chapter 1_function and limit

7/26/2019 Chapter 1_Function and Limit

1/87

CALCULUSCALCULUS


2/87

Objectives

Four ways to represent a function

Basis functions and the transformations of functions

Calculating limits of functions

limits at a point! limits involving "nfinity#

$erifying continuity of a function

Chapter %& Function and limit


3/87

%'%

Functions and

(heir )epresentations

FU*C("O*S A*+ L","(S


4/87

A function f is a rule that assigns to each element x in a

set Dexactly one element, called f(x), in a set E.

The set Dis called the domainof the function f.

The range of fis the set

of all possible values

of f(x) asxvaries

throughout the domain.

FU*C("O*

Fig' %'%'-! p' %.


5/87

The graph of f is the set of all points (x, y) in the

coordinate plane such that y = f(x) and x is in the domain

of f.

The graph of falso allows us to picture:

The domain of fon thex-axis

"ts rangeon the y-axis

/)A01


6/87

The graph of a function fis shown.a. Find the values of f() and f(!).

b. "hat is the domain and range of f #

f%# 2 -

f3# 4 5'6

+ 2 75! 68

"mf# 2 74.! 98

:;ample %/)A01


7/87

Find the domain and region of thefunctions ( if it is a function).

a.

b.

( ) for all natural numbers n.f n n=

):0):S:*(A("O*S

( ) is any real number such that larger than xg x

+"SCUSS"O*


8/87

There are four possible wa$s to represent a function:

Algebraicall$ (b$ an explicit formula)

%isuall$ (b$ a graph)

&umericall$ (b$ a table of values)

%erball$ (b$ a description in words)

):0):S:*(A("O*S OF FU*C("O*S


9/87

The human population of the world Pdepends on

the time t.

The table gives estimates of the

world population P(t) at time t,

for certain $ears.

'owever, for each value of the

time t,there is a corresponding

value of P, and we sa$ that

Pis a function of t.

:


10/87

When you turn on a hot-water faucet, the temperature T of

the water depends on how long the water has been

running.

raw a rough graph of Tas a function of the time t that has

elapsed since the faucet was turned on.

:;ample 9):0):S:*(A("O*S


11/87

A curve in thexy-plane is the graph of a function of

x if and onl$ if no vertical line intersects thecurve more than once.

(1: $:)("CAL L"*: (:S(


12/87

The reason for the truth of the %ertical *ine Test can be

seen in the figure.

(1: $:)("CAL L"*: (:S(


13/87

+f a function fsatisfies:

f(-x) =f(x), x D

then f is called an even function'

The geometric significance of an even function is that itsgraph is symmetric with respect to the y4a;is.

S=,,:()=& :$:* FU*C("O*

y = x4

movie
http://../phuoc%20vinh/Truong%20FPT/math%201/Calculus/Animation_Chap01/01_02_12a.htmlhttp://../phuoc%20vinh/Truong%20FPT/math%201/Calculus/Animation_Chap01/01_02_12a.html


14/87

+f f satisfies:

f(-x) = -f(x), x

D

then fis called an odd function.

The graph of an odd function is symmetric about the origin.

S=,,:()=& O++ FU*C("O*

y = x3 y = x5 y = x7

movie
http://../phuoc%20vinh/Truong%20FPT/math%201/Calculus/Animation_Chap01/01_02_12b.htmlhttp://../phuoc%20vinh/Truong%20FPT/math%201/Calculus/Animation_Chap01/01_02_12b.html


15/87

*et f is an odd function. +f (-,!) is in the graph of f then

which point is also in the graph of f#

a. (,!) b. (-,-!) c. (,-!) d. All of the others

Answer: c

:;ample


16/87

:;ample %%

uppose f is an odd function and g is an even function.

"hat can we sa$ about the function f.g defined b$ (f.g)(x)f(x)g(x)#

/rove $our result.


17/87

A function fis called increasing on an intervalIif:

f(x) 0 f(x1) wheneverx 0x1 inI

+t is called decreasing onIif:

f(x) 2 f(x1) wheneverx 0x1 inI

"*C):AS"*/ A*+ +:C):AS"*/ FU*C("O*S


18/87

The function f is said to be increasing on theinterval 7a! b8, decreasing on 3b, c4, and

increasing again on 3c, d4.

"*C):AS"*/ A*+ +:C):AS"*/ FU*C("O*S


19/87

%'% >U"? >U:S("O*S

) +f f is a function then f(x51)f(x)5f(1)

a. True b. False

1) +f f(s)f(t) then s t

a. True b. False

) *et f be a function.

