[PDF] Calculus Volume 1

View - Download Calculus Volume 1

Previous PDF

Next PDF

Volume 1

CalculusgVolumegxg

SENIOR CONTRIBUTING AUTHORS

gggggg

OpenStaxg

RicegUniversityg

θxwwgMaingStreetgMStzμεg

HoustonsgTexasgμμwwεg

g g

©2018 Rice University.gTextbookgcontentgproducedgbygOpenStaxgisglicensedgundergagCreativegCommonsgAttributiongNont

g

- IfgyougnoncommerciallygredistributegthisgtextbookgingagdigitalgformatgoincludinggbutgnotglimitedgtogPDFgandgHTMLpsgtheng

- Ifgyougnoncommerciallygredistributegthisgtextbookgingagprintgformatsgthengyougmustgincludegongeverygphysicalgpagegtheg

followinggattributionωg

- Ifgyougnoncommerciallygredistributegpartgofgthisgtextbooksgthengyougmustgretaingingeverygdigitalgformatgpagegviewg

- Ifgyougusegthisgtextbookgasgagbibliographicgreferencesgpleasegincludeghttpsωvvopenstaxuorgvdetailsvbooksvcalculustvolumet

xgingyourgcitationug g g

Trademarks

g

PRINT BOOK ISBN-10 1-938168-02-X

PRINT BOOK ISBN-13 978-1-938168-02-4

PDF VERSION ISBN-10 1-947172-13-1

PDF VERSION ISBN-13 978-1-947172-13-5

Revision Number C1-2016-003(03/18)-MJ

Original Publication Year 2016

g gg

OPENSTAX

RICE UNIVERSITY

humangendeavorug g

FOUNDATION SUPPORT

g g g g

peopleg understoodg aboutg basicg economicsg theg happierg andg moreg prosperousg theyg wouldg beug Accordinglysg heg

g g othergareasgofgsocialgimportanceug g

contentg byg leveragingg disruptiveg technologiessg openg educationalg resourcessg andg newg modelsg forg collaborationg

g g g

Access. The future of education.

OpenStax.org I like free textbooks

and I cannot lie.Give $5 or more to OpenStax and we'll send you a sticker!

OpenStax is a nonprofit initiative,

which means that that every dollar you give helps us maintain and grow our library of free textbooks.

If you have a few dollars to spare,

visit

OpenStax.org/give

to donate.

We'll send you an OpenStax sticker

to thank you for your support! -56@9C:CBH9BHG )F9:579 <5DH9F IB7H=CBG5B8 F5D53;:;91@>5/A:/@5;:? :B1>?1A:/@5;:? D<;:1:@5-8-:0 ;3->5@495/A:/@5;:? <5DH9F %=A=HG $>1B51C;2-8/A8A? (41 595@;2-A:/@5;: (41 595@ -C? ;:@5:A5@E (41$>1/5?1125:5@5;:;2- 595@ <5DH9F 9F=J5H=J9G

125:5:3@411>5B-@5B1

(411>5B-@5B1-?-A:/@5;: 5221>1:@5-@5;:&A81?

1>5B-@5B1?-?&-@1?;24-:31

1>5B-@5B1?;2(>53;:;91@>5/A:/@5;:? (414-5:&A81 1>5B-@5B1?;2:B1>?1A:/@5;:? 9<85/5@5221>1:@5-@5;: 1>5B-@5B1?;2D<;:1:@5-8-:0 ;3->5@495/A:/@ 5;:? <5DH9F DD@=75H=CBGC:9F=J5H=J9G &18-@10&-@1? 5:1-><<>;D59-@5;:?-:05221>1:@5-8? !-D59--:0!5:59- (41!1-:*-8A1(41;>19 1>5B-@5B1?-:0@41'4-<1;2->-<4 595@?-@:25:5@E-:0?E9<@;@1?
<<8510#<@595F-@5;:$>;.819? KH<5@-8K?&A81 "1C@;:K?!1@4;0 :@501>5B-@5B1? <5DH9F "BH9;F5H=CB <<>;D59-@5:3>1-? (41125:5@1:@13>-8 (41A:0-91:@-8(41;>19;2-8/A8A? :@13>-@5;:;>9A8-?-:0@41"1@4-:31(41; >19 'A.?@5@A@5;: :@13>-8?:B;8B5:3D<;:1:@5-8-:0 ;3->5@495/

A:/@5;:?

