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Thiết Kế Bài Giảng Giải Tích 12 Tập 2
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I hiet ke bai giang GIAI TICH ] 2
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^ N H A XUAT BAN HA N6 I
TRAN VINH
THIET KE BAI GIANG GIAI TICH
TAP HAI
NHA XUAT BAN HA NOI
Chi/dNq III
NGUYEN HAM - TICH PHAN VA UNC DUNG
Pha n 1
NHJtXG VAX D E CUA CHMfONG
I. NOI DUNG
Noi dung chinh cua chucung 3 :
Nguyen ham : Dinh nghia ; tinh chat; cac nguyen ham ccf ban ; cac phucmg phap tinh nguyen ham.
Tich phan : Dinh nghia ; cac tinh chat cua tich phan ; cac phuang phap tinh tich phan.
" Lftig dung cua tich phSn : Bai toan dien tich, bai toan thi tich.
n . MUC TIEU
1. Kien thiirc
Nam dugc toan bo kien thiic co ban trong chuong da neu tren, cu the : Nam viing dinh nghia nguyen ham, cac nguyen ham co ban, cac tinh chat ciia nguyen ham.
• Dinh nghia tich phan, cac tinh chat ciia tich phan, ung dung ciia tich phan, moi quan he giiia tich phan va nguyen ham.
M6t s6' ling dung tich phan trong hinh hoc : Tinh dugc dien tich hinh phang, the tich vat the trong khong gian.
2. KT nang
van dung cac nguyen ham co ban de tinh cac nguyen ham.
Van dung thanh thao cong thiic Niuton - Laibonit de tinh tich phan. Moi
quan he giiia dao ham va nguyen ham.
Van dung tich phan de tinh dien tich hinh phang va the tich ciia vat the. 3. Thai do
Tu giac. tich cue, dgc lap va chii dgng phat hien ciing nhu ITnh hoi kien" thiic trong qua trinh hoat dgng.
Cam nhan dugc su cSn thiet cua dao ham trong viec khao sat ham so. Cam nhan dugc thuc te cua toan hgc, nhat la doi vdi dao ham.
PHa n 2.
CAC BAI SOA]!!^
§1. Nguyen ham
(tiet 1, 2, 3, 4, 5)
I. MUC TIEU
1. Kien thurc
HS nam duac :
Nh6 lai each tinh dao ham cua ham sd.
• Dinh nghia nguyen ham.
• Cac tinh chat ciia nguyen ham.
Mot so' nguyen ham co ban.
Cac phuong phap tinh nguyen ham : Phuong phap doi bien sd va phuong phap nguyen ham tiing phan.
2. KT nang
HS tinh thanh thao cac nguyen ham co ban.
Tinh dugc nguyen ham dua vao phuong phap doi bien sd va phuong phap nguyen ham tiing phan.
3. Thai do
Tu giac, tich cue trong hgc tap.
Biet phan biet ro cac khai niem co ban va van dung trong tiing trudng hgp cu the. " Tu duy cac va'n de cua toan hgc mot each Idgic va he thdng. n . CHUAN BI CUA GV VA HS
1. Chuan bj ciia GV
Chuan bi cac cau hoi ggi mo.
Chuan bi pha'n mau, va mdt sd dd diing khac.
2. Chuan bj cua HS
Can dn lai mot sd kien thiic da hgc ve dao ham.
ra. PHAN PHOI THCJI LUONG
Bai nay chia lam 5 tiet:
Tiet 1 : Tic dau den hit miic 2 phdn I.
Tiet 2 : Tiep theo den het phdn I.
Tiet 3 : Tiep theo den het muc I phdn II.
Tiet 4 : Tiep theo den het phdn II.
Tiet 5 : Bdi tap
IV TIEN TRINH DAY HOC
A. DAT VAN OE
Cau hoi 1
Xet tinh diing - sai cua cac cau sau day :
a) Ham sd y = In(cosx) cd dao ham y' = -tanx.
b) Ham sd y = In(cosx) cd dao ham y' = -cotx.
Cau hoi 2
Chohamsdy = 3''""
a) Hay tinh dao ham cua ham sd da cho.
b) Chiing minh rang ham sd y = x3''"'' cd dao ham la y' = 3''"" G V :
Ham y = xS^'"" ggi la nguyen ham ciia ham sd y' = 3^'"" B. BAi Mdl
I NGUYEN HA M VA TINH CHAT
HOATDONC1
1. Nguyen ham
• Thuc hien f \ 1 trong 5'
Hoat dgng cua GV
Hoat dong cua HS
Cau hoi 1
Tim mot ham sd F(x) F(x) = 3x2
Cau hoi 2
Tim mot ham sd F(x)
FYY ^ — r vx; —
cos X
• GV neu dinh nghia :
ma ma
Ggi y tra loi cau hoi 1
GV ggi mot vai HS tra Idi. Bai toan nay cd nhieu dap sd.
Tong quat : F(x) = x^ + C trong do C la hang sd bat ki.
Ggi y tra Idi cau hoi 2
Lam tuong tu cau a.
In X
F(x) = - ^
cos X
Cho hdm sof(x) xdc dinh tren K
Ham soF(x) duac ggi Id nguyen hdm cda hdm sof(x) tren K neu F '(x) - f(x) vai mgi x e K
• GV neu va thuc hien vf du 1, GV cd the lay mdt vai vi du khac. HI. Tim nguyen ham ciia ham sd y = x.
H2. Tim nguyen ham cua ham sd y = x
H3. Tim nguyen ham cua ham sd y = x
H4. Tim nguyen ham ciia ham sd y = x"
4
• Thuc Men f\2 trong 5'.
Hoat dong ciia GV
Cau hoi 1
Tim mot ham sd F(x) ma F(x) = 2x.
Cau hoi 2
Tim mot ham sd F(x) ma
V{x)=-.
