1 + r)T -t .
T
C (T ) = c (t) (1 + r)T -t dt.
0
t = 0, c (t)
T
c (t) (1 + r)-t ,
T
C (0) = c (t) (1 + r)-t dt.
0
t0
Ä :
0
C (Ä) = C (t0) (1 + r)Ä-t .
T
C (T ) = c (t) (1 + r)T -t dt
0
T T
C (0) = C (T ) (1 + r)-T = (1 + r)-T c (t) (1 + r)T -t dt = c (t) (1 + r)-t dt.
0 0
t. K1 K2
t,
r.
1
K1 (t) = K1 (t1) (1 + r)t-t
2
K2 (t) = K2 (t2) (1 + r)t-t ,
K1 (t1) , K2 (t2) > 0.
t,
1 2
K1 (t1) (1 + r)t-t = K2 (t2) (1 + r)t-t ,
2
(1 + t)t-t
1 2
K1 (t1) (1 + r)-t = K2 (t2) (1 + r)-t .
rc r,
c c
K1 (t1) e-r t1 = K2 (t2) e-r t2.
t,
K1 K2
t t .
K1 K2
t.
K1 K2
K1(t1)
2-t1
= (1 + r)t .
K2(t2)
r. K1 K2 r,
r
2-t1
2-t1
(1 + r)t = (1 + r )t ,
t2 = t1.
K1 K2 K1 (t1) K2 (t2)
 t1 t2 r
1
t2-t1
K1(t1)
r = - 1.
K2(t2)
Mj,
j = 1, 2, ..., m Nj, j = 1, 2, ..., n.
K1 K2
m
K1 (t) = Mj (t)
j=1
n
K2 (t) = Nj (t)
j=1
 R+ := [0, ") .
C : R+ R+
x " R+ C (x)
x C
x > 0
C (x)
c (x) :=
x
x
x c :
(0, ") R+
x0 " R+, "x > 0
C (x0 + "x) - C (x0)
"x
"x
x0.
C (x0 + "x) - C (x0)
C (x0) := lim ,
"x0
"x
x0. C C
"x
C (x0 + "x) - C (x0) H" C (x0) "x,
"x = 1
x0
C (x0) .
x0 c(x)
 x x e" 0
C(x) = x3 - 60x2 + 1528x C
C (x) = 3x2 - 120x + 1528.
x = 5
C (6) - C (5) = 7224 - 6265 = 959,
C (5) = 1003
x = 100,
C (101) - C (100) = 572 569 - 552 800 = 19 769,
C (100) = 19 528
x = 5 x = 100
C(x)
c(x) = = x2 - 60x + 1528.
x
c x = 30
x = 30
c(30) = 628 = C (30)
x0
c(x0) = C (x0).
x0 x0
C(x)
c (x0) = 0 Ô! = 0,
x
x=x0
C (x0)x0 - C(x0)
= 0,
x2
0
C(x0)
= C (x0),
x0
c(x0) = C (x0).
x e" 0
U(x)
 x U : R+ R+
x U
x
"U
U (x) = lim .
"t0
"x
U
U
Z(x)
x
Z : R+ R+
Z(x) = U(x) - C(x) dla x e" 0,
U(x) C(x) x
x0
C (x0) = U (x0) x0
x0
p(x) = 40 - 0.03x, x
x
C(x) = 0.01x2 + 20x + 225
C (x) = 0.02x + 20,
Z (x) = (xp (x) - C (x)) = 20x - 0.04x2 - 225,
Z (x) = -0.08x + 20.
x0
U (x0) = C (x0),
x0 = 250.
f : (a, b) R (a, b) ‚" R+ x0 " (a, b) "x
(x0 + "x) " (a, b)
f x0 "x
"y f(x0 + "x) - f(x0)
:= ,
y f(x0)
 f(x0) = 0

"x
x0
x0
f x0, x0+"x
f(x0 + "x) - f(x0) x0
·
f(x0) "x
Ex ,"xf
0
f x0
lim Ex ,"xf
0
"x0
Ex f
0
"x = 0.01x0 = 1% · x0,
f(x0 + "x) - f(x0)
Ex f H" Ex ,"xf = · 100%.
0 0
f(x0)
Ex f
0
f x 1%
f(x0) = 0

x0
Ex f = f (x0) .
0
f(x0)
f(x0 + "x) - f(x0) x0 x0
lim Ex ,"xf = lim · = f (x0) .
0
"x0 "x0
"x f(x0) f(x0)
x f p%
x0 q%
q H" pEx f.