:@13>-8?&1?A8@5:35::B1>?1(>53;:;91@>5/A:/@5;:? <5DH9F DD@=75H=CBGC:"BH9;F5H=CB >1-?.1@C11:A>B1? 1@1>95:5:3*;8A91?.E'85/5:3 *;8A91?;2&1B;8A@5;:E85:0>5/-8'4188? >/ 1:3@4;2-A>B1-:0'A>2-/1>1- $4E?5/-8<<85/-@5;:? !;91:@?-:01:@1>?;2!-?? :@13>-8?D<;:1:@5-8A:/@5;:?-:0 ;3->5@ 49?
D<;:1:@5-8>;C@4-:01/-E -8/A8A?;2@41E<1>.;85/A:/@5;:?

DD9B8=L -56@9C:"BH9;F5@G

DD9B8=L -56@9C:9F=J5H=J9G

DD9B8=L +9J=9KC:)F95@7I@IG

"B89L (45?#<1:'@-D.;;75?-B-58-.812;>2>11-@4@@ 3/;:@1:@/;8

4>E MHCalculus Volume 1

:G ,I>G0M:Q K>LHNK<> 1ABL M>QM;HHD P:L PKBMM>G MH BG:L> LMN=>GM :<<>LL MH AB@AJN:EBMR

E>:KGBG@F:M>KB:EL

F:BGM:BGBG@AB@A>LMLM:G=:K=LH?

:<:=>FBMHGH6CIH(D9B,H5L ,I>G0M:Q BL : GHGIKH?BM ;:L>= :M /B<> 2GBO>KLBMR :G= BM`L HNK FBLLBHG MH BFIKHO> LMN=>GM :<<>LL MH >=N<:MBHG ,NK ?BKLM HI>GER EB<>GL>= @> M>QM;HHD P:L IN;EBLA>= BG :G= HNK EB;K:KR A:L LBG<> L<:E>= MH HO>K ;HHDL ?HK @> :G= - TL NL>= ;R ANG=K>=L H? MAHNL:G=L H? LMN=>GML ,I>G0M:Q 1NMHK

HNK EHPKLHG:EBS>= E>:KGBG@ MHHE

BL

;>BG@ NL>= BG @> L MAKHN@AHNM MA> KLABIL PBMA IABE:GMAKHIB< ?HNG=:MBHGL :G= HNK

:EEB:G<> PBMA HMA>K >=N<:MBHG:E K>LHNK<> HK@:GBS:MBHGL ,I>G0M:Q BL ;K>:DBG@ =HPG MA> FHLM KL MH E>:KGBG@

6CIH(D9B,H5LGF9GCIF79G

IGHCA=N5H=CB

Calculus Volume 1BL EB<>GL>= NG=>K : K>:MBO> HFFHGL MMKB;NMBHG &GM>KG:MBHG:E 5 EB<>GL>

PAB:GL

MA:M RHN <:G =BLMKB;NM>

K>FBQ :G= ;NBE= NIHG MA> GM :L EHG@ :L RHN IKHOB=> :MMKB;NMBHG MH ,I>G0M:Q :G= BML GM <:NL> HNK ;HHDL :K> HI>GER EB<>GL>= RHN :K> ?K>> MH NL> MA> >GMBK> ;HHD HK IB MA> L> FHLM

