Muon the, cac em hay danh thcri gian nhat dinh de lam cac bai tap trong sach, sau do doi chieu va kiem tra lai ket qua thiTc hien... H^m so xac dinh tren K... Sir bien thien ciia hkm so.
Trang 2NGUYEN oCfC CHI
I A I T I C H 12
XAXG CAO
THU VlifJ l\m BINH TH'JAN
NHA XUAT BAN DAI HOC Q U 6 c GIA HA
Trang 3LCJl NOI DAU
G I A I B A I T A P D A I S O 12, duac bien soan v6i muc dich giiip hoc
sinh doi chieu va kiem tra lai cac ket qua khi thtfc hien giai cac bai tap trong sach giao khoa Muon the, cac em hay danh thcri gian nhat dinh de lam cac bai tap trong sach, sau do doi chieu va kiem tra lai ket qua thiTc hien
G I A I B A I T A P D A I S O 12, phu huynh c6 the suf dung de kiem tra con minh trong viec hoc tap va luyen tap cac kien thufc va ky nang
Trang 4TJTNG D U N G D A O H A M D E K H A O S A T V A V E
D6 T H I C U A H A M S O
§1 TiNH DdN DIEU CUA HAM SO
ivdl D U I \ cAi^ I^ I O f
1, D i n h l i : Gia sil hkm so f c6 dao M m tren khoang I
• Neu f '(x) > 0, Vx e I t h i h ^ m so f dong bien tren khoang I
• N§'u f '(x) < 0, Vx e I t h i h ^ m so' f nghich bien t r e n khoang I
• Neu f '(x) = 0, Vx e I t h i h ^ m so f khong d6i t r e n khoang I
Chii y: Khoang I neu duoc thay bkng mot doan hoac mot niira khoang t h i
phai bd sung gia thiet " H a m so' lien tuc t r e n doan hoac niifa khoang do"
2 Vi#c x e t c h i e u b i e n t h i e n ciia h a m so' c6 dao h a m c6 the chuyen ve viec
xet dau dao h a m ciia h a m so' do
Trang 5V§y hkm so d6ng bien t r g n moi khodng ( - 0 0 ; ~) vk ( 1 ; nghich bien
Vay h a m so' dong bien t r e n m6i khoang ( - 0 0 ; - 73 ) va (0; 73 ), nghich
bien tren m o i khoang ( - 73 ; 0) va (73 ; + 0 0 )
X + 1
H a m so xac d i n h t r e n M\{-11 (-2x - 2)(x + 1) - ( - x ' - 2x + 3) - 2 x ' - 2x - 2x - 2 + x ' + 2x - 3
._ - x ' - 2x - 5 (x +
Vay h a m so nghich bien t r e n m§i khodng ( - 0 0 ; - 1 ) va (-!;•+«)
, (Bdi 3 trang 8 SGK) Gidi
a) f(x) = x'^ - Gx"^ + 17x + 4
H a m so xac d i n h t r e n R
Ta CO f '(x) = 3x^ - 12x + 17
f '(x) > 0, Vx e R Vay h a m so' dong bien t r e n R b) f{x) = x^ + X - cosx - 4
H a m so xac d i n h t r e n R
• Ta c6 f '(x) = 3x^ + 1 + sinx > 0, Vx e R Vay h ^ m so dong bien t r e n R
(Bdi 4 trang 8 SGK) Gidi
ftx) = ax - x^
H a m so xAc dinh t r e n R
Ta c6: f '(x) = a - Sx^
Trang 6* N6u a < 0 t h i f '(x) < 0, Vx e R H ^ m so nghich bi§'n t r g n
* Ng'u a = 0 t h i f '(x) < 0, Vx e R DAng thiJc chi xay ra k h i x = 0
Vay h a m so' nghich bien t r e n R
* Neu a > 0 t h i f '(x) = 0 cs x = ± j
-V3 Bang bien t h i e n
- /—; I — Vay a > 0 khong thoa dieu ki§n de toan
H ^ m so' dong bien t r e n
V£iy vdi a < 0, hkm so nghich bien t r e n R
5 (Bdi 5 trang 8 SGK) Gidi
fix) = - x^ + ax^ + 4x + 3
3
H ^ m s6' x^c d i n h t r e n R
f '(x) = x^ + 2ax + 4, c6 A' = a^ - 4
* Neu a^ - 4 < 0 hay - 2 < a < 2 t h i f '(x) > 0, Vx e R H ^ m so dong bien tren R
• Neu a = 2 t h i f '(x) = (x + 2f > 0,\fx* - 2 H a m so dong bien t r e n R
• Neu a = - 2 t h i f '(x) = (x - 2f > 0, Vx 2 H ^ m so dong bien t r e n M
• Neu a > 2 hoac a < - 2 t h i f '(x) = 0 c6 hai nghiem x i , X2 (xi < X2)
Bang bien t h i e n
f ' ( x )
f(x)
H a m so nghich bien t r e n (xj; X2) khong thoa de toan
Vay - 2 < X < 2 t h i h a m so da cho dong bien tren M
Trang 7H a m so nghich bien t r e n m o i doan
Vay h a m so' nghich bien t r e n R
Chufng m i n h tuong tyf sinx > x, Vx < 0
x^
b) X e t hkm so g(x) = cosx + — - 1
H ^ m so l i e n tuc t r e n nufa khoang [0; +ao), g'(x) = - s i n x + x
Theo cdu a, g'(x) > 0, Vx > 0, do d6 h ^ m so g dong big'n t r e n [0; + « ) ,
Va t a CO g(x) > g(0), Vx > 0
x ' Nghia la cosx + 1 > 0 , V x > 0
2 TCr d6 suy r a v d i m p i x < 0, t a c6:
(-x)^
cos{-x) + 1 > 0 hay cosx + 1 > 0 , V x < 0
2 2 x"
Vay cosx > 1 - y , Vx 0
c) Xet h ^ m so h(x) = x - — - sinx
6
CO h'(x) = 1 - — - cosx Theo cau b) h'(x) < 0, Vx * 0
Do do h a m so h nghich bien t r e n K h(x) < h(0), Vx > 0 x^ x^
J, (Bai 9 trang 9 SGK) Gidi
Xet h a m so' f(x) = sinx + tanx - 2x lien tuc tren niira khoang
Vay sinx + tanx - 2x > 0
hay sinx + tanx > 2x, Vx e
10 (Bai 10 trang 9 SGK) Gidi
t + 5 a) N a m 1980, t a c6 t = 10
Trang 8c) Toe dp t a n g dan so vac n a m 1990 1^
120 (20 + 5)'
Toe do tSng dan so' v^o n&m 2008 1^
120 (38 + 5)'
Vao n a m 1996 toc dp tang dan so ciia t h i tra'n l a 0,125
1 D i n h nghia: Gia suT h a m so' f xac dinh tren tap c M va XQ e y
* X Q dixac gpi la mot d i e m ciJ^c d a i ciia h a m so' f neu:
Ton t a i mot khoang (a; b) chiJa XQ sao cho (a, b) c 7
va f(x) < f ( x o ) vdi m p i x e (a, b) \
