De Thi Vi tích phân a1 Hki 1213

De Thi VTP A1 HK I 1213.dvi

2 THI VI TCH PHN A1HC K I - NHM 01NM HC: 2012 - 2013Ngy thi: 02/12/2012Thi gian lm bi: 90 phtTRNG I HC CN THKHOA KHOA HC T NHINB MN TON 1

1

TRNG I HC CN THKHOA KHOA HC T NHINB MN TON 1

THI VI TCH PHN A1HC K I - NHM 01NM HC: 2012 - 2013Ngy thi: 02/12/2012Thi gian lm bi: 90 pht

NI DUNG THI

( thi gm 07 cu1 c in trn 01 trang2)

Cu 1. (a) Dng nguyn l thay v cng b tng ng tnh gii hn: A = lim ln cos x .

(b) Dng quy tc LHospital tnh gii hn: B = lim

x et2 dt

0x .

x0 x2

Cu 2. Cho hm s f (x) = x3 cos 2x. Tnh f (38)(0).

x+ e2t2 dt0

Cu 3. Mt on dy AB di 3 m c ct thnh hai on ti im C. on AC c b thnh hnh vung, on BC c b thnh tam gic u. Ta nn chn im C nh th no tng din tch ca hai hnh l b nht.

Cu 4. Tnh din tch min phng gii hn bi ng y = x sin x, y = 0 v x = 0, x = .

Cu 5. Mt vt th c y l hnh trn bn knh 2 m. Nu ct vt th bi nhng mt phng vung gc vi mt ng knh c nh ca y ta c thit din l tam gic u. Hy tnh th tch ca vt th

n.x3n

nCu 6. Tm min hi t ca chui .(1)n

n=0

8 .(n + 1)

2Cu 7. Tnh tng ca chui ly tha 1 + x + x2

x3 xn++ ... +3n

+ ...

Cn Th, ngy 23 thng 11 nm 2012Cn b ging dy

L HOI NHN

1Thang im: 1,00 im/cu.2p n c cng b trn website Khoa Khoa hc t nhin vo chiu ngy 03.12.2012.im c nhp vo ti khon sinh vin vo sng ngy 08.12.2012.Phc kho bi thi: t 14 gi 00 n 16 gi 30 ngy 08.12.2012 ti VP. BM. Ton, Khoa Khoa hc t nhin.Mi thc mc v im bi thi sau ngy 08.12.2012 u khng c gii quyt.

P N1 21Cu 1. (a) Khi x 0 ta c: ln cos x cos x 1 2 x . Suy ra A = 2 .x2

2(b) B = lim e

= lim

1 = 0.

2x e2x

3

x ex

38Cu 2. f (38)(x) = . Ck (x3)(k). cos(38k)(x).k=0 Cho x = 0 ta c f (38)(0) = 0

2Cu 3. t BC = x v AC = 10 x, vi 0 < x < 10

Cnh ca hnh vung l 10 x4x

v din tch hnh vung (3 x)16x3

Cnh ca tam gic u

v chiu cao ca tam gic.

326

Din tch tam gic u x3 36

Tng din tch S(x) =

(3x)2

+ 16

x23. Ta tm x S(x) nh nht.36

St(x) =

x 38

x3+18

. im dng: St(x) = 0 = x =

81 36311

1, 6951.

Stt(x) =

9 + 43, Stt( 72

81 363

) > 0. Suy ra S(x) nh nht khi x 1, 6951.11

Vy BC 1, 6951 v AC 8, 3049

Cu 4. S = 0

|x sin x|dx =0

x sin xdx.

Dng tch phn tng phn ta c S = (sin x x cos x)|0 = (vdt).Cu 5. Chn h trc ta Oxyz vi O l tm ca y, mt phng Oxy cha mt y v trc Ox cha ng nh c nh ca mt y ca vt th. Ct vt th bi nhng mt phng vung gc vi trc Ox v cch O mt khong xta c thit din c cnh l 2r2 x2 v chiu cao l 3.r2 x2 vi r = 2 vx [r, r].32232

Din tch thit din: S(x) = 2 (r2

x ) = 2 (4 x ).