K>E>O:GM MH MA> G>>=L H? RHNK #>>E ?K>> MH K>FBQ MA> GM ;R :LLB@GBG@ RHNK LMN=>GML <>KM:BG KL :G= L>

BG RHNK LREE:;NL

BG MA> HK=>K MA:M RHN IK>?>K 5HN <:G >O>G IKHOB=> : =BK> L> P>;

OB>PH?RHNK;HHD

&GLMKN MA> HIMBHG H? :MBG@ : = O>KLBHG H? MA>BK ,I>G0M:Q ;HHD 1A> KLBHG <:G ;> F:=>

:O:BE:;E> MH LMN=>GML BG EHPBK <:FINL ;HHDLMHK> 3BLBM RHNK ;HHD I:@> HG ,I>G0M:QHK@

?HKFHK>BG?HKF:MBHG FF5H5 EE ,I>G0M:Q M>QM;HHDL NG=>K@H : KB@HKHNL K>OB>P IKH<>LL %HP>O>K

EBD> :GR IKH?>LLBHG:E@K:=> M>QM;HHD

>KKHKL

LHF>MBF>L H< HNK ;HHDL :K> P>; ;:L>=

P> <:G F:D> NI=:M>L I>KBH=B<:EER PA>G =>>F>= I>=:@H@B<:EER G><>LL:KR &? RHN A:O> : LM LN;FBM BM MAKHN@A MA> EBGD HG RHNK ;HHD I:@> HG ,I>G0M:QHK@ 0N;C>K >QI>KML K>OB>P :EE >KK:M: LN@@>LMBHGL ,I>G0M:Q BL = MH K>F:BGBG@ MK:GLI:K>GM :;HNM :EE NI=:M>L

LH RHN PBEE :ELH ?BG= : EBLM

H?I:LM>KK:M:LHGRHNK;HHDI:@>HG,

I>G0M:QHK@

CFA5H

5HN<:G:<<>LLMABLM>QM;HHD?HK?K>>BGP>;O

B>PHK-!#MAKHN@A,I>G0M:QHK@

:G=?HK:EHP6CIH,294D9DB 0>9D:6 " :ELB@G>= ?HK MA> MRIB<:E MPH HK MAK>>L>F>LM>K @>G>K:E <:E

BG ?>:MNK>L MH

>GA:G<> LMN=>GM E>:KGBG@ 1A> ;HHD @NB=>L LMN=>GML MAKHN@A MA> IML H? <:EEIL MA>F NG=>KLM:G=

AHP MAHL> IML :IIER MH MA>BK EBO>L :G= MA> PHKE= :KHNG= MA>F !N> MH MA> A>GLBO> G:MNK> H? MA> F:M>KB:E

P> :K> H??>KBG@ MA> ;HHD BG MAK>> OHENF>L ?HK ?E>QB;BEBMR :G= >??BG KL ?NGEBFBML =>KBO:MBO>L :G=

BGM>@K:MBHG

CJ9F5;95B8G7CD9

,NKCalculus Volume 1M>QM;HHD :=A>K>L MH MA> L :G= L>JN>G<> H? FHLM @>G>K:E <:EL G:MBHGPB=> 4> A:O>

PHKD>= MH F:D> <:EK>LMBG@ :G= :<<>LLB;E> MH LMN=>GML PABE> F:BGM:BGBG@ MA> F:MA>F:MB<:E KB@HK BGA>K>GM BG MA>

LN;C> BG FBG=

MA> GM H? MA> MAK>> OHENF>L H?CalculusA:O> ;>>G =>O>EHI>= :G= :KK:G@>= MH IKHOB=> : EH@B<:E IKH@K>LLBHG ?KHF ?NG=:F>GM:E MH FHK> :=O:G<>= IML ;NBE=BG@ NIHG PA:M LMN=>GML A:O> :EK>:=R