K h i do f(xo) dupe gpi l a gia t r i cufc d a i cua h a m so f
* Xo difpc gpi l a mot d i e m cxic t i e u cua h a m so' f neu:
Ton t a i mot khoang (a; b) chufa XQ sao cho (a; b) e D
v a f(x) > f(xo) v d i m p i x e (a; b) \
K h i do f(xo) dupe gpi l a g i a t r i cij^c t i e u ciia h a m so' f
* D i e m cifc dai va diem cUc tieu gpi chung la d i e m cu'c t r i
Gia t r i cUe dai va gia t r i cUc tieu gpi chung la cu'c t r i
* Neu X o l a diem C L C C t r i cua h a m so f t h i ( X Q ; f ( x o ) ) l a diem cu'c t r i ciia do
Gik sijf h a m so f lien tuc t r e n khoang (a; b) chiJa diem x o v a c6 dao h a m
t r e n cae khoang (a; X Q ) ( X Q ; b) K h i do
a) Neu f '(x) < 0 vdi m p i x e (a; X Q) v a f '(x) > 0 vdi m p i x e ( x o ; b) t h i h a m
so' f dat cvtc tieu t a i diem X Q
b) Neu f '(x) > 0 vdi mpi x e (a; X Q ) v a f '(x) < 0 vdi m p i x e ( X Q ; b) t h i h a m
so dat cue dai t a i diem X Q
4 Q u i t S c tim c\ic t r i
* Q u i t ^ c 1:
• T i m f '(x)
• T i m cac diem X i (i = 1, 2, ) t a i do dao h a m cua h a m so' hhng 0 hoac
h a m so' lien tuc nhung khong c6 dao h a m
• X e t da'u f '(x) neu f '(x) ddi da'u k h i x qua diem Xj t h i h a m so' dat cUc t r i
t a i X j
• f ' ( X )
X i
f ' ( x ) f(x)
Vay h a m so da cho dat cUc dai t a i diem x = - 3 , gia t r i eUe dai l a f(-3) :
H k m so dat cUc tieu t a i d i l m x = - 1 , gia t r i cUc tieu cua h a m so l a f l - 1 ) =
3
H a m so' xac d i n h t r e n R
Ta C O f '(x) = x^ - 2x + 2 > 0, Vx e K H a m so dong bien t r e n R , khong c6 cUc t r i
Trang 9H a m so' dat ciTc tieu t a i diem x = 1, gia t r i ci/c tieu la f ( l ) = 2
Vay h a m so' dat cifc dai t a i x = - 1 , gia t r i cUc dai fi-1) = 1 va dat ciic tieu
t a i x = 0, gia t r i cifc tieu fiO) = 0
15 f) fix) = x^ - 3x + 3
H a m so dat cifc tieu t a i x = - 72 , gia t r i c\ic tieu la -2
H ^ m so dat c\ic dai t a i x = 72 , gia t r i cUc dai 1^ 2
H a m so dat cifc dai t a i x = 0, gia t r i cxic dai: 2 72
c) Ap dung qui tAc 2
Trang 10* Vi y " — + kn = 4 s i n = 4 s i n I 3J = - 4 s i n
= _ 4 ^ = - 2 V 3 < 0
Do d6 h a m so dat ciTc dai t a i c6c diem x = - — + k 7 t , k e Z
• • 6 Gia t r i c\Xc dai y ( - — + k 7 r ) = - - + k 7 i - sin
Ngoai r a y " = 2cosx + 4cos2x
• V i y"(k7i) = 2cosk7r + 4cos2k7i = 2cosk7i + 4 > 0, V k s Z
Do do h a m so da cho d a t cUc tieu t a i cac d i e m x = k n v a gid t r i cUc t i e u
y(k7t) = 3 - 2cosk7x - cos2k7r = 2 - 2cosk7r = 2 - 2 ( - l ) ' ' (k e Z)
Gidi he phiTOng t r i n h 3a + 2b = 0
a + b = l
a? + 2b = 0 -2a - 2b = - 2 3a + 2b = 0
• f "(0) = 6 > 0 Vay h k m so dat ciTc tieu t a i diem x = 0 va f(0) = 0
• f "(1) = - - 1 2 + 6 = - 6 < 0 Vay ham so dat cifc dai tai diem x = 1 va f(l) = 1
14 (Bcii 14 trang 17 SGK) Giai
l + a + b + c = 0 (3)
4a - 2b + c - 8 = 0 -4a + b + 12 = 0
a + b + c + l = 0
G i a i he phUcfng t r i n h :
a + b + c + l = 0
o ] - 4 a + b + 12 = 0 3a - 3b - 9 - 0
a + b + c + l = 0
o -12 + b + 12 = 0
a = 3 Kiem t r a l a i k e t qua
f(x) = X-' + 3x^ - 4
f ' ( x ) = 3x? + 6x, f '(x) = 0 o X =
a + b + c + l = 0 -4a + b + 12 = 0
Trang 11H a m so dat ciTc t r i t a i dig'm x = - 2 gi^ t r i ciTc t r i 1^
f(-2) = - 8 + 12 - 4 = 0
Do f ( l ) = 1 + 3.1 - 4 = 0
Vay d i e m A ( l ; 0) thuoc do t h i ham so f
15 (Bai 15 trang 17 SGK) Giai
(x - m ) ' y' = O o x = m - l hoSc x = m + 1
Vay vdi moi gia t r i ciia m, h a m so da cho dat ciTc dai t a i diem x m - 1 v;
dat ciTc tieu t a i diem x = m + 1
§3 GIA TR! L6N NHAT, GIA TRj NHO NHAT
C U A HAM SO
I \ 6 l D U I \ C A l ^ r^iiof
1 D i n h n g h i a : Gia stf ham so f xac dinh t r e n tap 7 ('J cz R)
* Neu ton t a i diem XQ € V sao cho: f(x) < f(xo) vdi moi x,, e V
Thi so M = f(Xo) goi la gia t r i IdTn n h a t cua ham so f t r e n V
K i hieu M = maxf(x)
X 7
* Neu ton t a i diem x,, s 7 sao cho: f(x) > fixo) vdi moi x e 7
T h i so m - f(xo) goi 1^ gia t r i nho n h a t cua h^m so f t r e n 7
K i hieu m = m i n f ( x )
• r ' ^ ^ ^ i t a c tim gia t r i Idn n h a t , gia t r i nho n h a t
* T i m cac diem x,, x-,, x^ thuoc (a; b) t a i do h^m so f c6 dao ham bfing
0 hoac khong c6 dao ham
, T i n h f(xi), f ( x 2 ) , fix J , f(a), f(b)
* So s^nh cdc gia t r i t i m di^cfc _ So Idn nhat trong cac gia t r i do la gia t r i Idn nhat ciia f t r e n [a; b] _ So nho nhat trong cac gia t r i do la gia t r i nho nhat ciia f tren [a, b]
J B A I T A P
16 (Bdi 16 trang 22 SGK) Giai
f(x) = sin''x + cos''x
Ham so xac dinh t r e n K
* f(x) = (sin^x + cos^x)^ - 2sin^xcos^x = 1 1 (2sinxcosx)
Trang 12Bang bien thien
, (4x + 5)(x + 2) - ( 2 x ' + 5x + 4)