X
Hoat dong ciia HS
Ggi y tra loi cau hoi 1
GV ggi mot vai HS tra Idi. Bai toan nay cd nhieu dap sd.
Tong quat : F(x) = x^ +C trong dd C la hang sd bat ki.
Ggi y tra loi cau hoi 2
Lam tuong tu cau a.
F(x) = hix + C.
H5. Tim nguyen ham ciia ham sd y = sin x.
H6. Tim nguyen ham cua ham sd y = cosx.
1
H7. Tim nguyen ham ciia ham sd y 2Vx
N/2
H8. Tim nguyen ham ciia ham sd y = x
• GV neu dinh li 1:
Neu F(x) Id mot nguyen hdm cua hdm sof(x) tren K thi vai moi hang so C, hdm soG(x) = F(x) + C cUng Id mot nguyen hdm cda f(x) tren K H9. Biet ham sd cd mdt nguyen ham la y = sin x. Hay tim nguyen ham cua ham sd dd. HIO. Biet ham sd cd mdt nguyen ham la y = cosx. Hay tim nguyen ham cua ham sd dd. 1
Hll. Biet ham sd cd mdt nguyen ham la y = ^^ '^ . Hay tim nguyen ham cua ham sd dd.
H12. Biet ham sd cd mdt nguyen ham la y = ^ . Hay tim nguyen ham ciia ham sd dd. • Thuc hien Sgr 3 trong 5'.
Hoat dgng ciia GV
Cau hdi 1
Hoat dgng ciia HS
Ggi y tra loi cay hoi 1
Tinh dao ham ciia ham sd : y = G(x).
Cau hoi 2
Hay ket luan.
GV neu dinh li 2:
{G{x)y = [Fix) + C]' = F'(x) + C' = fix),xeK.
Ggi y tra loi cau hoi 2 GV tu ket luan.
Neil F(x) Id mot nguyen hdm cua hdm sof(x) tren K thi moi nguyen hdm cua f(x) tren K deu co dgng F(x) + C, vai C Id mot hang so. De chiing minh dinh li, GV neu va'n 66 de HS chiing minh.
GV neu ki hieu :
^f(x)dx - Fix) + C.
• GV neu chii y trong SGK.
• De thuc hien vi du 2, GV cd the neu cac vi du khac hoac cho HS tu neu vi du va dat cac cau hdi sau :
H13.Tinh J3xdx.
H14.Tinh Jkdx.
H15.Tinh f-dx.
Hld.Tinh f-^d x
•'2Vx
HOAT DONG 2
2. Tinh chat ciia nguyen ham
• GV neu tinh chat 1:
( lf{x)dx)' = fix) ; jf'ix)dx - fix) + C.
HI7. Hay chiing minh cac tinh chat tren.
H18. Tinh ftanxdx.
• GV neu va cho HS thuc hien vi du 3 hoac cd the lay nhiing vi du khac. • GV neu tinh chat 2 :
\kfix)dx = k \fix)dx
De chiing minh tinh chat nay, GV cSn dua ra cac cau hdi sau : HI9. Tinh dao ham hai ve.
H20. Chiing minh dao ham hai ve bang nhau.
• GV neu tinh chat 3 :
j[fix) ± gix)]dx = \fix)6x ± jgix)dx.
• Thuc hien "pt 4 trong 5'
Hoat dgng cua GV Hoat dgng ciia HS
Cau hoi 1
Tinh dao ham cua ham so d mdi ve.
Cau hoi 2
Hay lam tuong tu ddi vdi trudng hgp dau trir.
Ggi y tra loi cau hoi 1
[\fix)Ax+ \gix)6x\
= [\fix)6x) +[\gix)dx] ^fix) + gix).
Ggi y tra loi cau hoi 2 [lfix)dx - jgix)dx] = fix) - gix).
• GV neu va thuc hien vi du 4. GV cd the thay bdi vi du khac. H21. Tinh J (cos x + sin x)dx .
H22. Tinh [(cos x + tan x)dx .
H23. Tinh [(cosx - vx)dx .
H24.Tinh |(x^+x + l)dx.
10
nOATiyDNG3
3. Sii ton tai nguyen ham
• GV n6u dinh li 3:
Moi hdm sof(x) lien tuc tren K deu co nguyen hdm tren K
• Thuc hidn vi du 5:
2
H25. Chiing minh ham sd y = x ^ cd nguyen ham. Tinh nguyen ham cua ham sd dd:
H26. Chiing minh ham sd y = —z— cd nguyen ham. Tinh nguyen ham ciia ham sd dd. sin X
• GV cho HS tinh nguyfen ham va dien vao bang sau :
••"';:-v-.'^- fix) '"\--:.:,jr 0
ax«-^
1
x
e^
a*lna ia> 0,a^l)
cosx
—siruc
1
cos x
1
sin x
: fix) + C
11
• Thuc hien vi du 6 trong 5'
Cau a.
Hoat dgng ciia GV
Cau hdi 1
Tinh nguyen ham ciia ham so: y = ly}
Cau hoi 2
Tinh nguyen ham ciia ham sd: 1
Cau hoi 3
Tinh nguyen ham ciia ham so da cho.
cau b. HS tu tinh tuong tu.
Hoat dgng ciia HS
Ggi y tra loi cau hoi 1
[2x2dx = -x^
J 3
Ggi y tra loi cau hdi 2 1 ^ ' [-^L=dx- [x 3dx-3x3
Ggi y tra loi cau hoi 3 HS tu tinh.
II. PHUONG PHAP TINH NGUYEN HAM
HOAT DONG 4
1. Phuong phap doi bien sd
• Thuc hien ^ . 6 trong 5'
Hoat dgng cua GV
Hoat dgng ciia HS
Cau hdi 1
Dat u = X - 1, tinh du Cau hoi 2
Tinh |(x-l)'°dx.