0
x0 x p%
q% f(x0)
p q
f(x0 + x0) - f(x0) = f(x0).
100 100
f (x0 + "x) - f (x0) H" f (x0) "x,
x0 x0 f (x0 + "x) - f (x0)
Ex f = f (x0) H"
0
f (x0) f (x0) "x
p
"x = x0
100
q
· f(x0)
x0 100 q
Ex f H" = ,
0 p
f(x0) x0 p
100
q H" pEx f.
0
2x
f(x) = , x > 0
x + 8
x0 = 2
16
f (x) =
(x+8)2
1 16 8
Ex f = (x + 8) = .
0
2 (x + 8)2 x + 8
x0 = 2 E2f = 0.8 x0 = 2
1%, f 0.8%
2 · 2.02 4.04 202
f(x0 + 0.01x0) = f(2 + 0.02) = f(2.02) = = =
2.02 + 8 10.02 501
4
f(x0) = f(2) = = 0.4.
10
202
f(x0 + 0.01x0) 202 10 505
501
· 100% = · 100% = · · 100% = · 100% H" 100.798 403 2,
f(x0) 0.4 501 4 501
0.798 403 2%.
f.
pEx f
0
p%
f
f (x0 + "x) - f (x0) = f (x0) "x.
"x f
(x0, x0 + "x)
f(x)
f(x)
STYCZNA
STYCZNA
f(x)
0
f(x)
0
k k
X0
X0
f x0
f
x0
f x0 f (x0)
f
(x0, f(x0)) x1 k := x0 -x1
f(x0)
f (x0) =
k
x0 x0 f(x0) x0
Ex f(x0) = f (x0) = = .
0
f(x0) f(x0) k k
x0
x0 1 Ex f(x0) = > 1, x0 > k,
0
k
x0 1 Ex f(x0) =
0
x0
< 1, x0 < k
k
C : R+ R+ C(x)
x C
C(x) > 0
x
ExC = C (x).
C(x)
c
C (x)
ExC = .
c(x)
 c
x
Exc = c (x).
c(x)
Exc + 1 = ExC.
x x C(x) x2 xC (x) - C(x) x
Exc = c (x) = · = · = C (x) - 1 = ExC - 1.
C(x)
c(x) x C(x) x2 C(x)
x
Cena
q1
p2
q2
p1
x1 x2
Ilość
 q
p.
q (p)
q (p) q
p
c
p
p
1%
p q
p0 = 30 "p = 6
"p 6
= = 20%
p 30
p0 = 30 q = 200
6 p + "p = 36 q +
"q = 190 "q = -10
"q -10
= = -5%.
q 200
p0).
"q "p 1
: = - .
q p 4
p 1%
0.25%.
| |
c
(0; ")
" | | = 0
c
" | | < 1
c
1% 1%
" | | = 1
c
1% 1%
" | | > 1
c
1%
1%
" | | "
c
< 0
c
"q "p
: = -1
q p 4
| | < 1
c
w (d) w
d.
d w
"w "d
= : .
d
w d
w d
d
1%
| | < 0
d
" < 0,
d
"w < 0
"d > 0 "w > 0
"d < 0
" > 0,
d
"w < 0
" < 0 "w > 0 "d > 0
" [0, 1]
d
> 1
d
"
x
T1(x) = a · , x > 0 a, b > 0;
x + b
"
x - c
T2(x) = a · , x e" c a, b, c > 0;
x + b
"
x - c
T3(x) = a · x · , x e" c a, b, c > 0
x + b
x a, b, c
x
T1(x) = a · ,
x + b
x > 0 a, b > 0. T1
( )
f1 x
a
0
X
x
lim a · = a,
x"
x + b
T1 y = a
x0 + b ab b
Ex T1 = x0 · · = .
0
ax0 (x0 + b)2 x0 + b
Ex T1
0
b
lim = 0.
x"
x0 + b
( )
Exf1
0
1
0 X
T1
1 Ex T1 < 1 x0 > 0
0
b > 0 1%
1%
x - c
T2(x) = a · ,
x + b
x e" c a, b, c > 0. f2(x)
( )
f2 x
a
0
c
X
x > c
x - c
lim a · = a.
x"
x + b
T1
b + c
Ex T2 = x0 · , x0 > c.
0
(x0 - c)(x0 + b)
( )
Exf2
0
c
0
X
T2
b + c
lim x0 · = 0,
x"
(x0 - c)(x0 + b)
T2
x - c
T3(x) = a · x · ,
x + b
x e" c a, b, c > 0.
