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è 5 Åq: <Æ: s 2 £§:
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y 1 lq SX:
ëH]j 1 6 £§s ´úܼ ©, d¦o ×Ð ³ðl r¸. [y 2 &h]
(1) <Êú g(x)H x = a\"f pì r Ô¦ 0 pxstëß, <Êú f (x) x = g(a)\"f pì r 0 pxs 8¸, ½+Ë$í <Êú (f ◦ g)(x)H x = a\"f pì r Ô¦ 0 pxs. (F)
(2) <Êú f x = a\"f 5 Åqs, f H x = a\"f pì r 0 pxs. (F)
(3) ¿º <Êú f ü< g\ @/K, f ′^ = g′s, f = gs. (F)
(4) f (x) ½¨çß [−a, a]\"f 5 Åq l<Êú(odd func- tion)s,
∫ (^) a
−a
f (x)dx = 0s. (T)
(5) f (x) = x|x|H (0, 0)\"f /BG&h(inflection point)`¦ °ú H. (T)
(6) ëß{ 9 f ′(c) = 0s¦ f ′′(c) < 0 s, f H x = c\"f FG è°úכ(local minimum)`¦ °úH. (F)
(7) /BG y = x
2 x − 4 H ú¨î &hH(horizontal asymp- tote)`¦ °út ·ú§H. (T)
(8) y = |x|_ ¸<ÊúH z´Ãº^\"f 5 Åqs. (F)
(9) ¿º <Êú f (x), g(x) x = a\"f 5 Åqs, ½+Ë$í <Êú (f ◦ g)(x)¸ x = a\"f 5 Åqs. (F)
(10) <Êú f pì r 0 pxs, &hì r 0 pxs. (T)
ëH]j 2
x + √y = 1s ÅÒ#Q&. 6 £§<Êúpì r(implicit differentiation)`¦ s 6 x# dy dx , d
(^2) y dx^2 \¦ ½¨ r¸.[10&h]
·l) ª`¦ pì r ,
1 2 x−^1 /^2 +
y−^1 /^2 y′^ = 0
Õª QÙ¼Ð, y′^ = −
√y √ x "f, y′′^ =
x + √y 2
x =^
2 x
x
½Ó/BN@/<Ƨ pì r&hì r<Æ
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ëH]j 3 <Êú y =
x sin (^1) x x 6 = 0 0 x = 0 x = 0\"f pì r
0 pxt óøÍZ> r¸. [7&h]
·l) pì r_ &ñ _\ _K
f ′(0) = (^) hlim→ 0
h sin (^) h^1 h = lim h→ 0 sin
h
0 A_ FGôÇÉr >rF t ·ú§Ü¼Ù¼Ð, <Êú f H x = 0\"f p
ì rÔ¦ 0 pxs.
ëH]j 4 ~½Ó&ñ d 2 x − 1 − sin x = 0s &ñ SX > ôÇ >h_ z´ H¦ f¦ Ðsr¸. [8&h]
·l) f (x) = 2x − 1 − sin x ¿º, f (0) = − 1 < 0, f (1) = 1 − sin 1 > 0 s. ÅÒ#Q <ÊúH 5 Åq<Êú ½+Ë sÙ¼Ð, f ¸ 5 Åq<ÊúsÙ¼Ð, ×æçß°úכ &ño\ K f H \P 2 ;½¨çß (0, 1) s\"f H`¦ °úH. ëß{ 9 , Hs ¿º >h s©s, 7 £¤, αü< β <Êú f (x) = 0 ¿º Hs, Roll &ñ o\ _K, f ′(c) = 0 c \P 2 ;½¨ çß (α, β) s\ >rF # ôÇ. Õª Q, f ′(x) = 2 − cos xÐ ½Ó© 1 Ð ß¼. Õª QټР¸íH. "f, z´HÉr ôÇ >h ÷ rs.
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ëH]j 7 6 £§`¦ ½¨ r¸. [y 5 &h]
(1) (^) nlim→∞
n n^2 + 1^2
- n n^2 + 2^2
- · · · + n n^2 + n^2
·l)
n^ lim→∞
n n^2 + 1^2
n n^2 + 2^2
n n^2 + n^2
0
1 + x^2 dx
= tan−^1 x
1
0
π 4
√ dx e^2 x^ − 6 ·l) ex^ =
6 sec θÐ u¨ 8 ï rdÉr ∫ √^1 e^2 x^ − 6
dx =
√^1
dθ
=
θ + C
=
√^1
sec−^1 √ex 6
+ C
dx x^4 − x^2
·l) ï rd`¦ ÂÒì rì rúР¾º#Q Û¦ ∫ dx x^2 (x^2 − 1)
x^2
x − 1
x + 1
dx
=
x
ln
∣∣^ x^ −^1 x + 1
∣∣ + C
è 5 Åq: <Æ: s 2 £§:
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y 1 lq SX:
ëH]j 8 <Êú f (x) = 2 +
∫ (^1) /x
1
tan−^1 tdt_ x = 1\"f
_ +þAHd(linearization)¦ s 6 xK"f f (1.01)¦ ½¨
r¸. [5&h]
·l) f (1) = 2 s¦, f ′(x) = − 1 x^2 tan−^1 xsÙ¼Ð,
f ′(1) = − π 4.
"f, +þAHdÉr L(x) = − π 4 (x − 1) + 2.
Õª QÙ¼Ð, f (1.01) ≈ L(1.01) = 2 − π 400
ëH]j 9 <Êú y =
x \¦ x ≥ 1 ½¨çß\"f x»¡¤Ü¼Ð rr
& % 3 Ér { 9 ^_ ^&h`¦ ½¨ r¸. [5&h]
·l) { 9 ^_ ^&h V H
V =
1
π x^2 dx = π lim b→∞
∫ (^) b
1
dx x^2
= π lim b→∞
[
x
]b
1 = π
ëH]j 10 -q 30 ª^o=óøͦ [j 1 pxì rK"f ª =åQ¦ θëß pu ¦ 9 úÐ\¦ ëß[þt 9¦ ôÇ. éß_ &hs þj@/ ÷& H θ_ °úכ¦ ½¨ r¸. [10&h]
·l) &h A\¦ θ\ 'aôÇ dܼР?/
A = 100(1 + cos θ) sin θ 0 ≤ θ ≤ π 2
þj@/°úכ¦ ½¨ l 0 AK, ª =åQ°úכõ FG@/°úכ¦ q§KÐ )a . FG@/°úכÉr A′^ = 0 &hsټР½¨KÐ θ = π 3
. °úכ`¦ q§KÐ þj@/°úכÉr FG@/&h\"f °úH. "f ½¨ ¦ H θH π 3
