1
1˜Ÿ 4•Ø
1˜Ü© 4•ÐØ
1˜Ù Cþ†¼ê
§1. ¼êVg
1. )eت§¿xÑx‰Œµ
(1) −2 <
1
x + 2
(2) (x − 1)(x + 2)(x − 3) < 0
(3)
1
x − 1
< a
(4) 0 6 cos x 6
1
2
(5)
x
2
− 16 < 0
x
2
− 2x > 0
)µ
(1) x < −
5
2
½x > −
3
2
-
0
x
-1-2-3
bb
(2) 1 < x < 3½x < −2
-
0
x
-1-2 31 2
c cc
(3) a > 0ž§x < 1½x > 1 +
1
a
¶
-
0
x
1
1 +
1
a
c c
a < 0ž§1 +
1
a
< x < 1
-
0
x
1
1 +
1
a
cc
a = 0ž§x < 1
-
0
x
1
c
2
(4) 2kπ +
π
3
6 x 6 2kπ +
π
2
½2kπ −
π
2
6 x 6 2kπ −
π
3
(k ∈ Z)
-
0
x
(5) −4 < x 6 0½2 6 x < 4
-
0
x
-4 2 4
c c
2. y²eýéŠØªµ
(1) |x − y| > ||x|−|y||
(2) |x
1
+ x
2
+ x
3
+ ··· + x
n
| 6 |x
1
| + |x
2
| + ··· + |x
n
|
(3) |x + x
1
+ ··· + x
n
| > |x| − (|x
1
| + ··· + |x
n
|)
y²µ
(1) Ï|x||y| > xy§K(x − y)
2
> (|x| − |y|)
2
§u´|x − y| > ||x|−|y||
(2) ^êÆ8B{y².
(i) n = 2ž§d|x
1
+ x
2
| 6 |x
1
| + |x
2
|§(ؤá.
(ii) bn = kž(ؤá§=k|x
1
+ x
2
+ x
3
+ ··· + x
k
| 6 |x
1
| + |x
2
| + ··· + |x
k
|.
Kn = k + 1ž§|x
1
+ x
2
+ x
3
+ ···+ x
k+1
| 6 |x
1
+ x
2
+ x
3
+ ··· + x
k
| + |x
k+1
| 6 |x
1
| + |x
2
| +
··· + |x
k
| + |x
k+1
|
nþŒ•§é˜ƒg,ên§|x
1
+ x
2
+ x
3
+ ··· + x
n
| 6 |x
1
| + |x
2
| + ··· + |x
n
|þ¤á.
(3) |x + x
1
+ ··· + x
n
| > |x| − |x
1
+ x
2
+ x
3
+ ··· + x
n
| > |x| − (|x
1
| + ··· + |x
n
|)
3. )eýéŠØª§¿xÑx‰Œµ
(1) |x| > |x + 1|
(2) 2 <
1
|x|
< 4
(3) |x| > A
(4) |x − a| < η, η•~ê§η > 0
(5)
x − 2
x + 1
>
x − 2
x + 1
(6) 2 <
1
|x + 2|
< 3
)µ
(1) x < −
1
2
-
0
x
-1
b
(2) −
1
2
< x < −
1
4
½
1
4
< x <
1
2
-
0
x
1
2
-
1
2
ee e e
3
(3) A > 0ž§x < −A½x > A
-
0
x
A-A
ee
A < 0ž§x ∈ R
(4) a − η < x < a + η
-
a
0
x
a + ηa − η
ee
(5) ªdu
x − 2
x + 1
< 0§K−1 < x < 2
-
0
x
-1 1 2
b b
(6) −
5
3
< x < −
3
2
½−
5
2
< x < −
7
3
-
0
x
-1-2-3
ee ee
4. ¦e¼ê½Â•9§3‰½:þ¼êŠµ
(1) y = f(x) = −x +
1
x
½Â•9f(−1), f(1)Úf(2)¶
(2) y = f(x) =
√
a
2
− x
2
½Â•9f(0), f(a)Úf
−
a