12
Ggi y tra loi cau hoi 1 Ta cd du = u'dx = dx.
Ggi y tra loi cau hoi 2
[(x-iyOdx^[u'Odu = -ui'+ C
= l(x-i)"+C
U
caub.
Hoat dgng ciia GV
Cau hdi 1
Dat x = e' tinh dt.
Cau hoi 2
Tinh [ dx.
•' X
• GV neu dinh li 1:
Hoat dgng cua HS
Ggi y tra loi cau hdi 1
Ta cd t = Inx => dt = —dx
X
Ggi y tra loi cau hdi 2
[^^dx=[tdt=^2^C=^ln2 x + C J X J 2 2
Neil \fiu)du = Fiu) + C vdu = u(x) la hdm so co dao hdm lien tuc thi \fiuix))u 'ix) dx = Fiuix)) + C.
H27. Hay chiing minh dinh ii tren.
• GV neu he qua:
f 1
\fiax+ b)dx =—Fiax+ b) + C (a ^ 0).
J a
H28. Hay chiing minh he qua tren.
• GV cho HS thuc hien vi du 7. GV cd the thay bdi vi du tuong tu.
• Ddi vdi chii y trong SGK, GV neu va nhSii manh dieu nay :
Mgi bien sau khi thay ddi trong qua trinh tinh toan, song ket qua cud'i ciing phai la bien ban dau.
• Thuc hien vi du 8 trong 5'
13
Hoat dgng ciia GV
Cau hdi 1
Nen dat bien nao bdi bien u. Cau hdi 2
Tinh nguyen ham ciia ham sd da cho.
H29.Tinh f(3x + l)dx
H30.Tinh f(x + l)dx
H31.Tinh jtanxdx.
Hoat dgng ciia HS
Ggi y tra Idi cau hdi 1 Datu = x - 1.
Ggi y tra Idi cau hdi 2 HS tu tinh.
HOATD0NG5
2. Phifong phap nguyen ham tiimg phSn • Thuc hien -^ 7trong 5'
Hoat dgng ciia GV
Hoat dgng cua HS
Cau hdi 1
Tinh \ix cos xYdx.
Cau hdi 2
Tinh fcosxdx.
Cau hdi 3
Tinh [xsinxdx.
• GV neu dinh li 2 :
Ggi y tra loi cau hdi 1
Ta cd
Ux cos x)'dx = x cos x + Ci
Ggi y tra Idi cau hdi 2
[cosxdx = sinjc + C2,
Ggi y tra loi cau hdi 3
\x sin xdx = -x cos x + sin x + C
Neu hai hdm sou = u(x) vdv = v(x) co dao hdm lien tuc tren K thi \uix)v\x)dx = uix)vix) - ^u\x)vix)dx.
14
H32. Hay chiing minh dinh li tren.
• GV neu chii y : [udi; - uv - \vdu.
• Thuc hien vi du 9 trong 7' Day la vi du quan trgng, GV nen hudng dan cu the cau a.
Hoat dgng ciia GV
Cau hdi 1
Dat u va dv hgp If.
Cau hdi 2
Van dung dinh If, hay tfnh nguyen ham ciia ham sd tren.
caub.
Hoat dgng ciia GV
Cau hdi 1
Dat u va dv hgp If.
Cau hdi 2
van dung dinh If hay tfnh nguyen ham ciia ham so tren.
Cau c.
Hoat dgng cua GV
Cau hdi 1
Dat u va dv hgp If.
Cau hdi 2
Van dung dinh If hay tfnh nguyen ham cua ham sd tren.
Hoat dgng ciia HS
Ggi y tra loi cau hdi 1
Dat u = X, dv = e^'dx
Ggi y tra loi cau hdi 2
Jxe^'dx = xe'' - fe''dx = xe" -e " + C.
Hoat dgng cua HS
Ggi y tra loi cau hdi 1
Dat u = X, dv = cosxdx
Ggi y tra loi cau hdi 2
[x cos X d X = X sin X + cos x + C.
Hoat dgng ciia HS
Ggi y tra loi cau hdi 1
Dat u = Inx, dv = dx
Ggi y tra loi cau hdi 2
[in X d X = X In X - j dx = x I-nx x + - C. 15
• Thuc hien GL 8 trong 5'
GV cho HS tu dien vao bang. Ket qua nhu sau:
u
dv
jPix)e''dx P(x)
e^dx
[P(x)cosxdx
P{x)
cosxdx
HOAT DONG 6
JP(x) In xdx Inx
Prxjdx
TOM T ^ B^l H9C
1. Cho ham so fix) xac dinh tren K
Ham sd F(x) dugc ggi la nguyen ham ciia ham sd fix) tren K neu F 'ix) - fix) vdi mgi x e K
2. Neu Fix) la mdt nguyen ham cua ham sd fix) tren K thi vdi mdi hang so C, ham sd G(x) = Fix) + C ciing la mdt nguyen ham ciia fix) tren K h) Neu Fix) la mdt nguyen ham cua ham so fix) tren K thi mgi nguyen ham cua fix) tren K deu cd dang Fix) + C, vdi C la mdt hang sd.
3. ( jfix)dx)' - fix) ; lf'ix)dx - fix) + C.
\kfix)dx = k\fix)dx\; I \[fix)±gix)\dx = \fix)dx± \gix)dx.