16 3

Th tch vt th: V =2

S(x)dx =

(vtt)3

n.x3n

8n.(n + 1)Cu 6. Chui hi t ti x = 0, vi x = 0 ta c an(x) = (1)n .

=. an+1 (x) .

|x|3

Suy ra c(x) = lim .

.n .

. .

.an(x) . 8

Do khong hi t ca chui tha c(x) < 1 2 < x < 2.n

Ti x = 2 ta c chui

. phn k theo iu kin cn.n=0 n + 1

Ti x = 2 ta c chui

. (1)n=0

n nn + 1

phn k theo iu kin cn. Do min hi t

ca chui l khong (-2;2).. an+1 .

n .aCu 7. lim ..n

. = 1. Suy ra khong hi t ca chui l (1; 1)..

.x2 Ly x (1; 1) ta t S(x) = 1 + x + 2 +Suy ra St(x) = 1 + x + x2 + ... =1.

x3+ ... + 3

xnn + ....

S(x) =

1 xxSt(t)dt + S(0) = ln(1 x) + S(0).0

Ta c S(0) = 1 suy ra S(x) = 1 ln(1 x).

NI DUNG THI


4 THI VI TCH PHN A1HC K I - NHM 01NM HC: 2012 - 2013Ngy thi: 02/12/2012Thi gian lm bi: 90 phtTRNG I HC CN THKHOA KHOA HC T NHINB MN TON 2

5

ex sin 2xCu 1. (a) Dng nguyn l thay v cng b tng ng tnh gii hn: A = lim

1 .

(b) Dng quy tc LHospital tnh gii hn: B = lim

x tet2 dt

0x .

x0 x2

Cu 2. Cho hm s f (x) = x3 sin x. Tnh f (38)(0).

x+ te2t2 dt0

Cu 3. Mt on dy AB di 3 m c ct thnh hai on ti im C. on BC c b thnh hnh vung, on AC c b thnh tam gic u. Ta nn chn im C nh th no tng din tch ca hai hnh l b nht.

Cu 4. Tnh din tch min phng gii hn bi ng y = x sin x, y = x v x = 0, x = .

Cu 5. Mt vt th c y l hnh trn bn knh 4 m. Nu ct vt th bi nhng mt phng vung gc vi mt ng knh c nh ca y ta c thit din l tam gic u. Hy tnh th tch ca vt th.

Cu 6. Tm min hi t ca chui .(1)n (

n + 1).x3n+1n .

n=0

2

8 .(n + 2)

34

n+1

Cu 7. Tnh tng ca chui ly tha x

x x+... + (1) n+1 x

+ ...

2 34

n + 1


L HOI NHN

3Thang im: 1,00 im/cu.4p n c cng b trn website Khoa Khoa hc t nhin vo chiu ngy 03.12.2012.im c nhp vo ti khon sinh vin vo sng ngy 08.12.2012.Phc kho bi thi: t 14 gi 00 n 16 gi 30 ngy 08.12.2012 ti VP. BM. Ton, Khoa Khoa hc t nhin.Mi thc mc v im bi thi sau ngy 08.12.2012 u khng c gii quyt.


THI VI TCH PHN A1HC K I - NHM 01NM HC: 2012 - 2013Ngy thi: 02/12/2012Thi gian lm bi: 90 pht

P N

Cu 1. (a) Khi x 0 ta c: ex sin 2x 1 x sin 2x 2x2. Suy ra A = 2.xex21

(b) B = limx

xe2x

2 = lim

x x e

2 = 0.

3

38Cu 2. f (38)(x) = . Ck (x3)(k). sin(38k)(x).k=0 Cho x = 0 ta c f (38)(0) = C3 .6.( cos 0) = 6C33838

2Cu 3. t AC = x v BC = 10 x, vi 0 < x < 10

Cnh ca hnh vung l 10 x4x

v din tch hnh vung (3 x)16x3

Cnh ca tam gic u

v chiu cao ca tam gic.