f i x ) = — — 2x'' + 8x + 6
(x + 2)' ~ (x + 2)'
f (x) = 0 » 2x^ -i- 8x + 6 = 0 •» X = - 1 hoac x = -3 Bang bien thien:
t-5.1 1 + 4 "
11
3 Vay min f(xj = 2; max f(x) = —
x-io, I I x-ao: 11 3
0 fix) = X
-Ham so xac dinh tren (0; 2]
f'.(x) = 1 + = > 0, Vx 6 (0; 2L Bang bien thien
f'(x) fix) ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^
f(2) 2 i =
-2 -2 Vay max f(x) = —
x u l 0 ; 2 l 2
Ham so khong dat gia t r i nho nhat tren nOfa khoang (0; 2J
18 (Bai 18 trang 22 SGK) Gidi
a) y = 2sin^x + 2sinx - 1 H^m so xdc dinh tren M Dat t = sinx, - 1 < t < 1
y = fit) = 2t^ + 2t - 1
Tim gid t r i Idn nhS't wk gid t r i nh6 nhS't cua ham so y = fit) tren
doan [ - 1 ; 11,.do cung la gid t r i nhd nhat cua ham so da cho trSu
f ( t ) = 4t + 2; f ( t ) = 0 o 4 t + 2 = 0< T > t =
Trang 13Ta CO y = f(t) = - t ^ - - 1 + 5
2
1 1 l i f'(t) = - 2 t - - ; f'(t) = 0 c : > - 2 t - - = O c : t = - -
2 2 4 Bang bien t h i e n '*
Do do S(x) dat g i ^ t r i 16n nhat k h i M d v i t r i t r e n BC sao cho B M = ^ BC v^
gi^ t r i Idn n h a t ciia d i ^ n tich h i n h chuf nhat la —
8
20 (Bai 20 trang 22 SGK) Gidi
Tren moi don v i dien tich ciia mat ho c6 n con ca t h i sau mot vu, so' ca t r e n moi don v i dien tich mat ho t r u n g binh can nang
f(n) = nP(n) = 480n - 20n''^ (gam)
Xet ham so f(x) = 480x - 20x^ x e (0; +x) 'Bien so n e N* dUOc thay bkng bien x e (0; + « ) )
f (x) = - 4 0 x + 480 f'(x) = 0 » - 4 0 x + 480 = 0 « X = 12 Bang bien t h i e n
f(12) = 480.12 - 20.12^ = 2880
Trang 1524 (Bai 24 trang 23 SGK) Gidi
Goi M(x; x^) la diem bat k y cua parabol (.f)
f(x) dat gia t r i nho n h a t t a i diem x = - 1 , gia t r i nho n h a t la f ( - l ) = 5
Do do khoang each A M dat gia t r i nho n h a t k h i M d v i t r i diem M o ( - l ; 1) va
khoang each nho n h a t AM,, = Vs
25, (Bai 25 trang 23 SGK) Gidi
Van toe eua ca k h i bcfi ngUorc dong la (v - 6) knJgib
Thori gian de ca viiot khoang each 300 k m la 300
= 300 e 3v2 - 18v^ - v^
{ v - 6 ) ^
= 300 c. 2 X 1 ^ = 600 cv^ ^^-^^
( v - 6 ) ^ ( v - 6 ) ^ E'(v) = 0 <o V = 0 hoac v = 9 (v = 0 loai do v > 6)
g (Bai 26 trang 23 SGK) Gidi
S6' ngu'di n h i e m benh ke tii ngay xuat h i e n benh n h a n dau t i e n den ngay
y ii: fit) rz 90t - 3t^
y' = f "(t) = -6t + 90
y' = 0 « t = 15 Bang bien t h i e n
c) Toe dp t r u y e n b e ^ Idn hon 600
y > 600 c:^ sot - 3t^ > 600 e> t^ - 30t + 200 < 0 ci- (t - 15)^ < 5^
27 (Bdi 27 trang 29 SGK) Gidi
a) a) fix) = V 3 - 2 x
H a m so' xac d i n h t r e n [-3; IJ
- 1 f'(x) = = < 0 V d i moi x e
Trang 17§4 DO TH! CUA HAM SO
- PHEP TjNH TIEN HE TOA Dp
I\6l D U I \ CAIV I«ICf
1 P h e p t i n h t i e n h# t o a dp v a c o n g thijfc c h u y e n hp tpa dp
- Trong m a t ph^ng toa do Oxy cho diem
Kxo; yo), goi I X Y la h$ toa do mdi goc I va
hai true I X , l Y theo t h i l t u c6 cung vectot dan
vi i , j vdi hai true Ox, Oy
Goi M la diem bat k i eiia m a t ph^ng
(x; y) l a toa do diem M doi vdi he tpa do Oxy
(X; Y) l a toa do diem M doi v6i he tpa dp I X Y
[x = X + X ( ,
l y = Y + y„
C^e he thijfc t r e n goi la cong thiJc chuyen he tpa dp t r o n g ph6p t i n h tien
theo vecto 0 1
2 Phifofng t r i n h c u a do t h i do'i vdfi hp tpa dp m d i
Gpi y - f(x) l a phuong t r i n h cua do t h i {'^) do'i v 6 i he tpa dp Oxy, k h i do
phJOng t r i n h ciia do t h i {'f) doi v6i he tpa dp I X Y la Y = f(X + x,,) - yo
Cong thuTc chuyen h§ toa dp trong phep t i n h tien vectcf 0 1 \h
Phi/otng t r i n h duorng Parabol doi vdi he tpa dp I X Y la:
Cong thufc chuyen he tpa dp t r o n g phep t i n h tien vectof 0 1 la
Phucng t r i n h Parabol do'i vdi he tpa dp I X Y \k:
Y - - = - ( X + 1 ) ' - ( X + l ) - 3
2 2 hay Y = i ( X ' + 2 X + 1 ) - X - 1 - 3 + -
Cong thufc chuyen h f tpa dp trong phep t i n h tien vectcf 0 1 l a
Phuorng t r i n h Parabol doi vdi he tpa dp I X Y 1^:
8 X^ + - X + 2 n
4 64 16
Y = X + A - 4 X ' - X - — - — = ^ Y = - 4 X 1
8 16 16 d) y = 2x2 - 5
D i n h I X , = 0
y, = - 5 T a C O 1(0; - 5 ) Cong thiJc chuyen he tpa dp trong phep t i n h tien vectof 0 1 la Phuong t r i n h Parabol doi vdi he tpa dp I X Y la:
Y - 5 - 2 X ' - 5 hay Y = 2 X '
X
y
X
Y - 5
Trang 1830 (Bdi 30 trang 27 SGK) Giai
y = Y - 1 Phirang t r i n h diTcfng cong doi vdi h$ toa dp I X Y la
vay t r d n khoang (1; +oc) {'f) n k m phia t r e n (d)
^l (Bai 31 trang 27 SGK)' Gidi
fx = X - 2 Cong thilc chuyen he toa dp trong phep t i n h tien vector Ol la: _ y + 2
Phuong t r i n h diTdng cong Cf) doi vdi he toa dp I X Y la:
32. (Bai 32 trang 28 SGK) Gidi