5. Mgi ham sd fix) lien tuc tren K deu cd nguyen ham tren K 6.
1" fi^ la^dx = + C (a > 0, a ^ 1) J In a
Icosxdx = sinx+ C
16
J0dx = C \dx = X + C
fr"Hr r"+l -I-r' rr/3t H J a + 1
— dx = Inlxl + C Jx ' '
fe''dx = e* + C
jsinxdx = -cosx + C
—dx = tanx + C
•'cos X
—dx = -cotx + C •"sin X
7. Ne'u \fiu)du = Fiu) + C va M = u(x) la ham sd cd dao ham lien tuc thi jfiuix))u'ix)dx = Fiuix)) + C.
8. Neu hai ham sd u = uix)va.v - vix) cd dao ham lien tuc tren AT thi [u(x)v'(x)dx = u(x)v(x)- [u'(x)v(x)dx; iudv = uv - wdu
HOAT DONG 7
M9T SO C^a HOI TR^C NGHIEM ON T6P B^l 1 Cdu L Cho ham sd y = x Hay dien diing sai vao cac cau sau
(a) Ham so ludn cd nguyen ham.
(b) Ham sd chi cd mdt nguyen ham.
(c) Ham so chi cd nguyen ham la — x 4
(d) Ham sd cd vd sd nguyen ham dang — x + C. 4
Trd Idi.
D D
D D
a
D
b S
c S
d
D
17
Cdu 2. Cho ham sd y = Vx Hay dien diing sai vao cac cau
sau
(a) Ham sd ludn cd nguyen ham. Q (b) Ham sd chi cd mdt nguyen ham. [J
(c) Ham sd chi cd nguyen ham la — x ^
(d) Ham sd cd vd sd nguyen ham dang — x ^ + C. Trd Idi.
D D
a D
b s
c S
d D
Cdu 3. Cho ham sd y = x + cosx. Hay dien diing sai vao cac cau sau
(a) Ham sd ludn cd nguyen ham.
(b) Ham sd chi cd mdt nguyen ham.
1 7
(c) Ha m sd chi cd nguyen ham la — x + sin x.
(d) Ha m sd cd vd sd nguyen ham dang .—x^ + sin x + C. Trd Idi.
D D
D
D
a D
b S
c S
d D
Cdu 4. Ham sd nao sau day cd nguyen ham la 2x
(a)y=x'+2 ;
(c) y = 2 ;
Trd led. (c).
18
(b)y=2x; (d)y= VST.
Cdu 5. Ham sd nao sau day cd nguyen ham la Vx
(a)y= y = ^ ^ ; (b) y = -x2 ;
2Vx 3
(c)y = x2; (d)y=- .
X
Trd Idi. (a).
Cdu 6. Ham sd nao sau day cd nguyen ham la - cos 2x
(a) y = sin2x ; (b) y = —sin2x;
(c) y = -sin2x; (d)y=cos2x.
Trd Idi. (b).
Cdu 7. Ham sd nao sau day cd nguyen ham la cos 2x
(a) y = sin2x ; (b) y =—sin2x;
(c)y = -sin2x; (d)y=sin2x.
Trd Idi. (b).
Cdu 8. Ham so nao sau day cd nguyen ham la sin2x
(a) y = sin2x ; (b) y = —cos2x;
2
(c) y = -sin2x; (d)y=sin2x.
Trd Idi. (b).
Cdu 9. Ham so nao sau day cd nguyen ham la e"
(a)y = e''; (b)y=^e2''; (c)y = lnx; (d)y=e'"' '
Trd Idi. (a).
Cdu 10. Ham so nao sau day cd nguyen ham la In x
(a)y = lnx ; (b)y=- ; (c)y = -lnx ; (d)y=e'"' ' X
Trd Idi. (b).
19
HOAT DONG 8
naCTNG D^N Bfil T6P S9; a J.
38
HOAT DONG 7
MQT SO C^U MOI TR^C NGHIEM KhfiCH QUfiN
Hay dien dung sai vao 6 trong sau:
Cdu 1. Cho ham sd f(x) = x^ + x" -5x +3
1
(a) Khong ton tai f(x)d> D
1
(b) jf(x)dx 0
1
11 12
D
(c) Jf(x)dx = ^ 0
1
(d) rf(x)dx = l 0
Trd Idi
D D
(a) S
(b) D
(c) S
(d) S
Cdu 2. Cho ham sd y = 1
1
(a) Khong ton tai f(x)dx D 1
(b) ff(x)dx = 0 1
D
39
Z 1
(c) ff(x)dx= ff(x)d? D
Z I
(d) ff(x)dx = - ff(x)dx Q Trd Idi
(a) D
(b) D
(c) S
(d) D
Hay chpn khang dinh dung, trong cdc cdu sau:
1
Cdu 3. |lx + e''ldx bdng
0
(a)e ;
(c) 2e ;
Trd Idi. (b).
71
Cdu 4. sinxdx bang 0
(a) 1 ; (b) 2; Trd Idi. (b).
7t
Cdu 5. I cosxdx bang 0
(a)0; (b)l; Trd Idi. (a).
40
(b)i.(2e-l) ; (d) 3e.
(c) 3;
(c) 2 ;
(d)4. (d)3.
71
Cdu 6. tan xdx bang
0
(a)ln(cosl); (b)ln(sinl);
(c) -ln(cos 1) ; (d)ln(sinl).
Trd Idi. (c).
1
Cdu 7. ln(x)dx bdng
(a) 1 ; (b) 2; (c) - 1 ; (d) - 2
Trd Idi. (c).
Cdu 8. fe^^^'dx bdng
0
(a) — ^ ; (b)
2 J,
e^+e _ , e^- e
(c) —r- ; (d)
2 j
Tra /dj". (a).
HOAT DONG 8
HUdfiQ DJN 3^1 T6P SGK
Bai 1. Hudng ddn. Six dung cac tfnh chat cua tich phan
cau a. Hudng ddn. Dat 1 - x = t.
Ddpsd-^iS^-l).
100,xe[a; b], thi f/"(x) > 0.
Bai 7. Chiing minh Ifix) dx l\f(x)\ dx.