"e can find s and t such that st and f(s) is note6ual to f(t)

a. True b. False


20/87

%'.,A(1:,A("CAL ,O+:LS&

A CA(ALO/ OF:SS:*("AL FU*C("O*S

+n this section, we will learn about:

The purpose of mathematical models.

FU*C("O*S A*+ ,O+:LS


21/87

A mathematical model is a mathematical

description7often b$ means of a function or ane6uation7of a real-world phenomenon such as:

i8e of a population

emand for a product

peed of a falling ob9ect

*ife expectanc$ of a person at birth

ost of emission reductions

,A(1:,A("CAL ,O+:LS


22/87

"hen we sa$ that y is a linear function of x, wemean that the graph of the function is a line.

o, we can use the slope-intercept form of the e6uation

of a line to write a formula for the function as

where m is the slope of the line and b is the y-intercept.

( )y f x mx b= = +

L"*:A) ,O+:LS


23/87

A function Pis called a pol$nomialif

P(x) = anxn+ an-1xn-1+ + a2x2+ a1x+ a0

where n is a nonnegative integer and the numbers

a;, a, a1,


24/87

A rational function fis a ratio of two pol$nomials

where Pand are pol$nomials.

The domain consists of all values ofx

such that .

( )( )

( )

P xf x

Q x=

( ) 0Q x

)A("O*ALFU*C("O*S


25/87

()"/O*O,:()"CFU*C("O*S

f(x) sinx

f(x) = cos x

( , )D =

! 3-, 4

sin( 2 ) sin cos( 2 ) cos ;x k x x k x k Z + = + =


26/87

()"/O*O,:()"CFU*C("O*S

sin

tan cos

x

x x=

, ,2 2x

( , )R =

tan( ) tan ;x k x k Z+ =


27/87

()"/O*O,:()"CFU*C("O*S

The reciprocals of the sine, cosine, and tangentfunctions are

1cos

sin1

seccos

1cottan

ecx

x

xx

anxx

=

=

=


28/87

A function of the formf(x) =x a , where a is constant,

is called a power function.

0O@:) FU*C("O*S


29/87

The e;ponential functions are the functions of the

form , where the base a is a positive

constant.

The graphs of y 1x and y (;.!)xare shown.

+n both cases, the domain is and the range

is .

( ) xf x a=

( , )

(0, )

:


30/87

The logarithmic functions ,

where the base ais a positive constant, are the

inverse functions of the exponential functions.

( ) loga

f x x=

LO/A)"(1,"CFU*C("O*S

The figure shows the graphs of

four logarithmic functions with

various bases.


31/87

*abel the following graph from the graph of

the function $f(x) shown in the part (a)

$f(x)-1, $f(x-1), $-f(x), $1f(x), $f(-x)#

()A*SFO),A("O*S


32/87

=uppose c2 ;.

To obtain the graph of

y =f(x)+c, shiftthe

graph of y =f(x)

a distance c unitsupward.

To obtain the graph

ofy =f(x) -c, shiftthe graph of y =f(x)

a distance c units

downward.

S1"F("*/

@hy dont we consider the case c5


33/87

To obtain the graph ofy =f(x - c),shift the graph of

y =f(x) a distance c units to the right.

To obtain the graph

ofy =f(x " c),shiftthe graph of y =f(x)

a distance c units to

the left.

S1"F("*/


34/87


the function $f(x) shown in the part (a)

$f(x)-1, $f(x-1), $-f(x), $1f(x), $f(-x)#

*:@ FU*C("O*S F)O, OL+ FU*C("O*S


35/87


b$ shifting 1 units downward. b$ shifting 1 units to the right.

2y x= 2y x=


the function shown in the part (a):

$f(x)-1, $f(x-1),$-f(x), $1f(x), $f(-x)#

y x=


36/87

=uppose c2 .

To obtain the graph

ofy= cf(x), stretch

the graph of y = f(x)

verticall$b$ a factorof c#

To obtain the graph

ofy= (1!c)f(x),

compressthe graph

of y = f(x) verticall$ b$

a factor of c#

()A*SFO),A("O*S

'ow about the case c0#


37/87

+n order to obtain the graph of y f(cx),

compress the graph of y = f(x) hori8ontall$b$ a factor of c#

To obtain the graphof y f(x>c), stretch

the graph of y = f(x)

hori8ontall$ b$ a factor

of c.