2
¶
(3) s = s(t) =
1
t
e
−t
½Â•9s(1), s(2)¶
(4) y = g(α) = α
2
tan α½Â•9g(0), g
π
4
, g
−
π
4
¶
(5) x = x(θ) = s in θ + cos θ½Â•9x
−
π
2
, x(−π)
(6) y = f(x) =
1
(x − 1)(x + 2)
½Â•9f(0), f(−1)
4
)µ
(1) ¼ê½Â••X = (−∞, 0)
S
(0, ∞)§f(−1) = 0, f (1) = 0, f(2) = −
3
2
(2) ¼ê½Â••X = [−|a|, |a|]§f(0) = |a|, f(a) = 0, f
−
a
2
=
√
3
2
|a|
(3) ¼ê½Â••(−∞, 0)
S
(0, ∞)§s(1) =
1
e
, s(2) =
1
2e
2
(4) ¼ê½Â••
n
x
x ∈ R, x 6= kπ +
π
2
, k ∈ Z
o
§g(0) = 0, g
π
4
=
π
2
16
, g
−
π
4
= −
π
2
16
(5) ¼ê½Â••X = (−∞, ∞)§x
−
π
2
= −1, x(−π) = −1
(6) ¼ê½Â••X = (−∞, −2)
S
(−2, 1)
S
(1, +∞)§f(0) = −
1
2
, f(−1) = −
1
2
5. ¦e¼ê½Â•9Š•µ
(1) y =
√
2 + x − x
2
(2) y =
√
cos x
(3) y = ln
sin
π
x
(4) y =
1
sin πx
)µ
(1) ¼ê½Â••X = [−1, 2]§Š••
0,
3
2
(2) ¼ê½Â••
h
2kπ −
π
2
, 2kπ +
π
2
i
(k ∈ Z)§Š••[0, 1]
(3) ¼ê½Â••
1
2k + 1
,
1
2k
(k ∈ Z)§Š••(−∞, 0]
(4) ¼ê½Â••(n − 1, n)(n = 0, ±1, ±2, ···)§Š••(−∞, −1]
S
[1, +∞)
6. f(x) = x + 1, ϕ(x) = x − 2§Á)•§|f (x) + ϕ(x)| = |f(x) + |ϕ(x)|
)µd®•§f(x)ϕ(x) > 0=(x + 1)(x − 2) > 0§Kx > 2½x 6 −1.
7. f(x) = (|x| + x)(1 − x)§¦÷veˆªxŠµ
(1) f(0) = 0
(2) f(x) < 0
)µ
(1) ‡f(x) = 0§K|x| + x = 0½1 − x = 0§=x 6 0½x = 1
(2) Ï|x| + x > 0§K‡f(x) < 0§•‡1 − x < 0=Œ§=x > 1
8. ã1-5L«>³|V ! ½>{R
0
ÚŒC>{R|¤>´.3˜ãØ•žmS§A, Bü:m>ØV Œ±w
¤˜‡~þ.¦Ñ>6IÚŒC>{R¼êª.
)µd®•9ÔnÆ•£§V = I(R
0
+ R).
9. 3˜‡Î/NìS?,«M—§TÎ/Nì .Œ»´a§p•h§?M—pÝ´x£ã1-6¤. T
M—NÈV Úxƒm¼ê'XV = V (x)§¿Ñ§½Â•ÚŠ•.
)µd®•§V = πa
2
x§§½Â••[0, h]§Š••[1, πa
2
h]
10. ,/Y± ¡È´˜‡F/§Xã1-7§.°2’§>–•45
o
§CDL«Y¡§¦¡ABCD¡
ÈS†Yh¼ê'X.
)µd®•9ã§S = h(h + 2).
11. k˜•H¶³§X^Œ»•Ròűz¦¨ωlÝ„Ýl¶³SåL-Ô§¦-Ô.¡†/¡
ålsÚžmt¼ê'X£ã1-8¤.
)µd®•9ã§s = H − ωRt
t ∈
0,
H
ωt
12. y = f(x) =
1 + x
2
, x < 0
x − 1, x > 0
§¦f(−2), f(−1), f (0), f(1)Úf
1
2
.
)µd®•§f(−2) = 5, f (−1) = 2, f(0) = −1, f(1) = 0, f
1
2
= −
1
2
.