Bai 8. Neu m < /"(x) < M, x G [a ; 6], m, M la cac hang sd thi
b
mib - a) < f/•(x)dx < M(6 - a).
45
§3. Ifng dung cua tich phan trong hinh hoc (tiet 11, 12, 13, 14)
I. MUC TifeU
1. Kien thirc
HS nam dugc :
Khai niem dugc ling dung trong hinh hgc nhu the' nao?
Bai toan dien tich dugc tfnh nhu the' nao?
• Bai toan the tich dugc tinh nhu the' nao?
2. KT nang
- Tinh dugc dien tich va the tich mdt sd hinh co ban.
Hoan thien each tinh toan tich phan.
3. Thai do
• Tu giac, tich cue trong hgc tap.
Bi^t phan biet rd cac khai niem co ban va van dung trong timg trudng hgp cu the. - Tu duy cac vdn de ciia toan hgc mdt each Idgic va he thdng.
n. CHUAN BI CUA GV VA HS
1. Chuan bj ciia GV
• Chudn bi cac cau hdi ggi md.
Chudn bi cac hinh ttt hinh 51 den hinh 62.
Chudn bi phdn mau, va mdt sd dd diing khac.
2. Chuan bj cua HS
• Cdn dn lai mdt sd kien thiic da hgc d hai bai trudc.
OntapkT bai 2.
m . PHAN PHOI THdl LUONG
Bai nay chia lam 4 tiet:
46
Tiet 1 : Tit dau din hit muc 1 phdn I.
Tiit 2 : Tiep theo din hit phdn I.
Tiet 3 : Tii'p theo din hit muc 1 phdn II.
Tii't 4 : Tii'p theo din hit phdn II.
IV. TIEN TRINH DAY HOC
A. OAT VAN OE
Cau hdi 1
a) Neu cac tfnh chdt ciia tich phan.
b) Neu cac ndi dung co ban ciia phuong phap doi bien sd va phuong phap tfch phan tiing phdn.
Cau hdi 2
„2
Cho ham sd y = x + 1
x - 1
a) Hay pha bd dau gia tri tuyet ddi.
b)Tfnh f 3..2 x^-x + 1
Tl dx.
B. BAI MOI
I - TINH DIEN TICH HINH PHANG
• Thuc hien "pt 1 trong 5'.
Hoat dgng ciia GV
Cau hdi 1
Hay ve hinh va gidi han phdn hinh can tfnh dien tfch.
Cau hdi 2
Hay thiet lap cdng thiic tfnh dien tfch.
Hoat dgng ciia HS
Ggi y tra loi cau hdi 1
GV ggi HS len ve hinh va kdt luan phdn hinh ve. GV tham khao hinh 46.
Ggi y tra loi cau hdi 2
2
S= f|-2x-l|dx
1
47
Cau hdi 3
Tfnh dien tfch hinh dd.
Cau hdi 4
So sanh theo yeu cau bai toan.
Ggi y tra Idi cau hdi 3 S = 3.
Ggi y tra Idi cau hdi 4 Hai dien tich nay bdng nhau
HOAT DONG 1
1. Hinh phSng gidi han bdi dudng cong va true hoanh
• GV neu cac cau hdi sau:
HI. Dien tich cd the am dugc hay khdng?
H2. Qua hoat dgng 1 hay neu mdi quan he giiia dien tich va tfch phan. • GV treo hinh 51 va giai thich ddi vdi phdn hinh ndm dudi true hoanh. • GV neu dinh nghia
Gia sic hdm soy = f(x) lien tuc, khong dm tren dogn [a ; bJ. Ta dd bie't hinh thang cong gidi hgn bdi do thi cua fix), true hodnh vd hai dudng thdng x = a,x = b co dien tich S duac tinh theo cong thicc
S = jfix)dx..
GV su dung hinh 52 va neu dinh nghia
Dien tich S cda hinh phang gidi hgn bdi dd thi cua hdm sdf(x) lien tuc, true hodnh vd hai dudng thdng x = a,x = b (H.61) duac tinh theo cong thitc
H3. Neu phan dien tich ndm phia tren true hoanh thi ta sii dung cong thiic nao ? H4. Neu phan dien tich ndm phia dudi true hoanh thi ta sii dung cdng thiic nao ? • Thuc hien vi du 1 trong 4' (GV cd the Idy vf du khac), GV sii dung hinh 53.
48
Hoat dgng ciia GV Hoat dgng ciia HS
Cau hdi 1
Trong doan nao ham so nhan gia tri am?
Cau hdi 2
Trong doan nao ham so nhan gia tri duong?
Cau hdi 3
Thiet lap cdng thiic tfnh dien tfch hinh da cho.
Ggi y tr a Idi cau hdi 1
Trong doan [-1 ; 0].
Ggi y tr a Idi cau hdi 2
Trong doan [0 ; 2].
Ggi y tr a loi cau hdi 3
S =
2 0 2
S= j|x^|dx = f(-x^)dx + fx^dx
-1 -1 0
GV nen de HS tinh tiep.
H5. Tinh dien tfch hinh phdng gidi han bdi y = x , x = 0, x = 3 va true hoanh. H6. Tinh dien tfch hinh phdng gidi han bdi y = cosx , x = 0, x = 3 va true hoanh. H7. Tfnh dien tich hinh phdng gidi han bdi y = sin x , x = 0, x*= 3 va true hoanh. H8. Tinh dien tich hinh phang gidi han bdi y = In x , x = 0, x = 3 va true hoanh.
49
HOAT DONG 2
2. Hinh phdng gidi han bdi hai dudng cong
• GV sii dung hinh 54 dedio ta dien tich hinh phang trong trudng hgp nay - GV nen dat ten cac diem cua hinh 54 la giao ciia y = /i(x) va y = fiix) vdi cac dudng thdng x = a, x = b.
