Calculus - A New Horizon

1302 Pages·1998·293.3 MB·English

by  Howard Anton

GEOTUETRY FORMULAS
= rr.tr = [rrr]l {Illff ]rr.r ljcrsht. r=rr.r,,lbr\'. rrLLu\ -nrnrLlishl. / flr.utrlLrrn.c . = xr.l.n!rh
Para e ogranl Tr arg e Trapezo d Sector
.
/
-_r l -6- .^\
1
1-
1=!u+bth .t=;rr:.C=lzr
(tl
n raO a nsi
R grl I r.J ar Cy fder Rrght CrrcL ar Cone Any Cylnder or Pr sm wth Paralle Bases Sphere
/ffr
ffir
,,L_{
l =l;r''.S=.1;r'l
ALGEBRA FORMULAS
THE QUADRATIC
FORMULA THE BINOMIAL FORMULA
,;;.,;
i,,,'",',,*,,' r \ =\/ , , i+.,, :.,,1!r lt1_1.. ,
cqurti,rr ail L\ +( =0arc
lr.'
. /r + \ii,r -tdr rr rr":r" ,." '.ir!":.: '('l !!' ',rr. .1,n11r, L;y"
1,,
TABLE OF INTEGRALS
BASIC FUNCTIONS
, ,' ,t,,: :'l-' ' , n. 1,, a,=1' ,,
[ I lr,r
:. I l =ttl |+e tt. ,[ u,,,r, =,r,, r ! + ('
., /, .u, :, , t:. .l -u,,t, = lr \inr r c
, /.'",,a,= co\r-( ,., /'".,,r,:rn sec,,, t.,n! r(
=tn rrLn(l;r+ 1,i) +c
, r,, : \in,/ r ('
./.."., ,. 1""'"= rrr c\cr-cot! -('
u. ,.,,,t, = rn scc, , (' :tn ranlrr +C
1
- ,,--,r.i 'r ,r r ,,. .u-',,r, =,.or r,-tn' t-, ,:+r'
l" r f
'../..^ .r,,=a.o. , .r-rrr' '" /*. ",,=,(. , ,
, ,,.-,.f 1
,. ./ '"r',,r,=rtanLrr ln r'l+ rrr + c l- ,
RATIONAL FUNCTIONS CONTAINING POWERS OF a + bu lN THE DENOMINATOR
I ottllu+h/ll+C II udu - I I u I Ii.
-lhu Jta.b":' ht Lz,r I bu a bur
( l. Jt --"u:u td,u- oI tl i'"+4" -)oLa- bu' a no bu I) c III ut.dt+ub ut aI ln a+uhu +a
t unu t f a l I du t h u+bul
JI ta+bu)- --l-t1-t'nltd+rth1du -c JI ,2-=-, a ttu, ln , tt-
-- l --_-+ I
Jt r,",t ,l^u *-rl'I" T a:o u )'1 t" ")1( 6',7. JII 'tl/1 d) th \\.. -- ala I tu) I aI ln a - bu t'
RATTONAL FUNCTTONS CONTATNTNG a2 + u2 tN THE DENOMTNATOR (a > 0)
t/llu f nu I u nl
68. jI ar'ij =-ula' n -+C 70. JI u' a' 2u u+.11 C
I du 1 u+al f hu-, -l+
6r). JI _a-= _ut' n ta u-a)l+c -1. ,tI -o: -ltn4 lu- h2- ln. tu ) J ran --c
INTEGRALS oF r a2 + u2, '. a2-u2,r u2 - a2 AND THEIR RECIPRoCALS {a > o)
':. Il haJ)i o a,, - lJt +a -"7tn', - '/,' t o-' , 7;
--..\ . lr\'dr-, , a,dtt- u,, 1t,-'; a, -.ri,^u-Ju -ol-. 7(t.
-r ll -r,-Ea) ;a, :," -, -!.i" 1+c 7"7.
POWERS OF U MULTIPLYING OR DIVIDING ' A2 - U2 OR ITS RECIPROCAL
iB l,|G-,,td,:irt,, ",nF ,,,+f;,;"'1+c 8r. II _u-z du ,. ,/,'t ,,1 - o-i),in --.
*, J \la'-u
- I -7 ^ : "e- -."1" * 9- 82. JIt u\_/ad_r'- _u_' l-_ I - o+ J,tl u1lli.
x0. Ir .G-a, '5 1 -.in' -C tI: : Ju rF -;l I
.luua J Jal
u2 u2
-+C
POWERS OF U MULTIPLYING OR DIVIDING , + A2 OR TITIN RECIPROCALS
'2
u. I urlF + a ttu : !s1 ; n2f t2 ',5
:
s5. I u\,lF -z d,, !a, o,)'t. + c
,1,t
$,,./ _t,"1,ffi_. .)2.
