Calculus For Biology and Medicine: Pearson New International Edition

707 Pages·2013·21.543 MB·English

by  Claudia Neuhauser

C
a
l
c
u
l
u
s

f
o
r

B
i
o
l
o
g
y

a
n
d

M
e
d
i
c
i
n
e

N
e
u
h
a
u
s
e
r

T
h
i
r
d

E
d
i
t
i
o
n
Calculus
for Biology and Medicine
ISBN 978-1-29202-226-0
Claudia Neuhauser
Third Edition
9 781292 022260
Calculus
for Biology and Medicine
Claudia Neuhauser
Third Edition
ISBN 10: 1-292-02226-4
ISBN 13: 978-1-292-02226-0
Pearson Education Limited
Edinburgh Gate
Harlow
Essex CM20 2JE
England and Associated Companies throughout the world
Visit us on the World Wide Web at: www.pearsoned.co.uk
© Pearson Education Limited 2014
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted
in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without either the
prior written permission of the publisher or a licence permitting restricted copying in the United Kingdom
issued by the Copyright Licensing Agency Ltd, Saffron House, 6–10 Kirby Street, London EC1N 8TS.
All trademarks used herein are the property of their respective owners. The use of any trademark
in this text does not vest in the author or publisher any trademark ownership rights in such
trademarks, nor does the use of such trademarks imply any affi liation with or endorsement of this
book by such owners.
ISBN 10: 1-292-02226-4
ISBN 13: 978-1-292-02226-0
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library
Printed in the United States of America
12233455666693072930866612267145812321
P E A R S O N C U S T O M L I B R AR Y
Table of Contents
Chapter 1. Preview and Review
Claudia Neuhauser 1
Chapter 2. Discrete Time Models, Sequences, and Difference Equations
Claudia Neuhauser 62
Chapter 3. Limits and Continuity
Claudia Neuhauser 91
Chapter 4. Differentiation
Claudia Neuhauser 132
Chapter 5. Applications of Differentiation
Claudia Neuhauser 202
Chapter 6. Integration
Claudia Neuhauser 276
Chapter 7. Integration Techniques and Computational Methods
Claudia Neuhauser 327
Chapter 8. Differential Equations
Claudia Neuhauser 391
Chapter 9. Linear Algebra and Analytic Geometry
Claudia Neuhauser 434
Chapter 10. Multivariable Calculus
Claudia Neuhauser 505
Chapter 11. Systems of Differential Equations
Claudia Neuhauser 588
Appendix A Frequently Used Symbols
Claudia Neuhauser 661
Appendix B Table of the Standard Normal Distribution
Claudia Neuhauser 662
Answers to Odd-Numbered Problems
Claudia Neuhauser 663
I
666689997157
References
Claudia Neuhauser 687
Contributors to Calculus
Claudia Neuhauser 691
Derivatives and Integrals
Claudia Neuhauser 695
Index 697
II
1
Preview and Review
LEARNING OBJECTIVES
Thefirsttwosectionsofthischapterserveasareviewofalgebra,trigonometry,andprecalculus,
materialneededtomasterthetopicscoveredinthisbook.Section1.3reviewsgraphingfunctions
andintroducestheimportantconceptoftransformingfunctionsintolinearfunctions.Thesection
includesasubsectiononvisualizingverbaldescriptionsofbiologicalphenomena.
A Brief Overview of Calculus
IsaacNewton(1642–1727)andGottfriedWilhelmLeibniz(1646–1716)aretypically