• GV dua ra cac cau hdi sau:
H9. Dien tich hinh can tim la hieu ciia hai hinh nao?
HIO. Hay lap cdng thiic tinh dien tfch dd.
• Go/ Si,S2 Id dien tich cua hai hinh thang cong gidi hgn bdi true hodnh, hai dudng thdngx = a,x = b vd cdcdUdngcongy = fj(x), y = f2(x) tuang Ang. Khi do, dien tich S cua hinh D Id
l\flix)-f2ix)\dx.
GV neu chii y trong SGK va lay mdt vai vf du minh hga
Thuc hien vf du 2 trong 5' GV sur dung hinh 55 trong SGK
y = cosx
50
Hoat dgng cua GV
Cau hdi 1
Ta can tim giao diem ciia hai dudng cong trong doan nao? Cau hdi 2
Hay tim giao diem dd.
Cau hdi 3
Thiet lap cong thiic tfnh dien tfch dd
Hoat dgng cua HS
Ggi y tra Idi cau hdi 1
Trong doan [0 ; TI].
Ggi y tra loi cau hdi 2
COSX - sinx = 0 <=> x = — e [0 ; TI] Ggi y tra Idi cau hdi 3
S = Icos X - sin x| dx .
0
GV nen de HS tfnh tiep.
Ddp sd S = 2V2
• Thuc hien vi du 3 trong 5' GV cho HS tu ve hinh de xac dinh phdn hinh cdn tinh dien tfch. Tuy nhien bai toan nay khdng nhdt thiet phai ve hinh.
Hoat dgng cua GV Hoat dgng ciia HS
Cau hoi 1
Hay tim giao diem ciia hai dudng cong.
Cau hdi 2
Thiet lap cdng thiic tfnh dien tfch dd.
Ggi y tra Idi cau hdi 1
Giai phuong trinh fiix) - f2ix) = 0 cd ba nghiem Xi - -2, x^ = 0, X3 = 1. Ggi y tra Idi cau hdi 2
1
S = flx^ +x^ -2x|dx
- 2
f(x^+x^-2x)dx -2
f(x^+x^-2x)dx
Cau hdi 3
Tfnh dien tfch hinh da cho.
Ggi y tra Idi cau hdi 3
HS tu tinh tiep.
51
II - TINH THE TICH
• Thuc hien ^ ^ 2 trong 5'
Hoat dgng ciia GV
Cau hdi 1
Khd'i lang tru la gi?
Cau hdi 2
Nhdc lai cdng thiic tfnh the tich khd'i lang tru.
Hoat dgng ciia HS
Ggi y tra Idi cau hdi 1 HS tra Idi.
Ggi y tra loi cau hdi 2 HS tu tra Idi.
HOAT DONG 3
1. The tich ciia vat the
• GV sii dung hinh 56 de md ta the tich vat the.
• GV neu cdng thiic :
The tich V cua phdn vdt the Vgidi hgn bdi hai mat phdng (P) vd (Q) duac tinh bed cong thdc :
• Thuc hien vi du 4 trong 5' GV su dung hinh 57 trong SGK
S(x) = B
M N
52
Hoat dgng ciia GV
*Cau hdi 1
Hay ndu cdng thiic the tfch hinh lang tru.
Cau hdi 2-
Sir dung tfch phan de tfnh the tfch hinh lang tru.
Cau hdi 3
So sanh va ket luan.
Hoat dgng cua HS
Ggi y tra Idi cau hdi 1
S = -Bh.
3
Ggi y tra Idi cau hdi 2
/> A
V = f5(x)dx = f5dx = 5X|Q = Bh 0 0
Ggi y tra Idi cau hdi 3
HS tu so sanh.
HOAT DONG 4
2. The tich khoi chdp va khd'i chop cut
Hll. Neu cdng thiic the tich khd'i chdp va the' tfch khd'i chdp cut. • GV sii dung hinh 58 va neu van de :
Khi cat khd'i chdp bdi mdt mat phang song song vdi day, ta dugc thiet dien cd dien tfch S(x).
H12. Hay tfnh S(x).
HI3. Dua vao cdng thiic tfnh tfch phan, hay tfnh V
h 2 h
The tich Vcua hinh chop V - IS—-dx = —7-
0 h h^
• GV neu cong thiic tfnh the tich hinh chdp cut
h
V ^-^iB + 4BF + B').
53
(x^^
\ 3 ;
_ Bh
0 " 3
HOAT DONG 5
III - THE TICH KHOI TRON XOAY
HI4. Neu cdng thiic the tich khd'i chdp va the tfch hinh tru, hinh cdu. HI5. Hay neu mot sd hinh thuoc khd'i trdn xoay.
' Thuc hien ^_ 3 trong 5'
Hoat dgng cua GV
Cau hdi 1
Hinh tru la gi?
Cau hdi 2
Hinh cau la gi?
• GV neu bai toan :
H16. Khi nao ta cd mdt khd'itrdn xoay?
Hoat dgng cua HS
Ggi y tra Idi cau hdi 1 HS tra Idi.
Ggi y tra Idi cau hdi 2 HS tu tra Idi.
HI7. Khi cdt khd'i trdn xoay bdi mdt mat phang vudng goc vdi true thi ta dugc hinh gi? Hay tinh dien tich hinh dd.
• GV neu cdng thiic tong quat de tinh the tich khd'i trdn xoay: Thiet dien cua khd'i tron xoay tren tgo bdi mat phdng vuong goc vdi true Ox tgi x e [a ; b] la hinh tron co bdn kinh bang I f(x) I vd co b
dien tich Id S(x) - nf ix). The tich cua vdt the V == TI f (x)dx. a
• Thuc hien vf du 5 trong 5'
Hoat dgng cua GV
Cau hdi 1
Hay ve hinh va mo ta the tfch khoi trdn xoay can tfnh.
