082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017
516 Chapter 18 Taylor polynomials, Taylor series and Maclaurin seriessop 2(V) =I s(e 40V a −1) +40Is e 40V a (V −Va ) +1600I se 40V (V −Va a) 22Thecoefficientsneedtobecalculatedonlyonce.Afterthatthecalculationofacurrentvalue only involves evaluating a quadratic.EXERCISES18.31 (a) Obtainthe second-order Taylor polynomial,p 2 (x),generatedbyy(x) = 3x 4 +1aboutx = 2.(b) Verifythaty(2) = p 2 (2),y ′ (2) = p ′ 2 (2)andy ′′ (2) = p ′′ 2 (2).(c) Evaluate p 2 (1.8)andy(1.8).2 (a) Calculate the second-order Taylor polynomial,p 2 (x),generated byy(x) = sinx aboutx = 0.(b) Calculate the second-order Taylor polynomial,p 2 (x),generated byy(x) = cosx aboutx = 0.(c) Compare your resultsfrom (a) and(b)with thesmall-angle approximations given in Section6.5.3 Afunction,y(x),issuchthaty(−1) = 3,y ′ (−1) = 2andy ′′ (−1) = −2.(a) Statethe second-order Taylor polynomialgenerated byyaboutx = −1.(b) Estimatey(−0.9).4 Afunction,y(x),satisfiesthe equationy ′ =y 2 +xy(1)=2(a) Estimatey(1.3)usingafirst-orderTaylorpolynomial.(b) Bydifferentiatingthe equationwith respect tox,obtain an expression fory ′′ .Hence evaluatey ′′ (1).(c) Estimatey(1.3)usingasecond-order Taylorpolynomial.5 Afunction,x(t),satisfiesthe equationẋ=x+ √ t+1x(0)=2(a) Estimatex(0.2)usingafirst-orderTaylorpolynomial.(b) Differentiate the equation w.r.t.t and henceobtain an expression forẍ.(c) Estimatex(0.2)usingasecond-order Taylorpolynomial.6 Afunction,h(t),isdefined byh(t) = sin2t +cos3tObtainthe second-order Taylor polynomial generatedbyh(t)aboutt = 0.7 Thefunctionsy 1 (x),y 2 (x)andy 3 (x)are defined byy 1 (x) =Ae x ,y 2 (x) =Bx 3 +Cx,y 3 (x) =y 1 (x) +y 2 (x)(a) Obtainasecond-order Taylor polynomial fory 1 (x)aboutx = 0.(b) Obtainasecond-order Taylor polynomial fory 2 (x)aboutx = 0.(c) Obtainasecond-order Taylor polynomial fory 3 (x)aboutx = 0.(d) Canyou draw any conclusionsfrom youranswers to (a),(b)and(c)?Solutions1 (a) p 2 (x) = 72x 2 −192x +145(c) p 2 (1.8) = 32.68,y(1.8) = 32.49282 (a) p 2 (x) =x (b) p 2 (x) =1− x223 (a) p 2 (x) = −x 2 +4(b) p 2 (−0.9) = 3.19.This isan approximation toy(−0.9).4 (a) 3.5(b) y ′′ = 2yy ′ +1,y ′′ (1) = 21(c) 4.4455 (a) 2.61(b) ẍ=ẋ +2 √ t +1(c) 2.67
18.4 Taylor polynomials of the nth order 5176 p 2 (t) =1+2t −4.5t 2( )7 (a)A1 +x+ x22(b) Cx(c) A+(A+C)x+ Ax22TechnicalComputingExercises18.3Many technicalcomputing languageshave thecapability ofproducingaTaylor polynomial ofafunction.Insome languagesthisis available asadefaultwhereasin others, such asMATLAB ® ,thisisoffered via a toolboxfunctionwhichmayneedto beloaded orpurchased separately. Ifthe software youare usingdoes not have the capability to produce theTaylor seriesautomaticallyyou maychoose to dothispartmanuallyandthenplot the resultsusingthetechnicalcomputing language.1 (a) Calculate the second-order Taylor polynomial,p 2 (x),generatedbyy(x) =x 3 aboutx = 0.(b) Drawy(x)andp 2 (x)for −2 x2.2 (a) Calculate the second-order Taylor polynomial,p 2 (x),generated byy(x) = sinx aboutx = 0.(b) Drawy(x)andp 2 (x)for −2 x 2.3 (a) Calculate the second-order Taylor polynomial,( )1p 2 (x),generated byy(x) = sin aboutx = 3.x(b) Drawy(x)andp 2 (x)for1 x 5.4 (a) Calculate the second-order Taylor polynomial,p 2 (x),generated byy(x) = e cosx aboutx = 0.(b) Drawy(x)andp 2 (x)for −2 x 2.18.4 TAYLORPOLYNOMIALSOFTHEnTHORDERIf we know y and its first n derivatives evaluated at x = a, that is y(a), y ′ (a),y ′′ (a), ...,y (n) (a), thenthenth-orderTaylorpolynomial, p n(x), maybewrittenasp n(x)=y(a) +y ′ (a)(x −a) +y ′′ (x −a)2(a) +y (3) (x −a)3(a)2! 3!+···+y (n) (x −a)n(a)n!This provides an approximation toy(x). The polynomial and its firstnderivatives evaluatedatx = a match the values ofy(x) and its firstnderivatives evaluated atx = a,thatisp n(a) =y(a)p ′ n (a) =y′ (a)p ′′ n (a) =y′′ (a).p (n)n (a) =y(n) (a)