J uJu) -.'
I du I
tl7. l:: sec' u +C
I uJ u) - a)
,*r. l '/u1 -e'Itu : rrF ,? ou., \ +c
*,. /"- "di-,, -,,
",1"-'fr=-* (
Ju
INTEGRALS CONTATNTNG (a2 + u2)3 2,1u2 - u2)3 2,lu2 - u2)3 2 (r>O)
JII \a' Juu') ' a'\/a' u' ro0. I e'? + "2)1/2 d, : irz,' + si1,iF + a + $ r.r, + "/i' + ;t + c
I
J| g-:t :l+doutclttt u2 Ju: +a2 +( r:i. I e' - a1)1/2 ,tu = itz,' - s,; nlF a+$rn,+,/i'-,'1 +c
ee. l \a2 u2)3t2,ju :,f,12,'1 - s"'t.r'J -,c+ $ri"-' 1 +c
-:+
RECIPROCALS OF BASIC FUNCTIONS
| T1
ft. J[ 1+sin/,r a,:tin!+sec,+. ,t. .l ,.,r,du- tu+tn,rr,_co.a, C
$. [ --Ld,: corx+csc,+c :.,. / -l dri-, corr'+c,.,,+L
J lf sec!
,n. Lt1 , ,.,, d, . i,, r lrlcos!/ =.rn'., , ( 24. J[ 1+c|s c,/ du-u tanu+,e-u+e
1 |
2r. [ du:)n ral,tt+c ls. [ du:u tnt+,u\+c
.l t+e'
POWERS OF TRIGONOMETRIC FUNCTIONS
j'
ze. | "i,'1,a,: - i sin2x + c zz.. lco(uau: cotu-u+c
27. | ."", ,d,: !,+!"i"2,+c $. u+C
lsec2udu=ral
.
28. .,t. ,a, .o,u-c
lan2uau:tanu-u+c 1..,
,u. | .n",,1, - j. ""- ,.-" t'-t | ,^"- ,t, .r5. J/c oi ridr- n-|t .o''- , Iror" uou
j"""","'.,*f J
zo. |""*,a,: 1,""",0, $. I sec" uau: *".." ',."',* fi | *" ',a,
r. la',a":*,,"' ', l'''',a, n. : -)-csc,-2acor,+ fi | *",,a,
1."'y-,a,
PRODUCTS OF TRIGONOMETRIC FUNCTIONS
I \intm I ht, \$(n ht| t co'(n n) co\Ifl - n),/
.r8. JI sinzasrnza/a 2(n I n) r 2(n "-)TC 40. J 2n.nt )pn-n1
t \ntn t nt aco'" r"
39. .lI co"nu,o'nudu= tl-n + n) j ltm - n) t( 41. J[ "n-"*:,a,--]""-' ._n 'n-. n.t/,,"- !ucos ud,
_ srn'l rco."- , |
t, [.;n-,,o: ,au
n+D n+n
I
PRODUCTS OF TRIGONOMETRIC AND EXPONENTIAL FUNCTIONS
42. .I1I e" .|nhu Iu - A,'p-+a rD,'1" tinbu hco'bu| C n. .Il a' co,un at - a.-c+lD.,',a.o'bL,-a'tnbut c
POWERS OF u MULTIPLYING OR DIVIDING EASIC FUNCTIONS
44. I usinudu : sinu ucosu i C 51. I ue'au = e'ru tt+c
.t
$. 1,""",d,=cos,,+xsin,, +c .lt u" "" o" : ,'"' , .t[n" " ,"tn,
46. I u2 s)n, d, :2,sin,' + (2 a') cos t + c 53. J[ ,n't,-na'I-n a 1na JIu" o,o, ,
47. 1,'1,.s,,t,: 2!cosll + (x2 - 2)sin' +c 5,1. JI e,'d, '.t- rn-t.'t u. t " I tl,I '-e" dd
$r. | ,,:"i,,d,:-u'cosu+n I u' lcosudu 55. II a'd,u' - ,, -attt) "''--,l-n-at JI a d
n
$. .[ u'cosudu:u'sin,-,.f ,'-1 s;nuau 56. , . :lnLln4 +c
[,'n,a,:ffi,r*,,*,',*.
so.