326

Din tch tam gic u x3 36

Tng din tch S(x) =

(3x)2

+ 16

x23. Ta tm x S(x) nh nht.36

St(x) =

x 38

x3+18

. im dng: St(x) = 0 = x =

81 36311

1, 6951.

Stt(x) =

9 + 43, Stt( 72

81 363

) > 0. Suy ra S(x) nh nht khi x 1, 6951.11

Vy AC 1, 6951 v BC 8, 3049

Cu 4. S =

|x sin x x|dx =0

(x x sin x)dx.0x2

. 2

= ( 2 sin+cos )=Dng tch phn tng phn ta c Sxxx ...0

2 (vdt).

Cu 5. Chn h trc ta Oxyz vi O l tm ca y, mt phng Oxy cha mt y vtrc Ox cha ng nh c nh ca mt y ca vt th. Ct vt th bi nhng mt phng vung gc vi trc Ox v cch O mt khong xta c thit din c cnh l 2r2 x2 v chiu cao l 3.r2 x2 vi r = 4 vx [r, r].32232

Din tch thit din: S(x) = 2 (r

x ) = 2 (16 x ).

4 128 3

6

7

Th tch vt th: V =4

S(x)dx =

(vtt)3

n + 1).x3n

8n.(n + 2)Cu 6. Chui hi t ti x = 0, vi x = 0 ta c an(x) = (1)n ( .

=. an+1 (x) .

|x|3

Suy ra c(x) = lim .

.n .

. .

.an(x) . 8

Do khong hi t ca chui tha c(x) < 1 2 < x < 2. n + 1

Ti x = 2 ta c chui

. phn k theo iu kin cn.n=0 n + 2

Ti x = 2 ta c chui

. (1)n=0

n n + 1n + 2

phn k theo iu kin cn. Do min hi t

ca chui l khong (-2;2).

.. an+1 .

n .aCu 7. lim ..n

. = 1. Suy ra khong hi t ca chui l (1; 1)..

n+1

Ly x (1; 1) ta t S(x) =

x2 x3

+23

x44 ... + (1)

n+1 xn + 1 + ....

Suy ra St(x) = x x2 + x3 x4 + ... =

x

x1 + x

= 1

1.1 + x

S(x) =0

St(t)dt + S(0) = x ln(1 + x) + S(0).

Ta c S(0) = 0 suy ra S(x) = x ln(1 + x).


THI VI TCH PHN A1HC K I - NHM E01 v G01NM HC: 2012 - 2013Ngy thi: 02/12/2012Thi gian lm bi: 90 pht

NI DUNG THI


Cu 1. (a) Tnh gii hn A = limx0

x3

dt 1+t40.x3

(b) Tm a hm s f (x) = . axnux 1x2nux > 1

lin tc ti x = 1.

Cu 2. Khi mt bn kim loi hnh trn b un nng, bn knh ca n tng vi tc l 0, 02 cm/pht. Tnh tc bin thin ca din tch bn kim loi khi bn knh ca n ang l 20 cm. Nu tc ny khng i th sau bao lu bn knh ca n s l 25 cm.

Cu 3. Mt xe bus c sc cha ti a 60 hnh khch. Nu mt chuyn xe ch c hnh khch

th gi cho mi hnh khch l .3

x .2. Hy tnh s hnh khch trn mi chuyn xe 40

s tin thu c cho mi chuyn l ln nht. S tin l bao nhiu?

Cu 4. Mt vt th c y l tam gic u ABC cnh 1 m. Nu ct vt th bi mt phng vung gc vi ng cao AH ca y th ta c thit din l na hnh trn c ng knh l on giao tuyn ca mt phng thit din vi mt y. Hy tnh th tch vt th.

3Cu 5. Tnh di cung phng y = 1 xx x vi x [1; 4].

xn+2Cu 6. Tm min hi t v tnh tng ca chui .

n=1

n


L HOI NHN

5Thang im: T cu 1 n cu 5: 1,00 im/cu; cu 6: 2 im6p n c cng b trn website Khoa Khoa hc t nhin vo chiu ngy 03.12.2012.im c nhp vo ti khon sinh vin vo sng ngy 08.12.2012.Phc kho bi thi: t 14 gi 00 n 16 gi 30 ngy 08.12.2012 ti VP. BM. Ton, Khoa Khoa hc t nhin.Mi thc mc v im bi thi sau ngy 08.12.2012 u khng c gii quyt.


