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Study Guide on Applications of Derivaties, and Curve Skectching

8 Pages
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Department
Mathematics
Course Code
MATA32H3
Professor
Szegadyand Grinell

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Newton'sMethod
TheRecursionFormula:Ù
:
ë-
;
Ô
L
B
ñ
:
T5
;
Therefore:theNewton'sMethodFormula:
TáLT
á?5F
B
:
Tá?5
;
B
ñ
:
Tá?5
;
Example:ComputeSquarerootof2.
B
:
T
;
LT
6Ft
B
ñ
:
T
;
LtT
T5Lt
T6LtF6
8Lsäw
T7LsäwF4ä69
7
L7
6F5
56 L5;
56 Lsävsxxy
HigherOrderDerivatives:
T7
×
×ë
1
.
uT6
×
×ë
1
xT
×
×ë
1
x
×
×ë
1
r
×
×ë
1
r
B
:
T
;
LT
7
B
ñ
:
T
;
LuT
6
B
ññ
:
T
;
LxT
B
ñññ
:
T
;
Lx
B
:
8
;
:
T
;
Lr
×ì
×ë
LuT
6
×.ì
×
ë
.LxT
×/ì
×
ë
/Lx
×0ì
×
ë
0Lr
Example:Given:T6EU
6Ls,Find×.ì
×ë.
Step1:find1stderivative
tTEtUÛUñLr
×ì
×ë LFë
ì
Step2:Takethederivativeofthedy/dxfoundinstep1.
×.ì
×ë.LFì?Ïä
Ïã:ë;
ì.
Step3:SubstituteStep1'sdy/dxtostep2:
×.ì
×ë.LFì>ã.
ä
ì.LFì..
ì/LF5
ì/
Newton'sMethodandHigherOrderDerivatives
Octoberr18r10
1:36PM
\1 1
www.notesolution.com
Assumethatfisdifferentiableon(a,b)intervalandassumethatfhasa
minimumatpointxrr>atthispointthef'(x)=0
IncreasingandDecreasingFunction:
FisincreasingonanintervalI,ifforeverypair(x,y)intheinterval,and
letsayx<y,thereforef(x)<f(y)
FisdecreasingonanintervalI,ifforeverypair(x,y)intheinterval,and
letsayx<y,thereforef(x)>f(y)
Criteria:Iffisdefinedon(a,b)andf'>0,thenitisincreasing,andiffis
definedon(a,b)andf'<0,thenitisdecreasing.
Example:
B
:
T
;
L5
7
T7FTEs
B
ñ
:
T
;
LT
6FsF»\FsFs\ss\»
TFsrrrrrr r +
TEsrrrrr ++
:
TFs
;
:
TEs
;
+r+
f
Whenx=r1,itisarelativemaximum,andx=1,itisarelativeminimum
MaximumandMinimum:
Fhasarelativemaximumatpointa,ifthereisanopeninterval
containingasuchthat
B
:
T
;
Q
B
:
=
;
Foreveryxintheinterval
Fhasarelativeminimumatpointa,ifthereisanopeninterval
containinga,suchthat
B
:
T
;
R
B
:
=
;
Foreveryxintheinterval
Absolutemaximum=themaxvalueinthewholedomainofthe
function
Absoluteminimum=theminvalueinthewholedomainofthefunction
Important:iffhasarelativeextrema,andfisdifferentiableonan
intervalcontaina,thenf'(a)=0\
Tocheckextremas:
Letabeavaluesuchthatf'(a)=0
Ifasxincreasesthrougha,thesignoff'changesfrom+tor,thenaisa
relativemaximum,andifitisfromrto+,thenaisarelativeminimum.
CriticalValue:
Aisacriticalvalue,iff'(x)=0orf'(x)doesnotexist.
CriticalPoint:
Is(a,f(a)),suchthataisacriticalvalue
BasisofCurveSketching
Octoberr25r10
1:12PM
\2 1
www.notesolution.com
Findf'1.
Findallcriticalvaluesandnearvaluesofthedomain2.
FirstDerivativeTest:
SignChart:
Criticalvalueisdrawnwiththelines.
Example:y=|x|.
Criticalvalueis0,butitisaD.N.E.
f'(a)exists-
f'(a)doesnotexist,butfiscontinuousata-
f'(a)doesnotexist,andfisnotcontinuousata-
TypesofCriticalValues:
Example:ULTA
ë
UñLA
ëETA
ë
UñL
:
sET
;
Aë
Thenr1isthecriticalvalue,byanalyzingthefunction,r1istherelative
minimum.
Example:ULT
.
/
UñLt
u
¾
T
/
Thecriticalvalueis0,butthederivativeof0isnotexist
FunctionsonaclosedInterval:
Iffisacontinuousfunctionondefinedclosedintervalofa,b.fhasboth
absolutemaximumandminimum.
Test:
Findingf'
Computecriticalpointsandvaluesoff@criticalpointsandendpoints
Findmaxandminvalueamongtheseintervals
Concavity:
Concaveup:f">0
Concavedown:f"<0
Inflectionpoints:whenconcavitybehaviourchanges.
Iffchangesconcavityata,thenitisaninflectionpoint
Example:
B
:
T
;
LT
7FT
6Es
B
ñ
:
T
;
LuT
6FtT
B
ññ
:
T
;
LxTFt
B
ññ
l
s
u
p
Lr
ThenConcavedown1/3Concaveup
\3 1
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Description
NewtonsMethodandHigherOrderDerivatives OctoberH18H10 1:36PM NewtonsMethod -; TheRecursionFormula: L * 6 #; Therefore:theNewtonsMethodFormula: * 6 ?#; 6L 6 ?# . * 6 ?#; Example:ComputeSquarerootof2. $ * 6 L 6 . J * 6 L J6 6#L J $ 6$L J . &L IM $ % # #; 6%L IM . % L .$ #$L #$ L ILINNO HigherOrderDerivatives: 6 .K6 $.N6 .N .H .H * 6 L 6 % : ; $ * 6 L K6 *:6 L N6 * :6 L N *:&;:6 L H L K6 $ . .L N6 L N L H . Example:Given:6$- 7 L I,Find . Step1:find1stderivative J6 - J7 7 L H L . Step2:Takethederivativeofthedydxfoundinstep1. ?:; .L . . Step3:SubstituteStep1sdydxtostep2: . . > . . L . L . > L . # . . www.notesolution.com
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