Cau hdi 2
54
Hoat dgng cua HS
Ggi y tra Idi cau hdi 1 HS tu ve hinh.
Ggi y tra Idi cau hdi 2
Tinh the tfch khdi trdn xoay dd. 7t n V = Tt sin^xdx=— (l-cos2x)dx
0 0
HS tu tfnh tiep.
• Thuc hien vf du 6 trong 5'
Hoat dgng cua GV
Cau hdi I
Viet phuang trinh niia mat cau phfa tren true hoanh.
Cau hdi 2
Tfnh the tfch hinh cdu ban kfnh R.
Hoat dgng ciia HS
Ggi y tra Idi cau hdi 1
y = ^R'^ -X^ i-Re^ -\).
cau d. vl + sin2x = |sinX + cosx| = \'2 sin(x + —) 4
71
Sii dung phuang phap ddi bien so: dat x + — = t.
4
Ddp sd. 2V2.
Bai 6. Hudng ddn. Dua vao tinh chat ciia tich phan va cac phuong phap tinh tich phan.
^ n • 2 ^ ('l-cos2x^ cos2x cos4x 1
cau a. cos2x.sin X = cos2x = : V 2 ; 2 4 4
65
Ddp sd. n
cau b. Pha bd dau gia tri tuyet ddi bang each them can trung gian 0.
1= r(2'^_±|dx+ |l2^- —Vx
Jl 2M A 2^ J
Ddp sd , 1
In 2
cau c. Phan tich tii so thanh da thiic.
Ddpsd ^ + llln2 .
z
cau d. 1 1 x'-2x- 3 4
Ddp sd. - — In 3
1 1
x - 3 x+1
cau e. / = j(l + sin2x)Jx.
Tt Ddp so. 1 + ^
z
cau g. ^= j x^+2xsinx + l-cos2x dx.
„ , ,. Tl 5TI
Dap so. -3 - + - ^
Bai 7. Hudng ddn. Dua vao cong thiic tinh dien tich hinh phang va the tich hinh trdn xoay.
Cau a.
GV cho HS ve hinh .
66
HI. Tfnh cac can cua tfch phan.
H2. Tfnh dien tfch :
2
S = 2j|Vr^-(l-x)]dx.
0
caub.
2
V = 47rJ|Vl-x2-(l-x)j dx
HOAT DONG 3
OfiP ^N B6i T6P TR^C NGHIEM
1. (C). 2. (D). 3. (B). 4. (C). 5. a) (C). b) (B). 6.(D).
MOT SO DE KIEM TRA THAM KHAO
Del
Phdn I. Trac nghiem khdch quan (4 diem).
Cdu 1. Hay dien diing, sai vao d trdng sau day :
(a) (b) (c) (d)
xdx = X + C
j xdx = C .
fxdx = x^+C fxdx = -x2+C.
n D D D
Hay dien diing, sai vao d Cdu 2.
1
trdng sau day :
D
(a)
(b)
(c)
(d)
Cdu 3. (a)
68
ff (x)dx = 0
1
1
ff (x)dx = 1
1
b a
ff(x)dx= ff(x)dx a b
b a
ff(x)dx = - ff(x)dx a b
2
Ix dx bdng:
1
\ . .^\:
D
D
D
4 (c) cd nghiem — ; . , |
Cdu 4. Cho dudng cong f(x) = x^ Dien tich hinh phdng gidi han bdi f(x) = x true tung, true hoanh va x = 1 la
(a)f; (b)2;
4 5
(e) cd nghiem - ; (d) -
Phdn 2. Tu ludn (6 diem)
1. Tinh cac tfch phan sau:
;r 4 ^
a) (x sinxdx; b) f-^; dx . / 3^x^-3x + 2
2. Tinh dien tich hinh phang gidi han bdi f (x) = Vx^ - 3 , y = 2x va true hoanh.
Di2
Phdn I. Trac nghiem khdch quan (4 diem).
Cdu I. Hay dien diing, sai vao d trdng sau day.
(a) Mgi ham so deu cd nguyen ham [j (b) Ham sd lien tuc tren (a ; b) thi cd tich phan tren [a ; b] [j
2
(c) jsinxdx = COS2-COS 1 LJ 1
2
(d) jsinxdx = cos 1-cos2 [J 1
Tt
Cdu 2. jx cos xdx bang
0
(a) 2 ; (b)-2 ; (c) Tt (d)-7i. 69
71
Cdu 3. X sin xdx bdng 0
(a) 2 ; (b) -2;
e
Cdu 4. X In xdx bdng 0
1 9 1 9
(a)-e2; (b)-e?; 2 4 Phdn 2. Tu ludn (6 diem) 1. Tinh cac tich phan sau :
1
a) fx(l-x)^°°^dx ;
0
(C) Tt (d) -71,
(c) ^e2 (d) e^ O
TT
h) Jsin3xcos5xdx
2. Cho ham so y = f(x) lien tuc tren doan [a ; b] b b
D D
i) Chiing minh rang : f(x)dx = f(a + b-x)dx. a) ~ ' ^ • " ~ ^
a a
71
4
b)Tinh ln(l + tanx)dx.
0
Di3
Phdn 1. Trac nghiem khdch quan (4 diem).
Cdu 1. Hay dien diing, sai vao d trdng sau day : (a) f(x + l)dx = -x2+x + C •
(b) f(3x + l)dx = -x2+x + C. • 70
(c) f2xdx = x2+C U
(d) fxdx = x^+C. n
Cdu 2. Hay dien diing, sai vao d trdng sau day :
1
(a) J(x200«-x + l)dx = 0 D
1
1
(b) f(x20°«-x + l)dx = l D
0
1
(c) fx(l-x)dx-0 D
0
b a
(d) ff (x)dx - - ff (x)dx D
a b
2
Cdu3. \lx^-l\dx bang:
1
(0 I. |;
(c) 1 ; (d) f
71
Phdn 2. Tu ludn (6 diem)
1. Tinh cac tich phan sau:
\ cosx \x + \ ^
a) I 5—dx; b) dx.