a) Thifc h i f n ph6p t i n h t i e n vectcf O I vdi 1(1; 1) Cong thiJc chuyen h# toa do trong ph6p t i n h t i e n n^y 1^ X = X + 1
y = Y + 1 Phi/cfng t r i n h cua do t h i doi vdi hp toa dp I X Y 1^
+ 1 => Y = —
X + 1 - 1 X Dat Y = « x ) = -
Tap xac d i n h cua h a m so n^y 1^ 7 = R \: VX e 7, - X G 7
Thirc h i p n ph6p t i n h t i e n vectcf 0 1 vdi I ( - l ; 3)
Cong thiJc chuyen he toa dp trong phep t i n h t i e n n^y \k
Phiforng t r i n h ciia do t h i doi vdi he toa dp I X Y 1^:
Do d6 t a m doi xilng cua ham s6' da cho la I ( - l ; 3)
(Bai 33 trang 28 SGK) Giai
Cong thufc chuyen hp tpa dp trong phep t i n h t i e n vectcf 0 1 la
Trang 19=> Y = aX + axo + b + - axo - b=:>Y=aX + —
X X Dat Y = f(x) = aX +
T|ip xdc dinh cua ham so n^y la 7 = K \1
Vx e 7, -X 6 ( /
«-X) = a(-X) + -X a x - = -f(x)
Vay hkm so Y = aX + — la ham so le nen nhan goc toa do I la tarn doi xiiCng
§5 DJdNG TIEM CAN CUA DO THj HAM SO
1 Dadng th^ng y = yo difoc goi la diforng tiem can ngang (goi t^t la ti$m can
ngang) cua do thi ham so y = f(x) neu: lira f(x) = yo hoac lim f(x) = yo
y = flx)
2 Du6ng thing x = XQ dUdc goi la ducmg tiem can diJng (gpi tit la ti$m can
drfng) ciia do thi ham so y = f(x) neu:
limf(x) = + 0 0 hoac limf(x) =+oo hoac
limf(x) =-co hoac limf(x) = -3o
3 Difcfng thing y = ax + b (a # 0) dUcfc gpi la dudng tiem can xien (goi tit la
ti^m can xien) cua do thi ham so' y = f(x) neu:
lim [f (x) - (ax + b)] = 0 hoac lim [f (x) - (ax + b)] = 0
3
y = fix)
X
Chii y: De xdc dinh cac he so a, b trong phifong trinh diTdng tiem can xien
y = ax + b, ta CO the sut dung
, f(x) Cong thiTc a = lim
f(x) , b = lim [f (x) - ax]
(Khi a = 0 thi ta c6 tiem can ngang la y = b)
* Vi lim y = +co va lim y = - Q O
Nen ducfng thing x = — la difcfng tiem can dufng ciia do thi ham so
Trang 20c) y = X + 2
X - 3
Hkm so xdc dinh tren R \|
* V i lim y = + x va lim y = -oo
Nen dudng th^ng x = 3 1^ ti^m ogn diifng ciia do thi ham so' (khi x ^ 3
Ham so xac dinh tren K \j -—
i V i lim y = -00 vk lim y = +oo
Nen dxibng t h i n g x = - — la tifm can diJng ciia do thi hkm so (khi
7
4
1 7 Vay difdng thang y = - x - - la ti^m can xien ciia do thi ham so (khi
-Ham so xkc dinh tr§ri R \I
* Vi hm y = +oc vk lim y = -oo x-.<-n" x->i-i)*
Nen du6ng thing x = -1 la tiem can diltig cua do thi ham so (khi x ~> (-1)
vk khi X -oo)
x' + 2
b ) y =
x ' - 2 x Hkm so xac dinh tren R \; 21
• V i lim y = -oo vk lim y = +oo
Trang 21H a m so xac dinh tren (-00; - 1 ] [ 1 ; +00)
Tiem can xien c6 dang y = ax + b
V i l i m y = l i m ( x + Vx^ +1) = l i m
1 = 0
" ^ ^ x - V x ^ + l Vky dudng t h i n g y = 0 Ik t i ^ m can ngang cua d& t h i h k m so (khi x -> -00)
Trang 22V a y dudng t h i n g y = - x - i Ik tigm c a n xien cua do t h i h k m so (khi x -> -oc)
Trang 23Diforng thSng y = - x la t i e m can xien ciia do t h i h ^ m so' (khi x -oo)
Phufong t r i n h m ddi v6i h f toa dp IXY
Vay h a m so' Y - la ham so' le nen do t h i nhan go'c 1(3; 4) lam tam
X doi xiifng
39 (Bai 39, trang 36 SGK) Gidi
C6ng thufc chuyen he toa dp theo phep t i n h t i e n 01
c) Phiforng t r i n h cua (f) doi vdi he toa dp IXY
Trang 24Dat Y = fiX) = ^ , tap xac dinh 7 = K \: Vx € '/, - X e 7
• Elide 1: T i m tap xac d i n h cua hkm so
• B i f d c 2: X6t sir bien t h i e n cua hkm s6
* T i m gi(Ji h a n t a i v6 cifc vk gidi han v6 cifc (n^u c6) cua hkm so T i m
cac dudng t i g m can cua do t h i (neu c6)
* Lap bang bien t h i e n cua hkm so bao gom:
T i m dao hkm cua hkm so
- X6t dau dao hkm, x6t chieu bien t h i e n vk t i m cUc t r i cua hkm s6 (n§'u c6), dien k e t qua vko bang
• BvCdc 3: Ve do t h i cua h a m so
* Ve ckc dufcfng t i | m can cua do t h i (neu c6)
* Xac d i n h mpt so diem dac bi#t ciia do t h i (giao d i e m vdi cdc true toa
dp, neu phep toan khong philc tap)
* N h a n xet ve do t h i (chi ra true vk t a m doi xijfng cua do t h i (neu c6))
Neu hkm so y = f(x) c6 dao h a m cap hai t r e n mpt khoang chiia diem X Q ,
f "(xo) = 0 va f "(x) doi dau k h i x qua Xo t h i d i e m U(xo; f(xo) la mot d i e m
uo'n cua do t h i hkm so y = f(x)
* Ghi chu: Do t h i h a m so bac ba y = ax^ + bx^ + cx + d (a ;^ 0) h a m c6 mot
diem uon va diem do \k t a m doi xiifng ciia do t h i
3 H a m so' t r u n g phi^o^ng y = ax'* + bx^ + c (a 5* 0)
40 (Bai 40, trang 43 SGK) Gidi
a) Khdo sat sU bii'n thien va ve do thi ham s6 y = + 3x^ - 4
1- H k m so xac d i n h t r e n R
2- Su bien t h i e n cua h k m so:
a) Gidi han t a i v6 ciTc
l i m y = -00 va l i m y = +00 b) Suf bi§'n t h i e n , ciTc t r i
Ta c6: y' = Sx^ + 6x = 3x(x + 2)