5
13. x(t) =
0, 0 6 t < 10
1 + t
2
, 10 6 t 6 20
t − 10, 20 < t 6 30
§¦x(0), x(5), x(10), x(15), x(20), x(25), x(30)§¿xÑù‡¼êã/.
)µd®•§x(0) = 0, x(5) = 0, x(10) = 101, x(15) = 226, x(20) = 401, x(25) = 15, x(30) = 20
14. e]y´&‡-þx¼ê.UìeÛ5½§éuISײ&§U&‡-þ§z-20ŽAGe]8©§Ø
v20Žö±20ŽOŽ.&‡-þ360Ž±Sž§ÁÑù‡¼êLˆª§¿xѧã/.
)µd®•§y = f(x) =
8, 0 < x 6 20
16, 20 < x 6 40
24, 40 < x 6 60
15. óÀu)ì)˜‡nŧÙÅ/Xã1-9§Ñ¼ê'Xu = u(t)(0 6 t 6 20).
)µd®•9ã§u = u(t) =
1.5t, 0 6 t 6 10
30 − 1.5t, 10 < t 6 20
16. e¼êfÚϕ´Äƒ§•Ÿoº
(1) f(x) =
x
x
, ϕ(x) = 1
(2) f(x) = x, ϕ(x) =
√
x
2
(3) f(x) = 1, ϕ(x) = sin
2
x + cos
2
x
)µ
(1) Ïf½Â••(−∞, 0)
S
(0, +∞)§ϕ½Â••(−∞, +∞)§ùü‡¼ê؃.
(2) Ïf(x) = x, ϕ(x) = |x|§ùü‡¼ê¼êLˆªØ˜§Kùü‡¼ê؃.
(3) Ïϕ(x) = sin
2
x + cos
2
x = 1ð¤á§ùü‡¼êƒ.
17. y² éu †‚ ¼êf(x) = ax + b§ egCê Šx = x
n
(n = 1, 2, ···)|¤˜ê § Ké A¼ê
Šy
n
= f(x
n
)(n = 1, 2, ···)•|¤˜ê.
y²µx
m−1
, x
m
, x
m+1
´x
n
¥?¿3‡ƒê(2 6 m 6 n)
âK¿§2x
m
= x
m−1
+ x
m+1
qy
n
= f(x
n
) = ax
n
+ b§ Ky
m−1
= ax
m−1
+ b, y
m
= ax
m
+ b, y
m+1
= ax
m+1
+ b§ u ´2y
m
=
2ax
m
+ 2b, y
m+1
+ y
m−1
= ax
m+1
+ b + ax
m−1
+ b = 2ax
m
+ 2b§l2y
m
= y
m−1
+ y
m+1
qx
m−1
, x
m
, x
m+1
´x
n
¥?¿3‡ƒê§Ky
m−1
, y
m
, y
m+1
´y
n
¥?¿3‡ƒê§u´y
n
= f(x
n
)(n =
1, 2, ···)•|¤˜ê.
18. XJ-‚y = f(x)þ?˜^uÑpu§¤•l£ã1-10¤§y²Øª
f(x
1
) + f(x
2
)
2
> f
x
1
+ x
2
2
é
u¤kx
1
, x
2
(x
1
6= x
2
)¤á£…äkþãA5¼ê‰à¼ê¤.
y ² µ 3-‚þ?ü:A(x
1
, f(x
1
)), B(x
2
, f(x
2
))§ëAB§Ù¥:C(x
C
, y
C
)§Kf(x
1
) + f (x
2
) =
2y
C
, x
1
+ x
2
= 2x
C
q-‚þx
D
=
x
1
+ x
2
2
¤é:p‹I•y
D
= f
x
1
+ x
2
2
§Kx
C
= x
D
q-‚y = f(x)þ?˜^uÑpu§¤•l…x
1
, x
2
•u†l:§Ky
C
> y
D
=
f(x
1
) + f(x
2
)
2
>
f
x
1
+ x
2
2
éu¤kx
1
, x
2
(x
1
6= x
2
)¤á.
-
6
0
x
1
x
2
x
A
C
B
x
D
y
f(x)
19. y²eˆ¼ê3¤««mS´üNO\¼êµ
(1) y = x
2
(0 6 x < +∞)
(2) y = sin x
−
π
2
6 x 6
π
2
y²µ
