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The primary objectiveof this chapteris to introduceyou to numericaldiftbrentiation Specific objectivesand topicscoveredare ' ' ' ' ' ' the Understanding applicationof hish-accuracy numericaldifferentiation lbrmulas for equispaced data Knowing hclwto evaluatederivativestbr unequallyspaced data Understanding how Richardson extrapolation appliedfbr numerical is differentiation Recognizingthe sensitivityof numericaldifferentiationto dataerror Knowing how to evaluate derivativesin MATLAB with the dif f and qlradient functions Knowing how to generate plots and vector fields with MATLAB colrtour
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At the beginningof Chap 17, we usedcalculusto integratethis equationto determine the vertical distance the jumper has fallen after a time / z
z(t)
(192)
I9I INTRODUCTION BACKGROUND AND
Now suppose problemThat is, you were askedto dethat you were given the reverse termine velocity basedon the jumper's positionas a function of time Because is the init verseof intergration differentiationcould be usedto make the determination: r'(l): dz0) , at 1193)
Eq wouldbring us backto Eq ( 191) Substituting (192)into Eq (193)anddifferentiating Beyond velocity, you might also be askedto computethe jumper's acceleration To do this, we could either take the first derivativeof velocity, or the secondderivative of displacement:
du(t\
t\i/:-----:-:
d 2 z t tl
(194\
In eithercase resultwould be the
a(t) :S ,"ftt (
(1 9 5 )
Although a closed-formsolutioncan be developed this case,thereare other funcfor tions that may be difficult or irnpossible differentiate to analyticallyFurther,suppose that there was some way to measurethe jumper's position at various times during the fall Thesedistances along with their associated times could be assembled a table of discrete as valuesIn this situation,it would be usefulto differentiate discrete the datato determine the velocity and the acceleration both theseinstances, In numericaldifTerentiation methods are availableto obtain solutions This chapterwill introducevou to someof thesemethods
I9I
INTRODUCTION AND BACKGROUND
| 9 | I Whor ls Differentiorion
of Calculusis the mathematics change Because engineers scientists and must continuously deal with systems and processes clrange that calculusis an essential tool of our profession Standingat the heartof calculusis the mathematical conceptof differentiation According to the dictionary definition, to dffirentiate means"to mark off by differences; distinguish; to perceive difference or between"Mathematically, derivetthe in the tive, which servesasthe fundamentalvehicle for diff-erentiation, represents rate of change the variablewith respect an independent to of a dependent variable depicted Fig 19l, the As in mathematical definitionof the derivativebeginswith a difference approximation: Ay Ar
f (xi + Ar) - /(t;) Ar
(1 9 6 )
where -y and /(:r) are alternativerepresentatives the dependent variable and x is the for independent variableIf A:r is allowedto approach zero,asoccursin moving from Fig 191a to c, the difference a becomes derivative: -'
{r+0
f (xi + Ar) - /("r;) Ar
(t97)
NUMERICAL DIFFERENTIATION
+ l(r-, Ar; f(x)
r, + ar
F I G U R EI 9 I Thegrophicol zero opprorimolion defrnition o derivotive: Ax opprooches in goingfrom(o)to (c),thedifference of os becomes derivotive o whereh ldx [which can erlso designated l/ o f '(ri) ]' is the first derivative r'with be oS of r e s p e c t t o - r e v a l u a t e d a t r , A s s e e n i n t h e v i s u a l d e p i c tF og o9 l c , t h e d e r i v a t i v e i s ii n l f the slopeof the tangentto the curve at,rr The secondderivativereDresents derivativeof the first derivative the
#:*(#)
/(r * Ar,t') - /(;r r')
A,r+o
( 198)
Thus, the secondderivativetells us how fast the slopeis changingIt is commonlyrefened to as the cun,ature,because high value for the secondderivativemeanshigh curvature a Finallypartialderivatives usedfor lunctionsthat depend morethanonevariable are on Partialderivativescan be thoughtof as taking the derivativeof the functionat a pointwith For example,given a function/ that depends both r all but one variableheld constant on y, the partial derivativeof/with respectto{ at an arbitrarypoint (x, 1,)is detined and as
( 199)
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