creditedwiththeinventionofcalculusandwerethefirsttodevelopthesubjectsys-
tematically.
Calculushastwoparts:differentialandintegralcalculus.Historically,differential
calculus was concerned with finding lines tangent to curves and with calculating ex-
trema(i.e.,maximaandminima)ofcurves.Integralcalculushasitsrootsinattempting
to determine the areas of regions bounded by curves or in finding the volumes of
solids.Thetwopartsofcalculusarecloselyrelated:Thebasicoperationofonecanbe
consideredtheinverseoftheother.Thisresultisknownasthefundamentaltheorem
ofcalculusandgoesbacktoNewtonandLeibniz,whowerethefirsttounderstandits
meaningandtoputittouseinsolvingdifficultproblems.
Findingtangents,locatingextrema,andcalculatingareasarebasicgeometricprob-
lems,anditmaybesomewhatsurprisingthattheirsolutionledtothedevelopmentof
methods that are useful in a wide range of scientific fields. The main reason for this
historical development is that the slope of a tangent line at a given point is related
to how quickly the function changes at that point. Knowing how quickly a function
changesatapointopensupthepossibilityofadynamicdescriptionofbiology,suchas
adescriptionofpopulationgrowth,thespeedatwhichachemicalreactionproceeds,
thefiringrateofneurons,andthespeedatwhichaninvasivespeciesinvadesanew
habitat.Forthisreason,calculushasbeenoneofthemostpowerfultoolsinthemath-
ematicalformulationofscientificconcepts.Applicationsofcalculusarenotrestricted
tobiology,however;infact,physicswasthedrivingforceintheoriginaldevelopment
of calculus. In this text we will be concerned primarily with how calculus is used in
biology.
Inadditiontodevelopingthetheoryofdifferentialandintegralcalculus,wewill
considermanyexamplesinwhichcalculusisusedtodescribeormodelsituationsin
the biological sciences. The use of quantitative reasoning is becoming increasingly
moreimportantinbiology—forinstance,inmodelinginteractionsamongspeciesin
acommunity,describingtheactivitiesofneurons,explaininggeneticdiversityinpop-
ulations,andpredictingtheimpactofglobalwarmingonvegetation.Today,calculus
(Chapters2–11)andprobabilityandstatistics(Chapter12)areamongthemostim-
portantquantitativetoolsofabiologist.
From Calculus for Biology and Medicine, Third Edition. Claudia Neuhauser.
Copyright © 2011 by Pearson Education, Inc. All rights reserved.
1
2 Chapter 1 Preview and Review
1.1 Preliminaries
Thissectionreviewssomeoftheconceptsandtechniquesfromalgebraandtrigonom-
etrythatarefrequentlyusedincalculus.Theproblemsattheendofthesectionwill
helpyoureacquaintyourselfwiththismaterial.
1.1.1 The Real Numbers
Therealnumberscanmosteasilybevisualizedonthereal-numberline(seeFigure
a b
1.1),onwhichnumbersareorderedsothatifa < b,thena istotheleftofb.Sets
(cid:2)5(cid:2)4(cid:2)3(cid:2)2(cid:2)1 0 1 2 3 4 5 (collections)ofrealnumbersaretypicallydenotedbythecapitalletters A, B,C,etc.
Todescribetheset A,wewrite
Figure1.1 Thereal-numberline.
A x condition
={ : }
where“condition”tellsuswhichnumbersareintheset A.Themostimportantsets
incalculusareintervals.Weusethefollowingnotations:Ifa <b,then