THI VI TCH PHN A1HC K I - NHM G01 v E01NM HC: 2012 - 2013Ngy thi: 02/12/2012Thi gian lm bi: 90 pht

Cu 1. (a) A = lim

3x21+x12

= 1.

P N

x0

3x2

(b) f (x) lin tc ti x = 1limx1+

f (x) = limx1

f (x) = f (1) a = 1

Cu 2. Gi r(t) v S(t) l bn knh v din tch bn kim loi ti thi im t.Suy ra: S(t) = r2(t)(1). Ti t0 ta c r(t0) = 20 v rt(t0) = 0, 02. Ta tnh St(t0). o hm hai v ng thc (1) theo t ta c St(t) = 2r(t).rt(t).4Cho t = t0 ta c St(t0) = 5 .

Vy din tch bn kim loi ang tng vi tc 45

(cm /pht).

S 0t = St(t ) =

252 20245

= 281, 25 (pht).

x 2

Cu 3. S tin thu c vi x hnh khch l L(x) = x .3.40

vi x (0; 60]

Lt(x) = .3

. .3

x40

3x ..40

Lt(x) = 0 x = 40 hoc x = 120. Ta nhn x = 40 T bng bin thin ca L(x) ta c L(x) t GTLN khi x = 40. Suy ra s hnh khchtrn xe nn l 40 S tin ln nht thu c l: 160 (n v tin).

Cu 4. Chn h trc ta Oxyz vi O trng vi A; H

mt phng Oxy.

. 3.; 0 2

v mt phng y thuc

Ct vt th bi mt phng vung gc vi trc Ox, gi r l bn knh thit din tac:x 2rxAH = BC = r = 3 .2

12 Din tch thit din l S(x) = 2 r = Th tch vt th

x .6

32V =

3

.S(x)dx =x3. 2

3=(m3)

18.0 480

1 .

1 .

Cu 5. yt = 2

x x .1 .

1 .

,1 + yt2 =2

x + x.

Chiu di cung:

4

..4

l =

1

,1 + yt2dx = . x x + x3

. 10 (vcd).

3=..1

Cu 6. (a) an = 1 =l = limn n

an+1an = 1

Bn knh hi t ca chui l r =

l = 1 v khong hi t (r; r) = (1; 1).

Khi x = 1 ta c chui

. l chui phn k.n=1 n

Khi x = 1 ta c chui

.

n=1

nl chui hi t.n

Suy ra min hi t ca chui l [1; 1)(b) Vi x thuc khong hi t (1; 1) t S(x) = .

v S1(x) =

. .

Suy ra S(x) = x2S1(x).

n=1n

n=1 n

n 11

1St (x) =

. x n=1x

=.1 x

1S1(x) = St (t)dt + S1(0) = ln(1 x) + S1(0).0 V S1(0) = 0 nn S1(x) = ln(1 x).Vy S(x) = x2 ln(1 x).

10

11


THI VI TCH PHN A1HC K I - NHM E01 v G01NM HC: 2012 - 2013Ngy thi: 02/12/2012Thi gian lm bi: 90 pht

NI DUNG THI


Cu 1. (a) Tnh gii hn A = lim

x3

dt01+sin4 t.

x0 x3

(b) Tm a hm s f (x) = . x + anu x 1x2 nu x > 1

lin tc ti x = 1.

Cu 2. Khi mt bn kim loi hnh trn b un nng, bn knh ca n tng vi tc l 0, 02 cm/pht. Tnh tc bin thin ca din tch bn kim loi khi bn knh ca n ang l 25 cm. Nu tc ny khng i th sau bao lu bn knh ca n s l 30 cm.

Cu 3. Mt xe bus c sc cha ti a 60 hnh khch. Nu mt chuyn xe ch c hnh khch

th gi cho mi hnh khch l .3

x .2. Hy tnh s hnh khch trn mi chuyn xe 40

s tin thu c cho mi chuyn l ln nht. S tin l bao nhiu?