()A*SFO),A("O*S


38/87

+n order to obtain the graph of y -f(x),

reflect the graph of y = f(x) about thex-axis.

To obtain the graphof y f(-x), reflectthe graph of y = f(x)about the y-axis.

()A*SFO),A("O*S


39/87




$f(x)-1, $f(x-1),$-f(x), $1f(x), $f(-x)#

y x=


40/87


b$ reflecting about thex-axis. b$ stretching verticall$ b$ a factor of 1. b$ reflecting about the y-axis

y x=

2y x=

y x=



$f(x)-1, $f(x-1), $-f(x), $1f(x), $f(-x)#

y x=


41/87

= The figure illustrates these stretching

= transformations when applied to the

cosine

= function with c 1.

()A*SFO),A("O*S

:;ample


42/87

:;ample

uppose that the graph of f is given.

escribe how the graph of the function f(x-1)51

can be obtained from the graph of f.

elect the correctanswer.

a. hift the graph 1 units to the left and 1 units down.

b. hift the graph 1 units to the right and 1 units down.

c. hift the graph 1 units to the right and 1 units up.

d. hift the graph 1 units to the left and 1 units up.

e. none of these

Answer: c


43/87

= Two functions fand gcan be combined to formnew functions:

(f5 g)x f(x) 5 g(x)

(f? g)x f(x) ? g(x)

CO,B"*A("O*S OF FU*C("O*S

( )( )( ) ( ) ( ) ( )

( )

f f xfg x f x g x x

g g x

= =


44/87

*et h(x)f(g(x)).

) +f g(x)x- and h(x)x51 then f(x) is:

a. x5 b. x5@ c. x5 d. &one of them

1) +f h(x)x51 and f(x)x- then g(x) is:

a. x5 b. x5@ c.x5 d. &one of them

Answer: ) d 1) a


45/87

%'. >U"? >U:S("O*S

) +f f and g are functions, then

a. True b. False

1)

is

a. ! b. c. 1 d. &one of the others

f g g f=o o

( )(2)f go

)(xf

)(xg

.

x 1 2 " # $

2 1 0 1 2

$ # 2 " $


46/87

Answer: f(x), f( x), f(x)

%'. >U"? >U:S("O*S


47/87

%'-

(he Limit of a Function

L","(S

+n this section, we will learn:

About limits in general and about numericaland graphical methods for computing them.


48/87

+n general, we write

if we can maBe the values

of f(x) arbitraril$ close to $b$ taBingxto be

sufficientl$ close to a

but not eDual to a.

( )limx a

f x L

=

(1: L","( OF A FU*C("O*


49/87

"e write

if we can maBe the values of f(x) arbitraril$ close to $b$

taBingxto be sufficientl$ close to aandxless than a.

( )limx a f x L

=

O*:4S"+:+ L","(S +efinition .


50/87

imilarl$, %the right-hand limit of f(x) as x approaches a is

e&ual to $'and we write

( )limx a

f x L+

=

O*:4S"+:+ L","(S

( )2

limx

g x

( )2

limx

g x+

( )2

limx

g x

( )#

limx

g x

( )#

limx

g x+

( )

#limx

g x


51/87

%'9

Calculating Limits

Using the Limit Laws

L","(S

+n this section, we will:

Cse the *imit *aws to calculate limits.


52/87

[ ]".lim ( ) ( ) lim ( ) lim ( )x a x a x a

f x g x f x g x

=

[ ]1.lim ( ) ( ) lim ( ) lim ( )x a x a x a

f x g x f x g x

=

(1: L","( LA@S

[ ].lim ( ) lim ( )x a x a

cf x c f x

=

lim ( )( )#.lim if lim ( ) 0

( ) lim ( )

= x a

x a x a

x a

f xf xg x

g x g x

uppose that cis a constant and the limits

and exist. Then

lim ( )x a

f x

lim ( )x ag x


53/87

where nis a positive integer.

10.lim n nx a

x a

=

US"*/ (1: L","( LA@S

%.lim n nx a

x a

=

&.lim

xac = c

'.limx a

x a

=

[ ]$.lim ( ) lim ( )

nn

x a x af x f x =

11.lim ( ) lim ( )n nx a x a

f x f x =


54/87

"e state this fact as follows. +f fis a polynomialor a

rational function and ais in the domain of f, then

lim ( ) ( )x a

f x f a

=

+"):C( SUBS("(U("O* 0)O0:)(=


55/87

if and onl$ if

US"*/ (1: L","( LA@S

lim ( )x a f x L = lim ( ) lim ( )x a x af x L f x + = =

(heorem %


56/87

/rove that does not exist.