theopeninterval(a,b) x a < x <b
={ : }
and
theclosedinterval a,b x a x b
[ ]={ : ≤ ≤ }
Wealsousehalf-openintervals:
a,b) x a x <b and (a,b x a < x b
[ ={ : ≤ } ]={ : ≤ }
Unboundedintervalsaresetsoftheform x x >a .Herearethepossiblecases:
{ : }
a, ) x x a
[ ∞ ={ : ≥ }
( ,a x x a
−∞ ]={ : ≤ }
(a, ) x x >a
∞ ={ : }
( ,a) x x <a
−∞ ={ : }
Thesymbols“ ”and“ ”mean“plusinfinity”and“minusinfinity,”respectively.
∞ −∞
Thesesymbolsarenotrealnumbers,butareusedmerelyfornotationalconvenience.
Thereal-numberline,denotedbyR,doesnothaveendpoints,andwecanwriteRin
thefollowingequivalentforms:
R x < x < ( , )
={ :−∞ ∞}= −∞ ∞
Thelocationofthenumber0onthereal-numberlineiscalledtheorigin,andwe
canmeasurethedistanceofthenumberx totheorigin.Forinstance, 5is5unitsto
−
theleftoftheorigin.Aconvenientnotationformeasuringdistancesfromtheorigin
onthereal-numberlineistheabsolutevalueofarealnumber.
Definition Theabsolutevalueofarealnumbera,denotedby a ,is
| |
a ifa 0
a ≥
| |=(cid:2) a ifa <0
−
Forexample, 7 ( 7) 7.Wecanuseabsolutevaluestofindthedistance
|− | = − − =
betweenanytwonumbersx andx asfollows:
1 2
distancebetweenx andx x x
1 2 1 2
=| − |
Notethat x x x x .Tofindthedistancebetween 2and4,wecompute
1 2 2 1
2 4 | − 6|=6|,or−4 | ( 2) 4 2 6. −
|− − |=|− |= | − − |=| + |=
2
1.1 Preliminaries 3
Wewillfrequentlyneedtosolveequationscontainingabsolutevalues,forwhich
thefollowingpropertyisuseful:
Letb 0.Then
≥
1. Fora 0, a bisequivalenttoa b.
≥ | |= =
2. Fora <0, a bisequivalentto a b.
| |= − =
EXAMPLE1 Solve x 4 2.
| − |=
Solution If x 4 0,then x 4 2andthus x 6.If x 4 < 0,then (x 4) 2and
− ≥ − = = − − − =
thus x 2.Thesolutions,illustratedgraphicallyinFigure1.2,aretherefore x 6
= =
and x 2. The points of intersection of y x 4 and y 2 are at x 6 and
= = | − | = =
x 2.Solving x 4 2canalsobeinterpretedasfindingthetwonumbersthat
= | − | =
havedistance2from4.
y
6 (cid:2)x (cid:2) 4(cid:2)
2
5
4
3
1
(cid:2)1 1 2 3 4 5 6 7 x
(cid:2)1
Figure1.2 Thegraphofy x 4 andy 2.Thepoints
=| − | =
ofintersectionareatx 6andx 2.
= =
We write the solution of an equation of the form a b as either a b or
| | = | | =
a b,illustratedinthenextexample.
=−
EXAMPLE2 Solve 3x 1 1x 1.
|2 − |=|2 + |
Solution Either
3 1
x 1 x 1
2 − =− 2 +
(cid:3) (cid:4)
3 1
x 1 x 1 3 1
2 − = 2 + x 1 x 1
or 2 − =−2 −
x 2
= 2x 0
=
x 0
=
AgraphicalsolutionofthisexampleisshowninFigure1.3.
Returning to Example 1, where we found the two points whose distance from
4 was equal to 2, we can also try to find those points whose distance from 4 is less
than (or greater than) 2. This amounts to solving inequalities with absolute values.
LookingbackatFigure1.2,weseethatthesetofx-valueswhosedistancefrom4is
lessthan2(i.e., x 4 <2)istheinterval(2,6).Similarly,thesetofx-valueswhose
distancefrom4|is−grea|terthan2(i.e., x 4 > 2)istheunionofthetwointervals
( ,2)and(6, ),or( ,2) (6,| −). |
−∞ ∞ −∞ ∪ ∞
3
4 Chapter 1 Preview and Review
y
6
5
4
3
2 (cid:2)3x (cid:2) 1(cid:2)
2
1
(cid:2)1x (cid:3) 1(cid:2)
2
(cid:2)4 (cid:2)2 2 4 x
(cid:2)1
Figure1.3 Thegraphsofy 3x 1 andy 1x 1.