, 1 + sin X 0 ^ ~ •^
2. Tinh dien tfch hinh phdng gidi han bdi f (x) = x + sin x, y = 2x va true hoanh.
Di4
Phdn 1. Trac nghiem khdch quan (4 diem).
Cdu 1. Hay dien diing, sai vao d trdng sau day.
(a) Mgi ham so deu cd nguyen ham [_}
(b) Ham sd lien tuc tren (a ; b) thi cd tich phan tren [a ; b] |_| 2
(e) cos xdx = sin 2-sin 1 Q 1
2
(d) Jcosxdx = sinl-sin2 \_\ 1
7t
Cdu 2. jx sin xdx bdng
0
(a) 2 ; ••(b)-2;
(C) Tt (d) -71.
Tt
Cdu 3. x^ sin xdx bdng
0
(a)Tt2-4 ; (b)7t2 + 4;
(c)-7t^+4 (d)-71^-4
72
e
Cdu 4. In xdx bang
0
1 2
(a)e; (b) 0; (c)- e (d) e o
Phdn 2. Tu ludn (6 diem)
1. Tfnh cac tfch phan sau :
a) p(l-x)'°°d x ; b) jeos^ xdx
0 0
2.
3 ,
a) Tfnh JoTT' b) Tfnh jmin{Vx;x^^dx
HMfdNG DAI V
Del
Phdn I. Trac nghiem khdch quan (mSi cdu 1 diem)
Cdu 1.
(a) (b) (c) (d)
S S S D
Cdu 2.
(a) (b) (e) (d)
D S S D
Cdu 3. (b),
Cdu 4. (a).
Phdn 2. Tu luan (6 diem)
1. a) 7t^ - 2 sin 1 - cos 1 - 2;
b) Ta cd - 3 x + 2 x- 2 x- 1
'* 1
Dodd f-^^ dx = 21n2-ln3
3^x^-3x + 2
2. HS tu tinh..
Di2
Phdn 1. Trac nghiem khdch quan (mdi cdu 1 diem) Cdul.
(a) (b) (c) (d)
S D D S
Cdu 2. (b)
Cdu 3. (c).
Cdu 4. (b).
Phdn 2. Tu ludn (6 diem)
1. Tfnh eac tfch phan sau :
a) Dat 1-x = t, khi X = 0, t = 1; khi X = 1, t = 0.
1 0
jx(l-x)20«8dx = -j(l-t)t2008dt;
b) J sin 3x cos 5xdx = VI-1
8
2. Cho ham sd y = f(x) lien tuc tren doan [a ; b]
a) Dat t = a +b - X.
HS tu chiing minh.
b) Ta cd 1+ tanx = 1 + tan van dung cau a) ta cd I = — In 2 , 8
74
De3
Phdn I. Trac nghiem khdch quan (mdi cdu 1 diem)
Cdu 1.
(a) (b) (c) (d)
D D D S
Cdu 2.
(a) (b) (c) (d)
D S S D
Cdu 3. (c).
Cdu 4. (b).
Phdn 2. Tu ludn (6 diem)
1. Tfnh cac tfch phan sau:
71
. e cosx . Tt
a) I r—dx = —;
j l + sin'^x 4
b) fii±idx = l-31n2
2. HS tu tfnh.
De4
Phdn 1. Trdc nghiem khdch quan (mdi cdu 1 diem)
Cdu I.
(a) (b) (c) (d)
S D D S
Cdu 2. (c)
Cdu i. (a).
Cdu 4. (b).
75
Phdn 2. Tu ludn (6 diem)
1. Tfnh cac tfch phan sau :
a) |x^I-x)'"dx jx^a-. = '
530553
0
7C
b) fcos xdx = — J 15 0
3 f dx
2.a) \--. = ln2 + 21n3; J X +1
-2'
2 1 2
^ 1 2 b) JminjVx ;x^}dx= fx^dx+ fVxdj(
^ I X
1
76
ChirONq IV
SO PHLTC
Pha n 1
msta^G TAX D E CUA CHUUlVG
I. NOI DUNG
Ndi dung chinh cua chuang 4 :
- So phiic : Dinh nghia ; hai sd phiic bdng nhau; bieu dien hinh hgc ciia so phiic; md dun ciia sd phiic T sd phiic lien hgp.
Cac phep toan ve sd phiic : Phep cdng va phep trii; phep nhan cac sd phiic ; Tong va tfch hai sd phiic lien hgp ; phep chia hai sd phiic.
Phuong trinh bac hai ddi vdi he sd thuc : Can bac hai ciia sd thuc am; phuong trinh bac hai ddi vdi he so thuc.
n . MUC TIEU
1. Kien thurc
Ndm dugc toan bd kien thiic co ban trong chuang da neu tren, cu the : Ndm viing dinh nghia so phiic va cac phep toan ciia nd.
• Hieu dugc mddun cua so phiic va bieu dien mdi sd phiic tren mat phang tga do. Mdi quan he ciia hai so phiic lien hgp.
2. KT nang.
Van dung thanh thao cac phep toan.
• Tim dugc mddun cua mdt so phiic.
• Giai dugc phuang trinh bac hai cd nghiem phiic.
77
3. Thai do
Tur giac, tich cue, dgc lap va chii dgng phat hien ciing nhu linh hdi kien thirc trong qua trinh hoat dgng.
Cam nhan dugc su cdn thie't eiia dao ham trong viec khao sat ham sd. Cam nhan duoc thuc te' ciia toan hgc, nha't la ddi vdi dao ham.
78
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