POLYNOMIALS MULTIPLYING BASIC FUNCTIONS
57. Inot"""a,=)*V'- \t, r,t"'" +)t'ro"" .. lsisns artemate: +-+-.. ]
58. lnAl"in.ra, = -!p@)cosau+ \p'(u)s;nau+\p"(u)cosau- lsigns altemate in pairs after first term: + + - - + + - - . .l
5g_ ln{,t"o".,a,: )no,"^., * }, aw"., }o'rr"^., . [sisns a]temate in pairs: ++--++--..1
,C','rERS 0F ! l,4LlLT PL','l!'lc CR DlviDli'lG . n + bl 0ll ITI il:CIPqCCAL
1 ,t+ht ./t
++a ln (lr
-n Jd+fu+\,d
) atl)u
+{ rr
\r Llr
'.: :: I . I iic 2,.-! -r.- 0R --'S eEC PROCAL
1 l. '/;-: 'r"-',,.', ( )+c
i.#;: ;-+'" (+)
!'-nr.,',,
i t-. .
"
j-: tt
--
' ./ ",r / rl 'l a' lar rl
':l
li:']
rri:!er.rnJ
TRIGO NO fulETRIC iDENTiTIES
SIG .I iDEI]T]T ES
aoil Hr= aoid
a\a .i= Clai lci --r=\ae 4
SU PPtEI,]E I]T iDEiiT.T ES
aLrI r "r= Jrlr/
-l
J,r:1 - = rJn ,
lJnl]-Ian,i
I rirr ( Irn i
Ian.? trn ;
l.inr-,r:
1-unaunri
;AI :.ANGLE FORIT]iJLAS
:'rr lc = l..rsr d l
.r*la:1 :\in o
Onlcn(ns
A New Honrzorv
SrxrH Eonotl
HowRno Aruron
Drexel University
JOHN WILEY & SONS, INC.
New York Chichester Brisbane Toronto Singapore
Mathematics Editor: Barbara Holland
Associate Editor: Sharon Smith
Freelance Production Manager: Jeanine Furino
Production and Text Design: HRS Electronic Text l\4anagement
Copy Editor: Lilian Brady
Photo Editor: Hilary Newman
Electronic lllustration: Techsetters, lnc.
Typesetting: Techsetters, lnc.
Cover Design: l\,4adelyn Lesure
Cover Photo: O Dann Colfey/The lmage Bank
This book was set in Times Roman by Techsctte$, Inc.. and prinled and bound by Von Hoffmann Press, Irc- The
cover was printed by The Phoenix Color Corp.
Recognizing the impoftance ofpreserving what has been written, it is a policy of John Wiley & Sons,Inc. to have
books of enduring value published in the United States printed on acid-free paper. and we exert our best efforts
to that end.
The paper on this book was manufactured by a mill whose forest management programs include sustained yield
harvesting ofits timberlands. Sustained yield harvesting pinciples ensure that lhe numben of trees cut each year
does not exceed the amount of new growth.
Copyright O 1999, John Wiley & Sons, Inc. All rights reserved.
No part of this publication may be rcproduced, stored in a rclrieval system. or fansmilted in any form or by any
means, electronic, mechanical, photocopying. recording. scanning. or othenvise. except as permitted under
S ections 1 07 and 108 of the I 976 United States Copydghl Act, without either the prior written permission of the
Publisher. or authorization through payment of the appropriate per copy fee io the Copyright Clearance Center,
222 Rosewood Drive, Daflvers. MA 01923. [phone]. fax [phone]- Requests 1() the Publisher for
permission should be addressed to the Pe.missions Depatment. JohnWiley & Sons,lnc.,605 ThirdAvenue. New
York. NY 10158 0012, [phone], fax [phone]. E mailr PERMREQ@WILEYCOM.
l)eriv? is a registered trademark of Soft Warehouse. Inc.
Mdple is a registered tmdemark ofWaterloo Maple Software, Inc.