=|2 − | =|2 + |
Thepointsofintersectionareatx 0andx 2.
= =
Ingeneral,tosolveabsolute-valueinequalities,thefollowingtwopropertiesare
useful:
Letb >0.Then
1. a <bisequivalentto b <a <b.
| | −
2. a >bisequivalenttoa >bora < b.
| | −
EXAMPLE3 (a) Solve 2x 5 <3. (b) Solve 4 3x 2.
| − | | − |≥
Solution (a) Werewrite 2x 5 <3as
| − |
3<2x 5<3
− −
Adding5toallthreeparts,weobtain
2<2x <8
Dividingtheresultby2,wefindthat
1< x <4
Thesolutionisthereforetheset x 1 < x < 4 .Inintervalnotation,thesolution
canbewrittenastheopeninterva{l(:1,4). }
(b) Tosolve 4 3x 2,wegothroughthefollowingsteps:
| − |≥
4 3x 2
− ≥ 4 3x 2
3x 2 − ≤−
− ≥− or 3x 6
2 − ≤−
x x 2
≤ 3 ≥
Thesolutionistheset x x 2orx 2 ,or,inintervalnotation,( ,2 2, ).
{ : ≥ ≤ 3} −∞ 3]∪[ ∞
1.1.2 Lines in the Plane
Wewillfrequentlyencountersituationsinwhichtherelationshipbetweenquantities
canbedescribedbyalinearequation.Forexample,whenaweightisattachedtoa
helical spring made of some elastic material (and the weight is not too heavy), the
relationshipbetweenthelength y ofthespringandtheweightx is
y y kx (1.1)
0
= +
where y denotesthelengthofthespringwhennoweightisattachedtoitandk isa
0
positiveconstant.Equation(1.1)isanexampleofalinearequation,andwesaythat
x and y satisfyalinearequation.
4
1.1 Preliminaries 5
y Thestandardformofalinearequationisgivenby
Ax By C 0
+ + =
(x, y)
2 2 where A,B,andCareconstants, AandBarenotbothequalto0,andxandyarethe
y (cid:2) y twovariables.Inalgebra,youlearnedthatthegraphofalinearequationisastraight
2 1
(x1, y1) line.
Ifthetwopoints(x ,y )and(x ,y )lieonastraightline,thentheslopeofthe
1 1 2 2
x (cid:2) x
2 1 lineis
y y
m 2− 1
= x x
2 1
−
x
(SeeFigure1.4.)Twopoints(oronepointandtheslope)aresufficienttodetermine
Figure1.4 Theslopeofastraight theequationofastraightline.
line. Ifyouaregivenonepointandtheslope,youcanusethepoint–slopeformofa
straightlinetowriteitsequation,givenby
y y m(x x )
0 0
− = −
wheremistheslopeand(x ,y )isapointontheline.Ifyouaregiventwopoints,first
0 0
computetheslopeandthenuseoneofthepointsandtheslopetofindtheequation
ofthestraightlineinpoint–slopeform.
Lastly,themostfrequentlyusedformofalinearequationistheslope–intercept
form
y mx b
= +
wheremistheslopeandbisthey-intercept,whichisthepointofintersectionofthe
linewiththe y-axis;the y-intercepthascoordinates(0,b).
Wesummarizethesethreeformsoflinearequationsinthefollowingbox:
Ax By C 0 (StandardForm)
y +y m+(x =x ) (Point–SlopeForm)
0 0
− = −
y mx b (Slope–InterceptForm)
= +
EXAMPLE4 Determine,inslope–interceptform,theequationofthelinepassingthrough( 2,1)
−
and(3, 1).
−2
Solution Theslopeofthelineis
y m y2−y1 −12 −1 −32 3
x (cid:4) h = x x = 3 ( 2) = 5 =−10
2 1
− − −
Usingthepoint–slopeformwith( 2,1),wefindthat
−
3
k y 1 (x ( 2))
− =−10 − −
y (cid:4) k
or,inslope–interceptform,
3 2
h x y x
=−10 + 5
Wecouldhaveusedtheotherpoint,(3, 1),andobtainedthesameresult.
Figure1.5 Thehorizontalliney k −2
andtheverticallinex h. = WenowrecalltwospecialcasesthatweillustrateinFigure1.5:
=
y k horizontalline(slope0)
=
x h verticalline(slopeundefined)
=
5