y' = 0 o 3x(x + 2) = 0 c : > x = 0 hokc x = - 2
Trang 25+ Hkm so d6ng hi&'n t r e n m5i k h o a n g ( - o c ; -2) vk (0; +00), n g h i c h bie'i
4 1 (Bai 41, trang 44 SGK) Gidi
a) Khdo sat sU bien thien va ve y = -x^ + 3x^ - 1
42 (Bdi 42, trang 44 SGK) Gidi
a) Khdo sat sil bien thien va ve do thi y = ^ x ^ - x^ - 3 x - -
3 3 1- H a m so xac d i n h t r e n K
2- S u b i e n t h i e n ciia h a m so'
^) Gidri h a n t a i v 6 cifc:
• l i m y = -00 • l i m y = +00 b) S u b i e n t h i e n , cUc t r i :
• T a c6 y ' = x^ - 2 x - 3
y ' = 0 c:> X = - 1 h o a c x = 3
H a m so' d o n g b i e n t r e n cac k h o a n g ( - 0 0 ; - 1 ) v a (3; +00); n g h i c h b i e n t r e n
k h o a n g ( - 1 ; 3)
Trang 26* H^m s? dat cifc dai tai x = - 1 , gi^ t r i cue dai y ( - l ) = 0
* Hkm so dat cUc ti§'u tai x = 3, gi& t r i cUc tieu y(3) =
c) Bang big'n thien
khi X qua x = 1 Vay diem U 1, - — 1^ V 1,
-d i l m u6'n cua -do thi
• Giao diem cua do thi vdi true tung
3)
0; -5
NMn xet: Do thi hkm so nhan U
l^m tam doi xiifng
b) Khdo sat sUbiin thi&n dS thi id vey = - 3x + 1
1 H^m so xac dinh tren K
2 Sif bi§'n thien cua h^m so'
a) Gi6i han tai v6 cifc
• limy = -00 , limy = +oo
b) Sir bien thien, cUc t r i
• y' = 3 x ^ - 3 , y' = 0 » x = ±l
• H^m so dong bien tren cac khoang (-oo; -1) vk (1; +oo), nghich bien tren
khoang (-1; 1)
• Ukm so dat ciTc dai tai x - - 1 , gia t r i cUc dai y ( - l ) = 3
• Ham so dat cifc tieu tai x = 1, gik t r i cufc tieu y(l) = - 1
c) Bang bien thien
y" = 0 tai X = 0 va doi dau tii am sang
dUcJng khi x qua x = 0 Vay diem U(0; 1)
la diem uon cua do thi
• Giao diem cua do thi vdi true tung (0; 1)
Nhdn xet: Do thi hkm so nhan diem U(0; 1)
Ikm tam doi xufng
c) Khdo sat sU biin thiin dS thi y = - g? + - 2x - -
3 3
1 H^m s6' xkc dinh tren K
2 Si^ bien thi§n cua ham so"
a) Gi(Ji han tai v6 cUc
• limy = +=» ; limy = -oo
b) S\i bien thien, cuTc t r i
, / = -x^ + 2x - 2 < 0, Vx 6 R
* Ham so nghich bien tren E
c) Bang bien thien
Nhdn xet: Do thi ham so nh$n U 1; - 24) 23 lam tfim doi xiifng
d) Khdo sat sU biin thiin vd ve do thi hdm so y = x^ - 3x^ + 3x + 1
1- H^m so xac dinh tren M
2- Sir bien thien cua h^m so'
a) Gi6i han tai v6 ciTc
• limy = -00 , limy = +oo
b) Sir bien thien, cire t r i
• y' = 3x^ - 6x + 3, y' > 0, Vx l
• H^m so dong bien tren M \
c) Bang bien thien
y
— 0 0 '
Trang 27'3 D6 thi
y " = 6x - 6, y " = 0 • » X = 1
• y" = 0 khi X = 1 va y" doi dau t\i am sang
difotng Khi x qua x = 1, nen U ( l ; 2) 1^
diem uon cua do thi
• Giao diem cua do t h i vdi trucf tung (0; 1)
Nhdn xet: Do thi h^m so nhan U ( l ; 2)
l^m tSm doi xufng
43 (Bai 43, trang 44 SGK) Gidi
a) Khdo sat sU biin thiin va ve dS thi y = -x'' + 2x^ - 2
1 H^m so xAc dinh tr§n R
2 Sir bien thien ciia hkm so
a) Gidi han tai v6 cifc:
• limy = -00 ; limy = -oo
b) Sir bien thien, ciTc t r i :
• y' = -4x^ + 4x = -4x(x^ - 1)
y' = 0 khi X = 0 hoac x = ±1
• Ham so dong bien tren cdc khoang (-oo; -1) (0; 1); nghich bien tren
cdc khodng (-1; 0) va (1; +oo)
• Ham so dat cUc dai tai ode diem x = - 1 gia t r i circ dai y ( - l ) = 1 tai
X = 1, gid t r i ciTc dai y(l) = - 1
• Ham so dat circ tieu tai diem x = 0, gik t r i cUc t i l u la y(0) = -2
2 ' 9 la
3 9 hai diem uon cua do thi
Giao diem ciia do t h i vk true tung 1^ (0; -2)
Nhdn xet: Ham so' da cho la ham so chSn
nen nhan true tung lam true doi xufng
}j) Bien ludn so nghiSm cua phucmg trinh -x^ + 2x^ - 2 = m (1)
Dat y = -x* + 2 x ^ - 2 c6 do thi da ve d cau a)
y = m c6 do thi 1^ dudng thing
Nghi$m s8' ciia phuotng trinh (1) 1^ ho^nh do giao diem cua hai do thi
• Neu m > - 1 thi phUOng trinh (1) v6 nghi#m
• Neu m = - 1 thi phUcrng trinh (1) c6 hai nghiem
• Neu - 2 < m < - 1 thi phucmg trinh (1) c6 bon nghi$m
• Neu m = -2 thi phucJng trinh (1) c6 ba nghi|m
• Neu m < -2 thi phUcfng trinh (1) c6 hai nghiem
44 (Bdi 44, trang 44 SGK) Gidi
a) Khdo sat sU bien thien va ve do thi y = x^ - 3x^ + 2
1- Ham so xac dinh tren R
2 Sir bien thien cua ham so'
a) Gidi han tai v6 cUe
Trang 28• H^m so' dat cUc dai tai x = 0 gi& t r i cue dai y(0) = 2 H^na sfl' d^t CU(
h^m so chan nen nhan true tung
\k true dS'i xiirng
h) Khdo sat su bien thiSn vd ve dS thi y =-x^ - 2x^ + 1
1 H^m so x&c dinh tren K
2 Sii bien thi§n cua h^m s6'
a) Gidri han tai v6 ciTc
• limy = -00 ; limy = -co
b) Sir bi^n thien, cifc t r i
• y' = -4x^ + 4x = -4x(x^ + 1)
y' = 0 o X = 0
• Hkm s6' d6ng bi§'n tren khodng (-00; 0), nghich bi§'n tren khodng (0; +00)
• Rkm s6 dat cue dai tai x = 0, gik t r i eUc dai y(0) = 1
c) Bang bi§n thign
m L U Y f t M T A P
45, (Bcii 45, trang 44 SGK) Giai