- Page 485 and 486: 14.3 Integration by substitution 46
- Page 487 and 488: 14.4 Integration using partial frac
- Page 489 and 490: Review exercises 14 469∫ π/3(b)
- Page 491 and 492: 15 ApplicationsofintegrationContent
- Page 493 and 494: 15.2 Average value of a function 47
- Page 495 and 496: 15.3 Root mean square value of a fu
- Page 497 and 498: 15.3 Root mean square value of a fu
- Page 499 and 500: Review exercises 15 479(d) 0.7071(e
- Page 501 and 502: [Example16.1 Show that f (x) =xandg
- Page 503 and 504: Solution (a) We use the trigonometr
- Page 505 and 506: 16.3 Improper integrals 485iC 1 C 2
- Page 507 and 508: ∫ 2either of the integrals failst
- Page 509 and 510: 16.4 Integral properties of the del
- Page 511 and 512: 16.5 Integration of piecewise conti
- Page 513 and 514: 16.6 Integration of vectors 49316.6
- Page 515 and 516: Review exercises 16 49512 Given14 G
- Page 517 and 518: 17.2 Trapezium rule 497y{ {y 0 y ny
- Page 519 and 520: 17.2 Trapezium rule 499The differen
- Page 521 and 522: yABCDE17.3 Simpson’s rule 501quad
- Page 523 and 524: 17.3 Simpson’s rule 503Table17.6T
- Page 525 and 526: Review Exercises 17 505EXERCISES17.
- Page 527 and 528: 18 Taylorpolynomials,Taylorseriesan
- Page 529 and 530: 18.2 Linearization using first-orde
- Page 531 and 532: 18.2 Linearization using first-orde
- Page 533 and 534: 18.3 Second-order Taylor polynomial
- Page 535: 18.3 Second-order Taylor polynomial
- Page 539 and 540: 18.4. Taylor polynomials of the nth
- Page 541 and 542: 18.5 Taylor’s formula and the rem
- Page 543 and 544: We know |x| < 1, and so 4x51518.5 T
- Page 545 and 546: 18.6 Taylor and Maclaurin series 52
- Page 547 and 548: 18.6 Taylor and Maclaurin series 52
- Page 549 and 550: 18.6 Taylor and Maclaurin series 52
- Page 551 and 552: 18.6 Taylor and Maclaurin series 53
- Page 553 and 554: Review exercises 18 5337 (a) 0, 2,
- Page 555 and 556: 19.2 Basic definitions 535In engine
- Page 557 and 558: 19.2 Basic definitions 537Note also
- Page 559 and 560: 19.2 Basic definitions 539fromwhich
- Page 561 and 562: 19.3 First-order equations: simple
- Page 563 and 564: 19.3 First-order equations: simple
- Page 565 and 566: 19.3 First-order equations: simple
- Page 567 and 568: 19.4 First-order linear equations:
- Page 569 and 570: and so, upon integrating,so that(co
- Page 571 and 572: 19.4 First-order linear equations:
- Page 573 and 574: 19.4 First-order linear equations:
- Page 575 and 576: 19.4 First-order linear equations:
- Page 577 and 578: 19.4 First-order linear equations:
- Page 579 and 580: 19.5 Second-order linear equations
- Page 581 and 582: 19.6 Constant coefficient equations
- Page 583 and 584: 19.6 Constant coefficient equations
- Page 585 and 586: 19.6 Constant coefficient equations
516 Chapter 18 Taylor polynomials, Taylor series and Maclaurin series
so
p 2
(V) =I s
(e 40V a −1) +40Is e 40V a (V −Va ) +1600I s
e 40V (V −V
a a
) 2
2
Thecoefficientsneedtobecalculatedonlyonce.Afterthatthecalculationofacurrent
value only involves evaluating a quadratic.