( )
f3 x
0
c b+c
X
a · (x2 + 2bx - bc)
T3(x) = > 0,
(x + b)2
 T3
T1 T2
x0(x0 + b) a(x2 + 2bx0 - bc) x2 + 2bx0 - bc
0 0
Ex T3 = · = .
0
ax0(x0 - c) (x0 + b)2 (x0 - c)(x - 0 + b)
( )
Exf3
0
1
c
0
X
T3
1 ((Ex f3) > 1 x0 > c),
0
Ex T3
0
x2 + 2bx0 - bc
0
lim = 1
x" - c)(x - 0 + b)
(x0
T3
1
1% 1%
x
T1(x) = a1 · , x > 0 a1, b1 > 0;
x + b1
x - c2
T2(x) = a2 · , x e" c2 a2, b2, c2 > 0;
x + b2
x - c3
T3(x) = a3 · x · , x e" c3 a3, b3, c3 > 0.
x + b3
( )
x
T
a2
a1
T1 T2 T3
0 c1 c2 X
b+c
T1, T2, T3
ci i = 1, 2, 3
ai i = 1, 2, 3
T1, T2, T3
a1
x > c1 c1
a2
x > c2 c2
 P
Ä„ R C Y
a, b, c, Ä…, ² Å‚.
a", Ä„ a" R - C,
 C = 75 + 10Q,
C Q.
Qd = Qs
Qd > 0
Qs > 0
P > 0
P1 > 0.
Qd = a - bP, P e" 0
Qs = -c + dP, P > P1
Qd = Qs,
a, b, c, d > 0.
I0
G0
C
Y
C = a + bY
Y = C + I0 + G0
a > 0 b " (0, 1) .
a
b 1,
b < 1.
Å» Å»
C Y
a + I0 + G0
Å»
Y = ,
1 - b
a + b (I0 + G0)
C = .
1 - b
n e" 1
n
Xi i - tej (i = 1, ..., n) ,
xij i - tej j - ta
Yi i - tej
i - tej
n
Xi = xij + Yi i = 1, ..., n.
j=1
 i - tej j - ta
j - tej
xij = aijXj i = 1, ..., n, j = 1, ..., n.
aij
n
Xi = aijXj + Yi i = 1, ..., n.
j=1
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
X1 a11 · · · a1n X1 Y1
ïÅ‚ śł ïÅ‚ śł ïÅ‚ śł ïÅ‚ śł
ïÅ‚ śł ïÅ‚ śł ïÅ‚ śł ïÅ‚ śł
= +
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
Xn an1 · · · ann Xn Yn
Å» Å»
X = [X1, ..., Xn]T , Y = [Y1, ..., Yn]T , A = [aij]i,j=1,...,n
Å» Å» Å»
X = AX + Y ,
Å» Å»
(I - A) X = Y .
A X = [xij]
Å»
(I - A) X
Å»
Y
Xi, xij Yi
i = 1, ..., n, j = 1, ..., n
xij
aij = ,
Xj
xij i-tej j-
xij d" Xj, aij d" 1.
j = 1, ..., n
n n
xij = Xj aij
i=1 i=1
j-
Xj, j -ta
n
Xj aij d" Xj, j = 1, ..., n
i=1
n
aij d" 1 j = 1, ..., n
i=1
Yj > 0, j - tej
n
aij < 1.
i=1
det (I - A) = 0, I - A

A
Å» Å»
X = (I - A)-1 Y .
Å» Å»
X Y
n e" 1
t
Xi (t) i - tej (i = 1, ..., n) t,
xij (t) i - tej j - ta t,
Yi (t) i - tej t
Å»
N *" {0} t X (t) = [X1 (t) , ..., Xn (t)]T
N *" {0} t xij (t) ,
Å»
N *" {0} t Y (t) = [Y1 (t) , ..., Yn (t)]T
t,
A
t + 1
t
Å»
Y (t)
t
Å» Å» Å»
Y (t) = S (t) + C (t) ,
Å» Å»
S (t) - S (t) = [S1 (t) , ..., Sn (t)]T ,
t + 1,
Å» Å»
C (t) - C (t) = [C1 (t) , ..., Cn (t)]T ,
t + 1
i = 1, ..., n Si (t)
j (j = 1, ..., n),
n
Si (t) = sij (t) , i = 1, ..., n,
j=1
sij (t) i - tej
j - tej
sij j-tej
t + 1,
sij (t) = zij (Xj (t + 1) + X (t)) i = 1, ..., n, j = 1, ..., n.
