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Various topics in mathematical analysis, including limits, derivatives, and integrals. Topics include limit laws, differentiation rules, and the Fundamental Theorem of Calculus. Examples include finding limits, derivatives, and integrals of functions using various techniques.
What you will learn
Typology: Thesis
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è 5 Åq: <Æ: s 2 £§:
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 pìr Ô¦ 0 pxstëß, <Êú f (x) x = g(a)\"f pìr 0 pxs8¸, ½+Ë$í <Êú (f ◦ g)(x)H x = a\"f pìr Ô¦ 0 pxs. (F)
(2) <Êú f x = a\"f 5 Åqs, f H x = a\"f pìr 0 pxs. (F)
(3) ¿º <Êú f ü< g\ @/K, f ′^ = g′s, f = gs. (F)
(4) f (x) ½¨çß [−a, a]\"f 5 Åq l<Êú(odd func- tion)s,
∫ (^) a
−a
f (x)dx = 0s. (T)
(5) f (x) = x|x|H (0, 0)\"f /BG&h(inflection point)`¦ °ú H. (T)
(6) ëß{ 9 f ′(c) = 0s¦ f ′′(c) < 0 s, f H x = c\"f FG è°úכ(local minimum)`¦ °úH. (F)
(7) /BG y = x
2 x − 4 H ú¨î &hH(horizontal asymp- tote)`¦ °út ·ú§H. (T)
(8) y = |x|_ ¸<ÊúH z´Ãº^\"f 5 Åqs. (F)
(9) ¿º <Êú f (x), g(x) x = a\"f 5 Åqs, ½+Ë$í <Êú (f ◦ g)(x)¸ x = a\"f 5 Åqs. (F)
(10) <Êú f pìr 0 pxs, &hìr 0 pxs. (T)
ëH]j 2
x + √y = 1s ÅÒ#Q&. 6 £§<Êúpìr(implicit differentiation)`¦ s 6 x# dy dx , d
(^2) y dx^2 \¦ ½¨ r¸.[10&h]
·l) ª`¦ pìr ,
1 2 x−^1 /^2 +
y−^1 /^2 y′^ = 0
ÕªQÙ¼Ð, y′^ = −
√y √ x "f, y′′^ =
x + √y 2
x =^
2 x
x
ëH]j 3 <Êú y =
x sin (^1) x x 6 = 0 0 x = 0 x = 0\"f pìr
0 pxt óøÍZ> r¸. [7&h]
·l) pìr_ &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
ìrÔ¦ 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, ׿çß°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. ÕªQ, f ′(x) = 2 − cos xÐ ½Ó© 1 Ð ß¼. ÕªQټР¸íH. "f, z´HÉr ôÇ >h ÷rs.
ëH]j 7 6 £§`¦ ½¨ r¸. [y 5 &h]
(1) (^) nlim→∞
n n^2 + 1^2
·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 ïrdÉr ∫ √^1 e^2 x^ − 6
dx =
dθ
=
θ + C
=
sec−^1 √ex 6
dx x^4 − x^2
·l) ïrd`¦ ÂÒìrìrúР¾º#Q Û¦ ∫ dx x^2 (x^2 − 1)
x^2
x − 1
x + 1
dx
=
x
ln
∣∣^ x^ −^1 x + 1
è 5 Åq: <Æ: s 2 £§:
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.
ÕªQÙ¼Ð, 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
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=óøÍ¦ [j 1 pxìrK"f ª =åQ
¦ θëß pu ¦9 úÐ\¦ ëß[þt9¦ 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