E>:KG>= :G= >FIA:LBSBG@ MP>>G MHIBMP>>G MA>HKR :G= :IIEB<:MBHGL 1A> @H:E H? >: >G:;E> LMN=>GML GHM CNLM MH K> IML ;NM PHKD PBMA MA>F BG P:RL MA:M PBEE ;> NL>?NE BG E:M>K L :G= ?NMNK>

<:K>>KL 1A> HK@:GBS:MBHG :G= I>=:@H@B<:E ?>:MNK>L P>K> =>O>EHI>= :G= O>MM>= PBMA ?>>=;:F:MB=N<:MHKL

=>=B<:M>=MHMA>IKHC>12-/1

A:IM>K#NG

A:IM>K)BFBML

A:IM>K!>KBO:MBO>L

A:IM>KIIEB<:MBHGLH?!>KBO:MBO>L

A:IM>K&GM>@K:MBHG

A:IM>KIIEB<:MBHGLH?&GM>@K:MBHG

/C@IA9

A:IM>K&GM>@K:MBHG

A:IM>KIIEB<:MBHGLH?&GM>@K:MBHG

A:IM>K1>LH?&GM>@K:MBHG

A:IM>K0>JN>G<>L:G=0>KB>L

A:IM>K-HP>K0>KB>L

A:IM>K-:K:F>MKB<"JN:MBHGL:G=-HE:K HHK=B

G:M>L /C@IA9

A:IM>K-:K:F>MKB<"JN:MBHGL:G=-HE:K HHK=B

G:M>L

A:IM>K3>

A:IM>K3>=#NG

A:IM>K*NEMBIE>&GM>@K:MBHG

A:IM>K3>

A:IM>K0>K!B??>K>GMB:E"JN:MBHGL

)985;C;=75@:CIB85H=CB

1AKHN@AHNMCalculus Volume 1RHN PBEE ?BG= >Q:FIE>L :G= >Q>KL MA:M IK>L>GM :L :G= M>L :L P>EE :L

FH=>KG :IIEB<:MBHGL :G= F>MAH=L !>KBO:MBHGL :G= >QIE:G:MBHGL :K> ;:L>= HG R>:KL H? QI>KB>G<> HG MA> I:KM

H? EHG@MBF> <:ELLHKL

LMKBOBG@ ?HK : ;:E:G<> H? G LN<<>LL?NE PBMA MA>BK LMN=>GML *HMBO:MBHG:E :IIEB<:MBHGL K BFIHKM:GM MHIBLL :G= >EE :L :K>:L

H? IARLB FBLMKR >G@BG>>KBG@ :G= K LG<>Student ProjectsBG >:K @BO> LMN=>GML HIIHKMNGBMB>L MH >QIEHK> BGM>K>LMBG@ LB=>EB@AML BG INK> :G= :IIEB>= F:MA>F:MBM>KFBGBG@ : L:?> =BLM:G<> ;>MP>>G MA> @K:G=LM:G= :G=

MA> MK: K:<>MK: MH <:E <>GM>K H? F:LL H? MA> $K:G= :GRHG 0DRP:ED HK MA> M>KFBG:E LI>>= H? : LDR=BO>KChapter Opening ApplicationsIHL> IKH;E>FL MA:M :K> LHEO>= E:M>K BG MA> K

NLBG@ MA> B=>:L K>= BG

MA:M K -KH;E>FL BG MA> AR=K:NEB< ?HK<> :@:BGLM MA> %HHO>K !:F :G= MA> E:MBO> BGM>GLBMR H? MPH >:KMAJN:D>LDefinitions, Rules,:G=Theorems:K>AB@AEB@AM>=MAKHN@AHNMMA>M>QM