US"*/ (1: L","( LA@S :;ample E

0limx

x

x


57/87

+f when x is near a (except possibl$ at a)

and the limits of fand g both exist asxapproaches a,

then

( ) ( )f x g x

lim ( ) lim ( )x a x a

f x g x

0)O0:)(":S OF L","(S (heorem .


58/87

The 6uee8e Theorem (the andwich Theorem or the /inching Theorem)

states that,if

whenxis near a (except possibl$ at a)

and .Then

( ) ( ) ( )f x g x h x

lim ( ) lim ( )x a x a

f x h x L

= = lim ( )x a

g x L

=

S>U::?: (1:O):, (heorem -


59/87

how that

&ote that we cannot use

This is because does not exist.

2

0

1lim sin 0.x

xx

=

US"*/ (1: L","( LA@S :;ample %%

2 2

0 0 0

1 1lim sin lim limsinx x x

x xx x

=

0

limsin(1! )x

x


60/87

'owever, since ,

we have:

11 sin 1

x

2 2 21sinx x x

x

US"*/ (1: L","( LA@S :;ample %%

TaBing f(x) -x1, and h(x) x1

in the 6uee8e Theorem,

we obtain:

2

0

1lim sin 0x

xx

=


61/87

%'9 >U"? >U:S("O*S

) +f

then

a. True b. False

lim ( ) 0, lim ( ) 0x x

f x g x

= =

( )lim oes not exist

( )x

f x

g x


62/87

%'9 >U"? >U:S("O*S

+f

a. True b. False

lim ( ) ( ) exists, then the limit must be () ()x f x g x f g

L","(S


63/87

%'3

Continuity

L","(S

+n this section, we will:

ee that the mathematical definition of continuit$

corresponds closel$ with the meaning of the word

continuit$ in ever$da$ language.

CO*("*U"(=


64/87

A function fis continuous at a number a if:

lim ( ) ( )x a

f x f a

=

CO*("*U"(= %' +efinition

&otice that :

f(a) is defined - that is,

a is in the domain of f exists.

lim ( )x a

f x

lim ( ) ( )x a

f x f a

=

CO*("*U"(=


65/87

+f fis defined near a - that is, fis defined on an open intervalcontaining a, except perhaps at a - wesa$ that fis

discontinuousat a if fis not continuous at a.

CO*("*U"(=

The figure shows the graph of a

function f.

At which numbers is fdiscontinuous#

"h$#

+efinition

CO*("*U"(=


66/87

A function fis continuous from the right

at a number a if

and fis continuous from the left at a if

lim ( ) ( )x a

f x f a+

=

lim ( ) ( )x a

f x f a

=

CO*("*U"(= .' +efinition

CO*("*U"(=


67/87

A function fis continuous on an interval if it is

continuous at ever$ number in the interval.

+f fis defined onl$ on one side of an endpoint of the interval,

we understand Dcontinuous at the endpointE to mean

Dcontinuous from the rightE or Dcontinuous from the left.E

CO*("*U"(= -' +efinition

CO*("*U"(=


68/87

+f fand gare continuous at aand cis a constant, then the

following functions are also continuous at a:

.f +g

2.f -g

. cf

".fg

#. ( ) 0f

i f g ag

CO*("*U"(= 9' (heorem

CO*("*U"(=


69/87

The following t$pes of functions are continuous atever$ number in their domains:

/ol$nomials

ational functions

oot functions

Trigonometric functions

CO*("*U"(= 6' (heorem

CO*("*U"(=


70/87

+f fis continuous at band then

+n other words,

+fxis close to a, then g(x) is close to bG and, since fis continuous at b, if g(x) is close to b, then f(g(x))

is close to f(b).

lim ( )x a

g x b

=

lim ( ( )) ( )x a

f g x f b

=

( )lim ( ( )) lim ( )x a x a

f g x f g x

=

CO*("*U"(= E' (heorem

CO*("*U"(= (h


71/87

+f gis continuous at aand fis continuous at g(a),then the composite function

is continuous at a.

This theorem is often expressed informall$ b$ sa$ing

Ha continuous function of a continuous function is

a continuous function.I

( ) ( ) ( ( ))f g x f g x=o

CO*("*U"(= ' (heorem

"*(:),:+"A(: $ALU: (1:O):, %5 (h


72/87

uppose that fis continuous on the closed interval

7a! b8and let be an$ number between f(a) and f(b),

where

Then, there exists a number cin (a, b)such that f(c) .