a) Khdo sat sU bien thien vd ve dS thi y = - 3x^ + 1
1 H^m so xac dinh tren iR
2 Sif bien thi§n ciaa h^m so'
a) Gidi han tai v6 ciTc
• Ham so dat cifc tieu tai x = 0, gik t r i cUe dai y(0) = 1, ham so dat eiJc
ti^u t a i x = 2, gik t r i eifc t i l u y(2) = -3
c) Bang bien thien
y" = 0 tai x = 1 v a d6i dSu khi x qua x = 1
n§n U ( l ; -1) la dilm u6n cua do thi
• Giao dilm cua d6 thi vdi true tung (0; 1)
Nhdn xet: Do thi nh^n d i l m U ( l ; -1)
l^m tarn dol xufng
b) Biin ludn s6 nghiem philang trinh
x-"* - 3x^ + m + 2 = 0 (1) o x^ - 3x^ + 1 = - m - 1
D a t y = x^ - 3x^ + 1 c6 do thi i'f) da ve d cSu a),
y = - m - 1 l a diT&ng thing (d) song song true hoanh
Nghiem cua (1) l a hoanh dp giao dilm cua (d) v a ( ^
Ta c6:
• Neu - m - 1 > 1 hoae - m - 1 < -3, nghia l a m < -2 hokc m >
phUdng trinh (1) c6 mpt nghiem
• Neu - m - 1 = 1 hoac - m - 1 = -3, nghia l a m = -2 hokc m =
phifong trinh (1) c6 hai nghiem
• Neu -3 < - m - 1 < 1, nghia Ik -2 < m < 2 thi phuong trinh (1) c6 ba nghifem
2 thi
2 thi
Trang 2946. (Bdi 46 trang 44 SGK) Gidi
a) Ho^nh dS giao diem cua d6 thi vk true ho^nh 1^ nghi$m phUcfng trinh
(x + l)(x^ + 2mx + m + 2) = 0
"x = -1
+ 2mx + m + 2 = 0 (1)
Do t h i h ^ m so e^t true hoknh t a i 3 d i l m phfin biet k h i chi k h i phuang
t r i n h (1) c6 h a i n g h i f m ph§n bi§t kh^c - 1 nghia 1^: ^ (trong do
fix) = x^ + 2mx + m + 2)
- m - 2 > 0 fm < - 1 ho&e m > 2
it 0 [m ?t 3 Vay m e (-oo; - 1 ) u (2; 3) u (3; +oo)
b) Khdo sat sU bien thien vd ve dS thi y = (x + l)(x^ - 2x + 1)
hay y = x^ - x^ - X + 1
1 H ^ m so' xdc d i n h t r e n K
2 Sif bien t h i e n h ^ m so'
a) Gi6i h a n t a i v6 cUc: • l i m y = -oo ; h m y = +oo
.3' 27J 1^ diem uon cna d6 thi / : 27 1 \6 X
• Giao diem eua do thi vdi true tung (0; 1)
X
• y = 0 <:> (x + l)(x - 1)^ = 0 o X = ±1 / -!• 3 ^
Do thi c^t true ho^nh tai ( - 1 ; 0) ( 1 ; 0)
Nhdn xet: Do thi nhfin di^m U Ikm tfim d6'i xufng
la' 27)
4 7 (Bdi 47, trang 45 SGK) Gidi a) Khdo sat sU bien thien vd ve do thi hdm so y = x'' - 3x^ + 2
1 Hkm so xac dinh tren M
2 Sii bi§'n thien cua ham so', a) Giofi han tai v6 cUc
• limy = +00 b) Sif bien thien, cUe tri:
• H ^ m so' dat eife t i l u t a i e^c diem ± - — , gid t r i c\ic tieu y
• Hkm so dat cifc dai t a i x = 0, gid t r i cUc dai Ik y(0) = 2
Nhdn xet: H a m so dfi eho Ik ham so "2 2
ehSn nen n h a n true tung la true do'i 1 xiing
Trang 30-b) B6 thi hdm s6 dd cho di qua diim ( XQ; yo) khi vd chl khi
Trang 31276 2 3 1 2^ 13
y = X + — + — => y = X + - —
9 9 36 9 12 Phi^cfng t r i n h tiep tuyen t a i diem uo'n U2
§7 KHAO SAT SL/ BIEN THIEN VA VE DO THj CUA
MOT SO HAM PHAN THlTC HQU TI
Moi Dui\ CAN mui
49 (Bai 49, trang 49 SGK) Gidi
a) Khdo sat si/ bien thien vd ve do thi y = x- 2
2x + l
1 H a m so xdc d i n h t r e n K \
2
2 Sir bien t h i e n cua h ^ m so'
a) Gidi h a n t a i v6 cUc, gidi han v6 cUc, dudng t i $ m cfin
X -> ' r va k h i X —>
v 2, 2> /
V i l i m y = +00 l i m y = -00 n6n dUdng t h i n g x = -— 1^ t i e m cfin dufng cua do t h i h ^ m so da cho k h i
• V i l i m y = —, n e n difdng t h i n g y = — l a ti§m can ngang cua do t h i h^m
• Giao diem cua do t h i v d i true tung (0; - 2 )
• Giao diem cua do t h i vdi true ho^nh (2; 0)
( \ Nh&n xet: Do t h i nhSn giao d i l m I cua hai t i e m can l a m t a m doi xufng
( I l\
b) ChuTng m i n h I — ; - 1^ t d m
doi xiJfng cua do t h i Cong
thijfc doi he t p a dp theo ph6p
5
Y =
-4X Dat Y = « X ) = -4X
5
, tap xdc d i n h ^ = K \ Vx e - X 6 V
5
Trang 32Hkm so Y = — Ik hkm s6 nen nh$n goc I
Vay giao d i e m I
4X ' 1 1^
2 Sir bi§n t h i d n cua h ^ m s6'
a) G i d i h a n t a i v6 cue, gidi h a n v6 cUc, dudjig t i f m cSn
* l i m y = - 0 0 l i m y = +oo , nen do t h i n h a n dudng t h ^ n g x = 1 1dm tieni
• Giao d i e m ciia do t h i vdi true tung (0; - 1 )
• Giao d i e m v d i true hodnh ( - 1 ; 0)
Nhdn xet: Do t h i n h a n giao diem 1(1; 1)
ciia h a i ti§m cdn Idfn t d m do'i xufng
b) Khdo sat sU biin thiSn vd ve dS thi y = 1
2x + 1
y = l - 3 x
1 H d m so xdc d i n h t r e n
2 Su bien t h i e n ciia h d m so'
a) Gidi h a n t a i v6 cUc, gidi h a n v6
cUc, dudng t i ^ m can
• V i l i m = +00 vd l i m y = - o o , nen dUdng t h i n g x = — Id t i | m edn diing
• Giao diem cua do t h i vdi true tung (0; 1)
I Giao diem cua do t h i vdi true hodnh
cua h a i ti§m c a n 1dm tarn doi xufng
51 (Bdi 51, trang 49 SGK) Gidi
a) Khdo sat sU biin thiSn vd ve dS thi y =
Trang 33c) B&ng bien thien
* Giao diem cua do t h i v6i true tung (0; 2)
Nhdn xet: Do t h i n h a n giao diem I ( - 2 ; 3)