BGK

ProofsH?MA>HK>FL

GG9GGA9BHGH<5HF9=B:CF79?9M7CB79DHG

&GKExamplesP:ED LMN=>GML MAKHN@A IKH;E>FL ;R IHLBG@ : JN>LMBHG

LM>IIBG@ HNM : LHENMBHG

:G= MA>G :LDBG@ LMN=>GML

MH IK: MA> LDBEE PBMA : a A>LMBHG 1A> ;HHD :ELH BGL :LL>LLF>GML :M MA> >G= H? >:K LH

LMN=>GML <:G :IIER PA:M MA>R`O> E>:KG>= MAKHN@A IK: IKH;E>FL *:GR >Q>KL :K> F:KD>= PBMA :[T]MH BG=B<:M> MA>R

:K> LNBM:;E> ?HK LHENMBHG ;R M>K E@>;K: 0RLM>FL 0 GLP>KL ?HK L>E>=

>Q>KL :K> :O:BE:;E> BG MA>Answer Key:M MA> ;: ;HHD 1A> ;HHD :ELH BGL :LL>LLF>GML :M MA> >G= H? >:

MAKHN@AIK:IKH;E>FL

5F@MCF@5H9HF5BG79B89BH5@G

Calculus Volume 1BL =>LB@G>= MH :< ;HMA ":KER :G= ):M> 1K:GL<>G=>GM:E :IIKH:L MH <:EGMB:E

:G= EH@:KBMAFB< ?NG BGMKH=N<>= BG?HKF:EER BG A:IM>K :G= IK>L>GM>= BG FHK> KB@HKHNL M>KFL BG A:IM>K

!B??>K>GMB:MBHG :G= BGM>@K:MBHG H? MA>L> ?NGK>= BG A:IM>KL ] ?HK BGLMKN MA>F PBMA

HMA>K MRI>L H? ?NGL> =BL

AHP>O>K

:K> BG L>I:K:M> L> LDBII>= ?HK BGLMKN?>K

EL:G=EH@:KBMAFL

CADF9<9BG=J95FHDFC;F5A

,NK :KM IKH@K:F BL =>LB@G>= MH >GA:G<> LMN=>GML` NG=>KLM:G=BG@ H? IML MAKHN@A :K :G= >??> BEENLMK:MBHGL

2Preface

This OpenStax book is available for free at htt4p://cnx.org/content/col11964/1.2 =B:@K:FL :G=IAHMH@K:IAL

88=H=CB5@F9GCIF79G

,HI89BH5B8=BGHFI7HCFF9GCIF79G

4>`O> = :==BMBHG:E K>LHNK<>L ?HK ;HMA LMN=>GML :G= BGLMKN

BGMMBG@ 0M:KM>= $NB=>L

:G BGLMKNLHENMBHG F:GN:E

:G= -HP>K-HBGM LEB=>L &GLMKNLHNK<>L K>JNBK> : O>KB?B>= BGLMKNPAB K>JN>LM>=

,I>G0M:Q;HHD

CAAIB=HM!I6G

,I>G0M:Q I:KMG>KL PBMA MA> &GLMBMNM> ?HK MA> 0MN=R H? (GHPE>=@> *:G:@>F>GM BG "=N<:MBHG &0(*" MH H??>K HFFNGBMR

%N;L HG ,"/ HFFHGL ] : IE:M?HKF ?HK BGLMKN :M>= K>LHNK<>L MA:M LNIIHKM ,I>G0M:Q ;HHDL

?K>> H? 1AKHN@A HNK HFFNGBMR %N;L BGLMKNBK HPG F:M>KB:EL HK =HPGEH:= K>LHNK<>L MH NL>

BG MA>BK HPG L

BGL

M>:KB:E

FNEMBF>=B:

:G= K>E>O:GM GM 4>

>G BGLMKN AN;L ?HK MA> LN;C>E>O:GM MH RHNK M>:L>:K >GKBL:G=MH>G@:@>PBMAHMA>K?:< NEMR

1HK>: HFFNGBMR%N;L

OBLBMKKK

C9F7CAACBG

CF; )5FHB9FF9GCIF79G

,I>G0M:Q -:KMG>KL :K> HNK :EEB>L BG MA> FBLLBHG MH F:D> AB@AJN:EBMR E>:KGBG@ F:M>KB:EL :??HK=:;E> :G= :<<>LLB;E> MH LMN=>GML