Marhemaica is a register€d Fademark of Wolfram Research,Inc.
tsBN 0 471 15306 0
Printed in the United States ofAmerica
r0 I I 7 6 5 4
3
H
urvo.a Anton obtained his B.A. trom Lehigh University, his M.A. from the
Aeour
Univelsity of Illinois, and his Ph.D. from the Polytechnic University of Brooklyn,
Hownno Aruror,t all in mathematics. In the early 1960s he workecl for Burroughs Corporation and
Avco Corporation at Cape Canaveral, Florida, where he was involved with missile
tracking problcms for the manned space program. In 1968 he joined the
Mathematics Department at Drexel University, where hc taught full time until
1983. Since that time he has been an adjunct professor at Drexel and has devoted
the majority oi his time to textbook writing and acriviries for mathematica] asso-
ciations. Dr. Anton was President of the Epentl Seclion of the Mathematical
Association of America (MAA), served on the board of Governors of that organi-
zation, and guided the creation of thc Student Chapters of the MAA. He has pub-
lished numerous research papers in Functional Analysis, Approximation Theory,
and Topology, as well as pedagogical papers on applications of mathematics. He
is best known lbr his textbooks in mathematics, which are among the most wide-
ly used in the world. There are currentJy more than ninety versions of his books,
including translations into Spanish, Arabic, PortugLLese, Italian, Indonesian.
French, Japanese, Chinese. Hebrew, and German. Dr. Anton has an avid interest in
computer technology as it relates to mathematical education rnd publishing. He
has devcloped pedagogical software fbr teaching calculus and linear algebra as
well as various sofiware programs for the publishing industry that automate the
production ol'fbur color mathenatical text and art. For relaxation he enjoys trav-
eling and photography.
To
My Wife Pat
My Children Brian, David, and Lauren
ln Memory of
My Mother Shirley
Stephen Girard (1750-1831)-Benefactor
Albert Herr-Esteemed Colleague and Contributor
Wh"n I U"gon writing the first edition of this calculus text almost 25 years ago, the task, though
daunting, was straightiorward in that the content and organization of a standard calculus course was nearly
universal-the chalienge for me at that time was to present the material in a livelier style and with greater
cla.rity rhan my predecessors. Since this calculus text is still among the most widely used in the world, I
take comfort that the goals I set for myself as a young w ter and mathematician have been achieved.
Howevel, times are changing, and the era of a standard and unive$al calculus course seems destined lbr
the repository of slide rules and three-cent stamps. We are witnessing a lot of experimentation with the con-
tent, organization, and goals of calculus-sonre of which has been successful and some of which has not.
Thus, my challenge in writing the sixth edition has been to create a text that has all of the strengths of (he
earlier editions, yet incorporates those new ideas that are clearly important and have withstood the objective
scrutiny of skilled and thoughtful teachers.
In preparing for this edition, I sought advice fiom outstanding teachers at a wide variety of institutions.
Needless to say, I received a diversity of opinioDs-some leviewers advised against any major changes,
arguing that the book was already clearly written and wo|king well in the classroom, while others felt that
major changes were required to ircorporate technology and rnake the book more contemporary. I listened
carelully, and the lively discussions that followed hclped me formulate my pbilosophy for the new edition.
Many of the specific changes are itemized in the pretace, but here are some of the general goals:
.
Add graphing calculator and CAS materials to the text in a way that will allow students who have
rhose roois ro use them but that will not prevent tbe text from being used by those students who do not
have access to that technology.
.
Place more emphasis on mathematical modeling and appiications.
.
Incorporate new examples and exercises that will be neaningf'ul to today's students and will more
accurately convey the role of caiculus in the real world.
.
Widen the variety of exercises to focus ntore on conceptual understandilrg through coniecture, multi-
step anaiysis, expository writing. and what-if anaiysis.
In addition, I wanted to provide some optional innovative materials that would capture the student's interest
and plovide the kind of prob!em-solving experience that he or she might find in a research or industrial set-
ting. This gave birth to atr exciting set of modules that we have called Erp lding tlrc C.tlculus Hori?.o .
These ruodules appear at the ends of selected chapters and each has an optional Internet component that we
hope will grcw dynamically over time with input from teachers and students.
In developing this edition I have stood firm on two principles lhat were adhered to in earlier editions:
. The text material is prcsented at a n'lathematical level that is suitable for students who will embark on
careers in engineering and science.
.
lt remains a primary goal of the text to teach the student clear, logical. mathematical thinking.
Informal discussions play an imponant notivational role in the exposition and are used extensively, but
cventually I want the studert to be able to read and understand the language of mathematjcs.
Although tbis edition has many changes and new features, they have been implemented in aJleible way
that will accommodate a wide vadety of teaching philosophies. Thus, I am confidcnt that professors who
have had positive experiences with earlier editions will be comfortable with this revision, and I am hopeful
that those prcfesson who are looking for a contemporary text with an established history ol success in the
classroom will be pleased with the innovations in this new edition.
Sincerely,
t-,
l"*t- r''^+"'''
Howard Anton