( ) ( )f a f b

"*(:),:+"A(: $ALU: (1:O):, %5' (heorem

"*(:),:+"A(: $ALU: (1:O):, : l


73/87

how that there is a root of the e6uation

between and 1.

*et .

"e are looBing for a solution of the given e6uation7

that is, a number cbetween and 1 such that f(c) ;.

Therefore, we taBe a , b 1, and ; in the theorem.

"e have

and

2" $ 2 0x x x + =

2( ) " $ 2f x x x x= +

"*(:),:+"A(: $ALU: (1:O):, :;ample

(1) " $ 2 1 0f = + = U:S("O*S


74/87

%'3 >U"? >U:S("O*S

) +f f()2; and f()0; then there exists a number c between and such that f(c);

a. True b. False

1) "hich is the e6uation expressing the fact that H f is continuous at 1I#

a. b.

c. d.

e.

2lim ( ) 0x

f x

=2

lim ( )x

f x

=

2lim ( ) 2x

f x

= lim ( ) (2)x

f x f

=

2lim ( ) (2)x

f x f

=


75/87

%'G

Limits involving infinity

@e will study&

"nfinity Limits

Limits at "nfinity

"*F"*"(: L","(S + fi iti 9


76/87

*et fbe a function defined on both sides of a, except

possibl$ at aitself. Then,

means that the values of f(x)

can be made arbitraril$ large

b$ taBingxsufficientl$

close to a, butnot e6ual to a.

( )limx a

f x

=

"*F"*"(: L","(S +efinition 9

"*F"*"(: L","(S + fi iti 3


77/87

*et fbe defined on both sides of a, except possibl$ at a

itself. Then,

means that the values of f(x)

can be made arbitraril$

large negative b$ taBingx

sufficientl$ close to a,

but not e6ual to a.

( )limx a

f x

=

"*F"*"(: L","(S +efinition 3

"*F"*"(: L","(S


78/87

imilar definitions can be given for the one-sided limits:

( )limx a

f x

= ( )limx a

f x+

=

( )limx a f x = ( )limx a f x+=

"*F"*"(: L","(S

+:F"*"("O*S


79/87

= xais called the vertical as$mptoteof f(x)

if we have one of the following:

+:F"*"("O*S

( )limx a

f x

= ( )limx a

f x+

=

( )limx a

f x

= ( )limx a

f x+

=

L","(S A( "*F"*"(=


80/87

L","(S A( "*F"*"(=

lim ( )x

f x L

=

if then ( )x M f x L > >

*et f be a function defined for ever$ x2a. Then

means that

+:F"*"("O*


81/87

The line $*is called the horiHontal asymptote

of f(x) if we have oneof the following:

+:F"*"("O*

( ) ( )lim limx x

f x L f x L

= =


82/87

2

1( )

2

xf x

x x

=

+

2

111

lim lim 11 22

1x x

x x

x x

x x

= =+

+

Find the as$mptotes of the function

( )( )2 2 2

21

1 11

2 ( 1)( 2 2)

1 lim

2 #x

x x xx

x x x x x

x

x x

+ +=

+ + +

+=

+

olution

$ is hori8ontal as$mptote


83/87

ompute

a.

b.

c.

d.

2

1limsin

lim( 1 )

limsin

lim( )

x

x

x

x

x

x x

x

x x

+

;

;

oes not exist

%'G >U"? >U:S("O*S


84/87

%'G >U"? >U:S("O*S

) Find

a. ; b. infinit$ c. d. oes not exist

limcosx

x

1) Find

a. ; b. infinit$ c. d. ose not exist

1lim cosx

xx

%'G >U"? >U:S("O*S


85/87

%'G >U"? >U:S("O*S

) +f

Then

a. True b. False

0 0lim ( ) , lim ( )x x

f x g x = =

0lim ( ) ( )* 0x

f x g x

=

@) A function can have twodifferent hori8ontal as$mptotes

a. True b. False

:;ercises


86/87

:;ercises

ection +. 1;,!,@,@J (p J,;)

ection +.1 : @J,!1,!@,K (p 1,1@)

+.@: L, 1J (p@@,@!)

+.K: 11,1@,1K (pKM)

@M page M1


87/87

(hanIs

chapter 1_function and limit

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