cua hai t i e m can 1^ t a m doi xufng
b) Chiing minh I Id tam dSi xvtng
Toa dp giao diem I cua hai t i e m can
cua do t h i la nghiem ciia he phuong
y = - m c6 do t h i la difcrng t h ^ n g (d) song song true hoanh
- Nghiem cua phifcfng t r i n h (1) \h hoanh do giao diem cua (d) \h {f), ta c6:
* Neu - m < - 7 hoac - m > 1 nghia la m > 7 hoac m < - 1 t h i phi/cfng t r i n h
(1) c6 hai nghiem phSn biet
* Neu - m = - 7 hoac - m = 1 nghia 1^ m = 7 hoac m = - 1 t h i phucfng t r i n h
(1) CO mot nghiem
* Neu - 7 < - m < 1 nghia la - 1 < m < 7 t h i phuong t r i n h (1) v6 nghiem
- 3x + 6
x - 1
52 (Bdi 52, trang 50 SGK) Gidi
a) Khdo sat sU bien thien vd ve do thi hdm s6 y =
1 Ham so xac d i n h t r e n M \
2 Sii bien thi§n ciia h a m so
a) Gidi h a n v6 cUc, gidi h a n t a i v6 cUc, di/dng t i f m can
la t i e m can xien ciia do t h i h a m so ( k h i x - » - o o va k h i x -> + Q O )
b) Sii bien t h i e n , cUc t r i :
, x^ - 2x - 3
y = y' = 0 o <=> X = - 1 hoac X = 3
- H a m so' dat cvtc dai t a i x = - 1 , gia t r i cifc dai y ( - l ) = - 5 , h a m so' dat
cifc tieu t a i x = 3; gia t r i c U c tieu y(3) = 3
• Giao diem cua do t h i vdi true tung (0; -6)
Nhan xet: Do t h i n h a n giao diem 1(1; - 1 )
cua hai t i e m c$n l a m t a m do'i xufng ,
b) Khdo xdt siX bien thien vd ve do thi cua
, , 2x^ - X + 1
ham so y =
1 - X
1- H a m so xac d i n h t r e n K \
2- Sir bien thi§n ciia h a m so
a) Gidfi han t a i v6 cifc, gidi han v6 c U c ,
Trang 34- H ^ m so dat cUc tieu t a i x = 0, gid t r i ciTc t i ^ u y(0) = 1
- H a m so dat cifc d a i t a i x = 2, gid t r i cifc d a i y(2) = - 7
+ Giao d i e m cua do t h i vdti true tung (0; 1)
Nhan xet: Do t h i nhgn giao d i l m 1(1; - 3 ) cua
hai ti^m cgn cua do t h i 1km tam d6i xilng
c) Khdo sat sU bien thiin vd ve dS thi cua
, 2 x ' + 3x - 3
ham s6 y =
x + 2
1 H k m so xkc d i n h t r e n K \
2 Sif bien t h i e n ciia h a m so'
a) Gidri h a n t a i v6 cUc, gidi h a n v6 cifc,
Nhdn xet: Do t h i nhan giao diem I ( - 2 ; - 5 ) cua
hai t i | m can ciia do t h i 1km tam doi xiiCng
d) Khdo sat sU bii'n thien vd ve dS thi hdm s6 y = -X + 2 + —^—
x - 1
1 H k m so xac d i n h t r e n 1 \
2 Sif bien t h i e n ciia h k m s6'
a) Gidfi h a n t a i v6 cifc, gidi h a n v6 cifc diTdfng ti§m can
Trang 355 3 (Bdi 53 trang 60 SGK) Gidi
Nhdn xet: D o t h i n h a n giao d i e m 1(2; 1) cua
h a i t i e m can cua do t h i l a m t a m d o i xilng
f l^
b) Phuang trinh tiep tuyen.tqi A
(giao diem cua do thi vd true tung)
4-1
y- 3- 2-
54 (Bdi 54 trang 50 SGK) Gidi
a) Khdo sat sU bien thien vd ve do thi (JO cua ham so
^k h i n h d o i xufng c u a do t h i ,W q u a t r u e h o ^ n h ( n 6 t v e c h a i n , c h a m )
Trang 3655 (Bai 55 trang 50 SGK) Gidi
a) Khdo sdt su bii'n thiin va ve dS thi hdm s6'
2
y = X r
X ]
1 H^m s6' xdc dinh tren R \l "
2 Su bien thien cua ham so'
a) Gidi han tai v6 cUc, gidi han v6 cifc, diforng ti§ni can
• lim y = +00 va lim y = —<»
• lim y = +00 lim y = nen dudng th^ng x = 1 la tiem can diiug
x->r x-»r
cua do thi ham so (khi x 1" va khi x -> 1^)
• lim[y - x1 = 0 , n%n diTdng th^ng y = x 1^ tiem can xien ciia do thi
X - * ± X
ham so (khi x -> -oo va khi x -» +x)
b) Su bid'n thien
2 y' = l + > 0, vdi moi x 1
Nhan x§t: Do thi nhan giao dilm 1(1; 1)
hai tiem can cua do thi Ikm tarn doi xilng
b) Phuang trinh tiep tuyi'n cua do thi (f)
Phifdng trinh tiep tuy§'n (d) cua do thi (?0 c6 dang y = ax + b
Do (d) di qua A(3; 3) nen 3 = 3a + b => b = 3 - 3a
De dudug thdng (d) la tiep tuyen ciia ('f) khi vli chi khi
A' = 0
a * 1 <=>
a' - 4a •+ 3 = 0
a ;^1 o a = 3
Vay tiep tuyen vdi ('f) qua A(3; 3) y = 3x - 6
66 (Bai 56 trang 60 SGK) Gidi
1/ Khao sdt sU bien thien vk ve do thi ('<0 ham so'
y =
X + 1
1 Ham so xkc dinh tren K \l
2 Su bien thien cua ham so a) Gidi han tai v6 cUc, gidi han v6 cUc, di/dng t i f m cgn
thi ham so' (khi x -> - x v^ khi x + x )
b) Su bien thien, cUc tri:
- Ham so dat cUc dai tai x = - 2 , gik t r i cUc dai y(-2) = -4
H^m so' dat cUc t i l u tai x = 0, gik t r i cUc tieu y(0) = 0
c) Bang bien thien
+ Giao diem cua do thi vk true ho&nh (0; 0)
Nhan x6t: Do thi ham so nhsin giao diem
K - l ; -2) cua hai ti#m ckn cua d6 thi l^m
tSm doi xUng
Trang 37b) Suy ra cdch ve do t h i CiO h a m so' y =
x + 1
* GiOr nguyen phan ciia do t h i CO nkm phia
t r e n true hoanh va lay doi xufng phan
ciia (VO n k m phia dUcSi true hoanh qua
S6' n g h i e m cua phuong t r i n h (1) b^ng so giao diem cua hai do t h i
2 SiJf t i e p x i i c c u a h a i dvCbng cong
* H a i dirdng cong y = f(x) y = g(x) tiep
