1B MN TON NG DNG - HBK-------------------------------------------------------------------------------------

BGT TON 1

BI 3: HM S - HM S CP TNH LIN TC

TS. NGUYN QUC LN (12/2006)

2NI DUNG---------------------------------------------------------------------------------------------------------------------------------

1- KHI NIM HM S. CC CCH XC NH HM S

2- HM S NGC

3- HM LNG GIC NGC

4- HM HYPERBOLIC

5- LIN TC TI 1 IM. LIN TC 1 PHA

6- PHN LOI IM GIN ON

7- HM LIN TC TRN ON. NH L TRUNG BNH

3HM S-----------------------------------------------------------------------------------------------------------------------------------

Min xc nh Df . Min gi tr Imf: {y=f(x), xDf}. VD y=sinx

RX RY Hm s y = f(x): X R Y R:

Quy lut tng ng x X y Y. Bin s x, gi tr y. Tng quan hm

s: 1 gi tr x cho ra 1 gi tr y

i lng A bin thin ph thuc i lng B:

i sng: Tin in theo s kwh tiu th, givng trong nc theo th gii

K thut: Ta cht im theo thi gian

Tng

quan

hm s

4CC CCH XC NH HM S-----------------------------------------------------------------------------------------------------------------------------------

Bn cch c bn xc nh hm s: M t (n gin) - Biu

thc (thng dng) Bng gi tr (thc t) th (k thut)

vM t: n gin, d pht hin tng quan hm s

Trng lng

Gi tin

20 gr

18.000

20 40 gr

30.000

VD: Bng cc ph gi th bng bu in i chu u

v Bng gi tr: Thc t, r rng, thch hp cc hm t gi tr

VD: Ph gi th bu in i chu u ph thuc vo trng lng

40 60 gr

42.000

5XC NH HM S QUA TH-----------------------------------------------------------------------------------------------------------------------------------

v Dng th: Trc quan. VD: Lng CO2 trong khng kh

6CC CCH XC NH HM S: BIU THC -----------------------------------------------------------------------------------------------------------------------------------

Quen thuc (dng hin): y = f(x)

VD: y = x2, hm s cp c bn

Dng tham s ( )( )

==

tyytxx

VD: x = 1 + t, y = 1 t / thng

: 1 t 1 (x, y)

VD: x = acost, y = asint ng trn

Dng n F(x, y) = 0 y = f(x)

VD: trn x2 + y2 4 = 0, 01916

22

=-+yx

Biu thc:

7HM S NGC -----------------------------------------------------------------------------------------------------------------------------------

fsong nh Phng trnh f(x) = y (*) c nghim x duy nht

( ) XYfYyyfxxfy "== -- ::)( 11 :ngc ham thc bieu

Tm hm ngc: Gii (*) (n x) Biu thc hm ngc x = f-1(y)

Hm s y = f(x): X Y tho tcht:

" y Y, $! x X sao cho y = f(x) f: song nh (tng ng mtmt)

VD: Tm min xc nh v min gi tr trn cc hm s

sau c hm ngc v ch ra cc hm ngc y = ex, y = x2 + 1

Ch : Cn thn chn X & YVD: y = f(x) = 2x + 1 f1 = ?

8HM LNG GIC NGC --------------------------------------------------------------------------------------------------------------------------------------

y = sinx: song nh t ??? ???

Hm ngc y = arcsinx t ??? ???

y = cosx arccosx; y = tgx arctgx; y = cotgx arcotgx

VD: Tnh a = arcsin(1/2): Dng phm sin-1 trn my tnh b ti

[ ] yxyxyx arcsinsin:1,1,2

,2

==-

-

pp

y = arcsinx: D = [1, 1], y :2

,2

-

pp abba == sinarcsin

p dng: Tnh cc tch phn bt nh +- 22 1/1/ xdxb

xdxa

9HM HYPERBOLIC --------------------------------------------------------------------------------------------------------------------------------

Chi tit hm hyperbolic: Xem Sch Gio Khoa

2shsinh

xx eexx--

==Hm sin hyperbolic:

Hm cos hyperbolic: Rxeexxxx

">+

==-

02

chcosh

Hm tang hyperbolic: xxxx

eeee

xxxx -

-

+-

===chshthtanh

Hm cotang hyperbolic:xx

xxxth1

shchcothcotanh ===

Cng thc vi hm hyperbolic: Nh cng thc lng gic,

nhng thay cosx chx, sinx ishx (i: s o, i2 = 1)!

10

BNG CNG THC HM HYPERBOLIC --------------------------------------------------------------------------------------------------------------------------------

1cossin 22 =+ xx 1shch 22 =- xx( ) yxyxyx sinsincoscoscos m= ( ) yxyxyx shshchchch =( ) xyyxyx cossincossinsin = ( ) xyyxyx chshchshsh =( ) xxx 22 sin211cos22cos -=-= ( ) xxx 22 sh211ch22ch +=-=

( ) xxx cossin22sin = ( ) xxx chsh22sh =

2cos

2cos2coscos yxyxyx -+=+

2ch

2ch2chch yxyxyx -+=+

2sin

2sin2coscos yxyxyx -+-=-

2sh

2sh2chch yxyxyx -+=-

Cng thc HyperbolicCng thc lng gic

VD: Tnh tch phn + 21 xdx

11

HM HYPERBOLIC TRONG K THUT --------------------------------------------------------------------------------------------------------------------------------

Thit k hnh dng vm, cp treo, iu khin robot

12

HM LIN TC ------------------------------------------------------------------------------------------------------------------------------------

Hm s cp (nh ngha qua 1 biu thc) lin tc xc nh

VD: Tm a hm lin tc ti x = 0:

=

=

0,

0,sin

xa

xxx

y

f(x) xc nh ti x0( ) ( )0

0lim xfxf

xx=

Hm f(x) lin tc ti x0: Hm lin tc/[a, b] (C): ng lin

Gin

on!

VD: Kho st tnh lin tc ca cc hm s:

11tg/ 2

2

+-+

=x

xxyax

xyb sin/ =

-