Bài tập Toán A2 TNKT

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✷✳✾✳ lim

n →∞

np(n+a) n = 0 (∀p > 0, a > 0)

✶

✷✳✶✵✳ lim

n →∞

n (1+p) n = 0, p > 0, α ❜➜t ❦ý

❈❤ó þ✳ ❚❤æ♥❣ t❤÷í♥❣✱ ✤➸ t➻♠ ❣✐î✐ ❤↕♥ ❝õ❛ ♠ët ❞➣② sè t❛ ❞ü❛ ✈➔♦ ❝→❝ ❦➳tq✉↔ ❝õ❛ ❝→❝ ❣✐î✐ ❤↕♥ tr♦♥❣ ❜➔✐ t➟♣ ✷✮✳

✷✳✻✳ lim

x →+∞

(2x − 3)20.(2x + 2)30(2x + 1)50

✷✳✼✳ lim

x →+∞

x4 + x + 18x3√

cos xsin2x

✸✳✼✳ lim

x →0

x2p

✸✳✶✶✳ lim

x →0

sin22x + sin xsin 3x

✸✳✶✹✳ lim

x →0

x− sin 5x + sin2x4x + arcsinx+x2

2x

✹✳✷✳ lim

x →+∞

3x2 − x + 12x2 + x + 1

 x2 1−x

✻

 1 sin3 x

✹✳✼✳ lim

x →0

arcsin x + 1sin x + 1

1 x

✹✳✽✳ lim

x →+∞

sin 1

x + cosx1x

✹✳✾✳ lim

x →0

2e

x

x+ 1 − 1

x2+1 x

✹✳✶✵✳ lim

x →0 +



1 1+e 1/x



a+ 2x ♥➳✉ x ≥ 0

✼

tr➯♥ t♦➔♥ trö❝ sè (−∞, +∞).

✻✳✷✳

f(x) =

2x + a ♥➳✉ x ∈ [0, 1]

t↕✐ x = 0✳

✽

✾

✶

.xa

❝✮ ❑❤↔ ✈✐ ❧✐➯♥ tö❝ t↕✐ x = 0✳

❇➔✐ ✹✳ ✹✳✶✳ ❈❤♦ ❤➔♠ sè f(x) = (x − a)ϕ(x) tr♦♥❣ ✤â ϕ(x) ❧➔ ❤➔♠

❧✐➯♥ tö❝ t↕✐ ✤✐➸♠ x = a✳ ❚➼♥❤ f′(a)✳

✹✳✷✳ ❈❤♦ ❤➔♠ sè f(x) = |x − a|ϕ(x) tr♦♥❣ ✤â ϕ(x) ❧➔ ❤➔♠ ❧✐➯♥ tö❝t↕✐ ✤✐➸♠ x = a ✈➔ ϕ(a) 6= 0 ❈ù♥❣ ♠✐♥❤ r➡♥❣ f(x) ❦❤æ♥❣ ❝â ✤↕♦ ❤➔♠t↕♦ x = a✳ ❚➼♥❤ f′

✶✼✳✶✳ | sin x − sin y| ≤ |x − y|, ∀x, y.

✶✼✳✷✳ | arctan x − arctan y| ≤ |x − y|, ∀x, y

x − ❝♦t❛♥x



❇➔✐ ✷✺✳ ❑❤↔♦ s→t ✈➔ ✈➩ ✤ç t❤à ❝→❝ ❤➔♠ sè ❝❤♦ ❜ð✐ ♣❤✉ì♥❣ tr➻♥❤t❤❛♠ sè s❛✉

✾

✶✳✶✷✳ R dx

p(x − a)(b − x)

✶✳✶✸✳ R p

(x − a)(b − x)dx

✶✳✶✹✳ R x2

p(a2 + x2)dx

❇➔✐ ✷✳ ❉ò♥❣ ♣❤✉ì♥❣ ♣❤→♣ t➼❝❤ ♣❤➙♥ tø♥❣ ♣❤➛♥ t➼♥❤ ❝→❝ t➼❝❤ ♣❤➙♥s❛✉

✻✳✺✳ R2

− 1 2

Z a +T a

f (x)dx =

Z T a

f (x)dx

✈î✐ a ❧➔ ♠ët sè tü❝ tò② þ✳

✶✹

❇➔✐ ✶✵✳ ❈❤♦ f, g ❧➔ ❝→❝ ❤➔♠ ❦❤↔ t➼❝❤ tr➯♥ [a, b]✳ ❈❤ù♥❣ ♠✐♥❤

Z b a

f (x)g(x)dx

2

≤

Z b a

f(x)dx

Z b a

g2(x)dx

❇➔✐ ✶✶✳ ❈❤ù♥❣ ♠✐♥❤ ✈î✐ ♠å✐ x > 0 t❛ ❝â

ex− 1 <

Z x 0



❇➔✐ ✶✷✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣

1 −

Z 1 0

x2e−1dx ≤

Z 1 0

dx

1 + x2e−x < 1 − 12

Z 1 0

✶✺✳✶✳ x = a(t − sin t), y = a(1 − cos t), 0 ≤ t ≤ 2π✳

x2 + 1dx20.7.

x2 − 2x − 120.9

x4 + 120.13

Z

3

√x

q5x3√3

3

dx✳

x10

Z5x2 + 6x + 9(x − 3)2(x + 1)2dx23.5

Z x3 + 3x2 + 5x + 7

x4 − 8x2 + 16dx23.13

✷✹✳✹✳ R dx

(x2 + x + 1)2✳

✶✾

✶✳✷✳ P∞

n=1

1(3n − 2)(3n + 1)

✶✳✸✳ P∞

n=1

n + 1(2n + 1)2(2n + 3)2

✶

n4n − 3

2n

✶✳✶✸✳ P∞

n=1

1n(n + 3)

n + 1

n3

✷✳✷✳ P∞

n=1

(n!)3(3n)!cos x tr➯♥ (−∞, +∞)✳

23

❇➔✐ ✹✳ ❚➻♠ ❜→♥ ❦➼♥❤ ✈➔ ♠✐➲♥ ❤ë✐ tö ❝õ❛ ❝❤✉é✐ ❤➔♠ ❧ô② t❤ø❛

✹✳✶✳ P∞

n=1

(n!)3(3n)!x

n

✹✳✷✳ P∞

n=1

n3 + 3n! (x − 1)2n

✹✳✸✳ P∞

n=1

xn

√n

✶✼✳✼✳ P∞

n=1

n3 + 3n! (x − 1)2n✳

❇➔✐ ✶✾✳ ❈❤♦ f(x) = P∞

n=1

1

nsin(nx), x ∈ [−π, π]✱ ❝→❝ ❤➺ sè ♥➔♦ tr♦♥❣ ❦❤❛✐tr✐➸♥ ❋♦✉r✐❡r ❝õ❛ ❤➔♠ sè f(x) tr➯♥ ✤♦↕♥ [−π, π] ❜➡♥❣ ✵❄

✷✵✳✸✳ f(x) =

2π + x, −π ≤ x < 0

13.5 +

15.7 +

✷✵✳✶✶✳ f(x) = ex ✈î✐ −1 < x < 1✳

✷✵✳✶✷✳ f(x) = sin x

2 ✈î✐ −π < x < π✳

✽