EXERCISES18.3
1 (a) Obtainthe second-order Taylor polynomial,
p 2 (x),generatedbyy(x) = 3x 4 +1aboutx = 2.
(b) Verifythaty(2) = p 2 (2),y ′ (2) = p ′ 2 (2)and
y ′′ (2) = p ′′ 2 (2).
(c) Evaluate p 2 (1.8)andy(1.8).
2 (a) Calculate the second-order Taylor polynomial,
p 2 (x),generated byy(x) = sinx aboutx = 0.
(b) Calculate the second-order Taylor polynomial,
p 2 (x),generated byy(x) = cosx aboutx = 0.
(c) Compare your resultsfrom (a) and(b)with the
small-angle approximations given in Section6.5.
3 Afunction,y(x),issuchthaty(−1) = 3,y ′ (−1) = 2
andy ′′ (−1) = −2.
(a) Statethe second-order Taylor polynomial
generated byyaboutx = −1.
(b) Estimatey(−0.9).
4 Afunction,y(x),satisfiesthe equation
y ′ =y 2 +x
y(1)=2
(a) Estimatey(1.3)usingafirst-orderTaylor
polynomial.
(b) Bydifferentiatingthe equationwith respect tox,
obtain an expression fory ′′ .Hence evaluate
y ′′ (1).
(c) Estimatey(1.3)usingasecond-order Taylor
polynomial.
5 Afunction,x(t),satisfiesthe equation
ẋ=x+ √ t+1
x(0)=2
(a) Estimatex(0.2)usingafirst-orderTaylor
polynomial.
(b) Differentiate the equation w.r.t.t and hence
obtain an expression forẍ.
(c) Estimatex(0.2)usingasecond-order Taylor
polynomial.
6 Afunction,h(t),isdefined by
h(t) = sin2t +cos3t
Obtainthe second-order Taylor polynomial generated
byh(t)aboutt = 0.
7 Thefunctionsy 1 (x),y 2 (x)andy 3 (x)are defined by
y 1 (x) =Ae x ,y 2 (x) =Bx 3 +Cx,
y 3 (x) =y 1 (x) +y 2 (x)
(a) Obtainasecond-order Taylor polynomial for
y 1 (x)aboutx = 0.
(b) Obtainasecond-order Taylor polynomial for
y 2 (x)aboutx = 0.
(c) Obtainasecond-order Taylor polynomial for
y 3 (x)aboutx = 0.
(d) Canyou draw any conclusionsfrom your
answers to (a),(b)and(c)?
Solutions
1 (a) p 2 (x) = 72x 2 −192x +145
(c) p 2 (1.8) = 32.68,y(1.8) = 32.4928
2 (a) p 2 (x) =x (b) p 2 (x) =1− x2
2
3 (a) p 2 (x) = −x 2 +4
(b) p 2 (−0.9) = 3.19.This isan approximation to
y(−0.9).
4 (a) 3.5
(b) y ′′ = 2yy ′ +1,y ′′ (1) = 21
(c) 4.445
5 (a) 2.6
1
(b) ẍ=ẋ +
2 √ t +1
(c) 2.67
Change language