Cu 4. Mt vt th c y l tam gic vung cn ABC ti A cnh huyn 1 m. Nu ct vt th bi mt phng vung gc vi ng cao AH ca y th ta c thit din l na hnh trn c ng knh l on giao tuyn ca mt phng thit din vi mt y. Hy tnh th tch vt th.

3Cu 5. Tnh di cung phng y = 1 xx x vi x [1; 4].

xn+3Cu 6. Tm min hi t v tnh tng ca chui .

n=1

n


L HOI NHN

7Thang im: T cu 1 n cu 5: 1,00 im/cu; cu 6: 2 im8p n c cng b trn website Khoa Khoa hc t nhin vo chiu ngy 03.12.2012.im c nhp vo ti khon sinh vin vo sng ngy 08.12.2012.Phc kho bi thi: t 14 gi 00 n 16 gi 30 ngy 08.12.2012 ti VP. BM. Ton, Khoa Khoa hc t nhin.Mi thc mc v im bi thi sau ngy 08.12.2012 u khng c gii quyt.


THI VI TCH PHN A1HC K I - NHM G01 v E01NM HC: 2012 - 2013Ngy thi: 02/12/2012Thi gian lm bi: 90 pht

P N

3x2 4

Cu 1. (a) A = limx0

1+sin x33x2= 1

x(b) f (x) lin tc ti x = 1 lim1+

f (x) =limx1

f (x) = f (1) a = 2

Cu 2. Gi r(t) v S(t) l bn knh v din tch bn kim loi ti thi im t.Suy ra: S(t) = r2(t)(1). Ti t0 ta c r(t0) = 25 v rt(t0) = 0, 02. Ta tnh St(t0). o hm hai v ng thc (1) theo t ta c St(t) = 2r(t).rt(t).Cho t = t0 ta c St(t0) = .Vy din tch bn kim loi ang tng vi tc (cm /pht).

S 0t = St(t ) =

302 252= 275 (pht).

x 2

Cu 3. S tin thu c vi x hnh khch l L(x) = x .3.40

vi x (0; 60]

Lt(x) = .3

. .3

x40

3x ..40

Lt(x) = 0 x = 40 hoc x = 120. Ta nhn x = 40 T bng bin thin ca L(x) ta c L(x) t GTLN khi x = 40. Suy ra s hnh khchtrn xe nn l 40 S tin ln nht thu c l: 160 (n v tin).

Cu 4. Chn h trc ta Oxyz vi O trng vi A; H

mt phng Oxy.

. 1.; 0 2

v mt phng y thuc

Ct vt th bi mt phng vung gc vi trc Ox, gi r l bn knh thit din ta c:x 2rAH = BC = r = x.2

12 Din tch thit din l S(x) = 2 r = Th tch vt th

x .2

12V =

. 12

.S(x)dx =x3 =(m3)

6.0 480

1 .

1 .

Cu 5. yt = 2

x x .1 .

1 .

,1 + yt2 =2

x + x

Chiu di cung:

4

..4

l =

1

,1 + yt2dx = . x x + x3

. 10 (vcd).

3=..1

Cu 6. (a) an = 1 =l = limn n

an+1an = 1

Bn knh hi t ca chui l r =

l = 1 v khong hi t (r; r) = (1; 1).

Khi x = 1 ta c chui

. l chui phn k.n=1 n

Khi x = 1 ta c chui

.

n=1

nl chui hi t.n

Suy ra min hi t ca chui l [1; 1)(b) Vi x thuc khong hi t (1; 1) t S(x) = .

v S1(x) =

. .

Suy ra S(x) = x3S1(x).

n=1n

n=1 n

n 11

1St (x) =

. x n=1x

=.1 x

1S1(x) = St (t)dt + S1(0) = ln(1 x) + S1(0).0 V S1(0) = 0 nn S1(x) = ln(1 x).Vy S(x) = x3 ln(1 x).

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De Thi Vi tích phân a1 Hki 1213