xuc nhau ki|?i vk chi k h i he phi/ang t r i n h
f ( x ) = g(x)
f'(x) = g'(x)
he phifcttig t r i n h 1^ hoanh do tiep diem
cua hai dudng cong do
* N h a n x6t: Du^ng t h ^ n g y = px + q la tiep tuyen ciia Parabol
y = ax^ + bx + c k h i \k chi k h i phUcfng t r i n h hoanh dp giao diem
£ix^ + bx + c = px + q hay ax^ + (b - p)x + c - q = 0 c6 nghiem kep
c6 n g h i e m vk n g h i f m ciia
^ B A I T A P
57 (Bdi 57 trang 55 SGK) Giai
a) Khao ski sif bien t h i e n ve do t h i (90 h ^ m so'
- H a m so dat cUc dai t a i x = - 1 , gid t r i ciTc dai f ( - l ) = 2
H a m so dat cifc tieu t a i x = 0, gia t r i c\ic tieu f(0) = 1
* Giao diem ciia do t h i v^ true tung (0; 1)
Nhan xet: D 6 t h i nhSn diem uon U ^ 1 f 1^ tarn doi xiing
Trang 38a) 00 h a n t a i v6 cifc, gidi h a n v6 cifc, difdng t i e m can
*• n y - va lira y = -oc, nen difdng t h a n g x = 1 1^ t i e m can diifng
•a*El<^,hi h a m so' ( k h i x - 1 " va k h i x -> -1*)
* v»n y = 2, nen Axibng t h ^ n g y = 2 la t i e m can ngang cua do t h i hkm so
* Giao diem cua do t h i wk true ho&nh
N h a n xet: Do t h i n h a n giao diem
I(—1; 2) ciia hai t i e m can cua do
t h i \h t a m do'i xilng
• • X
b) T i m gia t r i ciia m Phuoug t r i n h ciia dirdng t h a n g (dm)
y - 2 = m(x + 2) hay y = mx + 2m + 2 Hoanh dp giao diem ciia (dm) va di/cfng cong da cho la nghiem ciia phifoug trhili
DiicJng thang (dm) c^t diTdng cong da cho tai hai diem thuoc hai nhanh ciia
no k h i va chi k h i phUomg t r i n h (1) c6 hai nghiem X i , x^, va x, < - 1 < Xv
B a t X = t - 1, phifong t r i n h (I) \,va t h a n h :
m(t - 1)- + 3m(t - 1) + 2m + 3 = 0
o m(t- - 21 + 1) + 3mt - 3 m + 2m + 3 = 0 <o m t " + mt + 3 = 0 (2)
Phuang t r i n h (1) c6 hai nghienf x, Xy va xi < - 1 < x^
K h i phUdng t r i n h (2) c6 hai nghiem t ] , t^ va t, < 0 < t2 nghia la
Trang 39hay y = - x
61 (Bdi 61 trang 56 + 57 SGK) Gidi
Hoanh dp t i e p diem cua hai Parabol la nghiem cua h | phJdng t r i n h
g t a n a cung la nghiem cua (1)
1 Vay vdi moi a e hai Parabol tiep xiic nhau
Hoanh dp tiep diem 1^: x = ———
g t a n a Tung dp tiep diem: y = - • ^
la tiep diem cua hai Parabol vdi moi a e 0 ; ^
2
^ L U Y E I V T A P
62 (Bai 62 trang 57 SGK) Gidi
a) Kh^o sat sif bien t h i e n ve do t h i h a m so'
* Giao diem cua do t h j vk true tung (0; -1)
* Giao diem ciia do t h i va true hoanh (1; 0) Nhan xet: Do t h i nhan giao diem I ( - l ; 1)
cua hai tiem can cua do t h i la tarn doi xuCng
b) Chilng minh I ( - l ; 1) Ik tam doi xuiig ciia do t h i Cong thtfc doi he toa do theo phep tinh tien vectcf 0 1
|x = X - l '
ly = Y + l
Trang 40PhUOng trinh do thi do'i vdi h$ tea do I X Y
X - l + l
hay Y = - l ^ Y = ~^
-2, Dat Y = f(X) = t i p xac dinh V = R \: V X e V, - X € (/
X
I M m so Y = Ik hkm so IS n6n nhan goc toa dO I ( - l ; 1) lam tam dS'i xufng
A
Vay do thi h^m so da cho nhan giao diem I ( - l ; 1) ciia hai difdng ti^m can
cua do thi l a t a m do'i xuTng
63 (Bdi 63 trang 57 SGK) Gidi
a) Khao sat s; i-ien thien va ve do thi (.//)
2 Sif bien thien cua h a m so'
a) Gidi h a n tai v6 cifc, gidi han v6 cifc, difdng tiem can
* lim y = - M l i m y = +x, nen di/dng t h i n g x = - - la t i § m c^ln dufng
ciia do thi h^m so (khi x -)• - - va k h i x - » - - )
2 2
• lim y = ^ ' J^^n dirdng thang y = ^ la tiem can ngang ciia do thi ham
so' (khi x - « va k h i x -> +oo)
b) S u bie'n thien
-3 y' = T < 0, v(Ji moi X —
* Giao di^m cua do thi vdi true tung (0; 2)
# Giao diem cua do thi vdi true hoanh (-2; 0) Nhan x6t: Do thi nhan giao diem 11 - - ; -
^ 2 2j
hai ti§m can cua do thi lam tam d6'i xilng
b) Durdng t h i n g (dm) di qua diem (xo; yo)
k h i v a chi khi
yo = mxo + m + 1 <=> m(xo + 1) - (1 + yo) = 0 (1)
Do thi di qua diem (XQ; yo) v(Ji moi m khi va chi k h i (1) nghi^m diing vdi moi m, nghia l a x„ + 1 = 0
y o + i = o
Xo = -1 y„ = -1 Vay vdi moi gia tri cua m, dubng t h ^ g (dm) luon di qua diim co dinh A ( - l ; - 1 )
—1 + 2 1 Mat khac - 1 = _ , < = > - ! = —- o - 1 = - 1 ( d i n g thiic diing)
2 ( - l ) +1 -1 Vay A thuoc (.J^ hay du:dng t h i n g (dm) luon di qua diem co' dinh A ( - l ; - 1 ) cua (./(O k h i m bie'n t h i § n
c) Hoanh do giao diem ciia (dm) y = mx + m - 1 va (.>lO l a n g h i | m phuTcfng trinh
Hai n h a n h cua (./rO n i m d hai ben ciia dudng ti^m can duTng x = - ^'
Diem A thupc nhdnh trai cua (./(O vi XA = - 1 < - - ^ Dudng t h i n g (dm) da cho qua diem A ( - l ; - 1 ) cua nen (dm) c i t (.//) tai hai diem cimg mot nhanh cua (,//) klii va chi k h i phu'ong trinh (2) co nghiem x < - — v a x ; ^ - ! nghIa l a