zij
n n
Si (t) = sij (t) = zij (Xj (t + 1) - Xj (t)) i = 1, ..., n,
j=1 j=1
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
S1 z11 · · · z1n X1 (t + 1) - X1 (t)
ïÅ‚ śł ïÅ‚ śł ïÅ‚ śł
ïÅ‚ śł ïÅ‚ śł ïÅ‚ śł
=
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
Sn zn1 · · · znn Xn (t + 1) - X (t)
Z = [zij]i=1,...,n, j=1,...,n
Å» Å» Å»
S (t) = Z · X (t + 1) - X (t) .
t
Å» Å» Å» Å» Å» Å» Å»
(I - A) X (t) = Y (t) = S (t) + C (t) = Z · X (t + 1) - X (t) + C (t) ,
Å» Å» Å» Å»
Z-1 (I - A) X (t) = X (t + 1) - X (t) + Z-1C (t)
Å» Å» Å»
X (t + 1) = Z-1 - Z-1A + I X (t) - Z-1C (t) .
A Z (I - A) Z
Å» Å»
C (0) S (0) ,
Å» Å» Å»
Y (0) = S (0) + C (0) .
Å» Å»
X (0) = (I - A)-1 Y (0) .
Å» Å» Å»
X (1) = Z-1 - Z-1A + I X (0) - Z-1C (0) .
Å»
X (1) t = 1
Å» Å»
Y (1) = (I - A) X (1) .
Å»
Y (1)
Å» Å»
S (1) C (1)
Å» Å» Å»
Y (1) = S (1) + C (1) ,
Å» Å»
S (1) C (1)
Å» Å»
X (1) C (1)
Å» Å» Å»
X (2) = Z-1 - Z-1A + I X (1) - Z-1C (1) .
"
Å»
X (t) .
t=0
Å» Å» Å»
X (t + 1) = Z-1 - Z-1A + I X (t) - Z-1C (t) ,
Å»
X (0) = X0,
Å»
C (t) t = 0, 1, ....
Å»
C (t)
Å» Å» Å» Å»
Y (t) S (t) = Y (t)-C (t)
t = 0, 1, ....
Å»
S (t)
Å» Å» Å»
C (t) d" Y (t) = (I - A) X (t) t = 0, 1, ....
 t = 0, 1, 2, ...
{P (t)}"
t=0
t = 0, 1, 2, ...
Qs (t) t
t
Qd (t) t
t
P (t) t.
Qd (t) P (t)
Qd (t) e" 0.
Qs (t) P (t - 1)
Qs (t) e" 0.
Qd (t) = Ä… - ²P (t)
Qs (t) = -Å‚ + ´P (t - 1)
Qd (t) = Qs (t)
t = 1, 2, ... Ä…, ², Å‚, ´ > 0
{P (t)}"
t=0
Å‚ Ä…
d" P (t) d" t = 0, 1, 2, ....
´ ²
Å‚ Ä…
d"
´ ²
²Å‚ - Ä…´ d" 0
Ä…
-²
-Å‚
P1 e" 0,
´
{P (t)}"
t=0
Ä… - ²P (t) = -Å‚ + ´P (t - 1) ,
² = 0

´ Ä… + Å‚
P (t) = - P (t - 1) + .
² ²
t
´
Po (t) = c - , t = 0, 1, 2, ...,
²
c
Ps (t) = k,
´ Ä… + Å‚
k = - k + ,
² ²
Ä… + Å‚
k = ,
² + ´
² + ´ > 0.
t
´ Ä… + Å‚
P (t) = c - + , t = 0, 1, 2, ....
² ² + ´
P (0) = P0,
Ä… + Å‚
P (0) = c + ,
² + ´
P (0) = P0
t
Ä… + Å‚ ´ Ä… + Å‚
P (t) = P0 - - + , t = 0, 1, 2, ....
² + ´ ² ² + ´
P (t)
Å‚
Ä…
d" P0 d" .
´ ²
Ä…+Å‚
P0 = ,
²+d
Ä… + Å‚
P (t) = , t = 0, 1, 2, ....
² + d
Å‚
Ä…
d" P (t) d" t = 0, 1, 2, ...
´ ²
Ä…+Å‚
P0 >
²+´
Ä…+Å‚
P0 < .
²+d
Ä…+Å‚
´
< 1. {P (t)}" .
t=0
² ²+d
´
= 1. {P (t)}"
t=0
²
P0 t
P (t) = .
2Ä…+Å‚ - P0 t
²+d
Ä…+Å‚
.
²+d
´
> 1. t
²
m-
m-
 t
m-