:G= BGLMKNO>KRPA>K> 1A>BK MHHEL BGM>@K:M> L>:FE>LLER PBMA HNK ,I>G0M:Q MBME>L :M : EHP LL MA> I:KMG>K

K>LHNK<>L?HKRHNKM>QM

OBLBMRHNK;HHDI:@>H

G,I>G0M:QHK@

6CIHH<95IH ,9B=CF7CBHF=6IH=B;5IH!K 0MK:G@ K><>BO>= ABL -A! ?KHF 2 ) BG :G= A:L ;>>G M>:F:MBO>K LBG<> %BL :E

M>QM;HHD BL HG> H? >E>O>G MA:M A> A:L IN;EBLA>= :G= BL MA> ;:LBL ?KHF PAB>G =>KBO>= :G= NI=:M>=

I:LM/AH=>L0KLBMR

Edwin ÒJedÓ Herman, University of Wisconsin-Stevens Point !K %>KF:G >:KG>= : 0 BG *:MA>F:MBR *N== HEE>@> BG :G * BG *:MA>F:MB:G= : -A! BG *:MA>F:MB 2GBO>KLBMR H? ,K>@HG BG %> BL GMER : -KH?>LLHK :M MA> 2GBO>KLBMR H?

4BLO>GL -HBGM %> A:L FHK> MA:G R>:KL H? >QI>KB>G<> M>:@> F:MA>F:MB

BL : LMN=>GM K>L>:K

F>GMHK

BL>QI>KB>G<>=BG=>O>EHIF>GM

=>LB@G :G=

BL:ELH:G:OB=;H:K=@:F>=>LB@G>K:G=IE:R>K

cCBHF=θIH=Bℓ εIHKBG>;;HMM (>ND: HEE>@> +BM:3BK@BGB:BE: #:R>MM>OBEE>0M:M>2GBO>KLBMR

0A>KB'HR=

/HEEBGL HEE>@> 'HR:MB!>;G:MA

4BGHG:0M:M>2GBO>KLBMR

3:E>K>>#:E=NMH

-:EF>: HEE>@> 'HL>IA):D>R +>P*>QB2GBO>KLBMR 'NEB>)>O:G=HLDR #K:FBG@A:F0M:M>2GBO>KLBMR !:OB=*< NG>

4BEEB:F'>P>EE HEE>@>

*BEE>*>KKBP>:MA>K

KHGQOBEE>%B@A0 (BKLM>G/*>LL>K HEHK:=H0M:M>2GBO>KLBMR-N>;EH

E?K>=(*NES>M

#EHKB=:0M:M> HEE>@>:M':

4BEEB:F/:=NEHOBMBK>=

#EHKB=:0M:M> HEE>@> :M': "KB<:*/NMM>K

KBSHG:0M:M>2GBO>KLBMR

!:OB=0FBMA

2GBO>KLBMRH?MA>3BK@BG&LE:G=L

"E:BG>1>KKR

0:BGM'HL>IA`L2GBO>KLBMR

!:OB=1HK:BG %:FIMHG2GBO>KLBMR rσJ=σKσFG *:KP:G;N0:PP: #EHKB=:0M:M> HEE>@> :M': (>GG>MA'>KG:K=

3BK@BGB:0M:M>2GBO>KLBMR

'HAG>R>KL

2GBO>KLBMRH?*:KRE:G=

A:KE>LN>AKE>

#K:GDEBG*:KLA:EE HEE>@> *:MMA>P :MA>R

4H??HK= HEE>@>

*BE HA>G %H?LMK:2GBO>KLBMR

4BEEB:F!>0:E:S:K

KHP:K= HNGMR0F

*NKK:R"BL>G;>K@

2GBO>KLBMRH?*:LL:MMLFA>KLM

quotesdbs_dbs22.pdfusesText_28