082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

532 Chapter 18 Taylor polynomials, Taylor series and Maclaurin series
REVIEWEXERCISES18
1 Afunction, f (t),and itsfirstand second derivatives (a) Calculatey ′ (1),y ′′ (1)andy (3) (1).
dy
dx + y2
2 =xy y(1)=2 sin(kx)
(c) lim 1 −cos(kx)
are evaluated att = 3. Find the second-order Taylor
polynomial generated by f aboutt = 3.
(b) Statethe third-orderTaylor polynomial generated
byy(x)aboutx = 1.
f(3)=2 f ′ (3)=−1 f ′′ (3)=1 (c) Estimatey(1.25).
2 Afunction,s(t),andits firstderivative are evaluated
att =3;s(3) =4,s ′ (3) = −1.
(a) Find the first-orderTaylor polynomial generated
bysatt=3.
(b) Estimates(2.9)ands(3.2).
(c) Ifadditionally weknows ′′ (3) = 1, use a
second-order Taylor polynomial to estimate
s(2.9)ands(3.2).
3 Obtain alinear approximation to
y(t) =at 4 +bt 3 +ct 2 +dt +earoundt =1.
4 Find a quadratic approximation toy =x 3 nearx = 2.
5 Find linear approximations to
8 (a) Findthe third-orderTaylor polynomial generated
byy(x) = e −x aboutx = 1.
(b) Statethe third-ordererror term.
(c) Findan upperbound forthe error termgiven
|x| < 1.
9 (a) Findaquadraticapproximation toy(x) = sin 2 x
aboutx = 0.
(b) Statethe remainder termoforder2.
(c) Statean upper boundforthe remainderterm
given |x| < 0.5.
10 (a) Findacubicapproximation toy(x) =xcosx
aboutx = 0.
(b) Statethe error termoforder3.
(a) z(t) =e t neart =1
(c) Statean upper boundforthe errorterm given
(b) w(t) =sin3t neart =1
|x| < 0.25.
(c) v(t) =e t +sin3tneart =1
6 Thefunction,y(x),satisfiesthe equation
y ′′ +xy ′ −3y=x 2 +1
11 Giventhaty(x) =x 2 ,
(a) Calculate the Taylor seriesofy(x) aboutx =a.
(b) Calculate the Maclaurin series ofy(x).
y(0) =1 y ′ (0) =2
12 Findthe Maclaurinexpansion ofy(x) = ln(2 +x) up
(a) Evaluatey ′′ (0).
to andincluding the terminx 4 .
(b) Differentiate the equation.
(c) Evaluatey (3) (0).
13 Byconsideringthe Maclaurinexpansions ofsin(kx)
andcos(kx),kconstant,evaluate ifpossible
(d) Write down acubic approximation fory(x).
sin(kx)
(a) lim
(e) Estimatey(0.5).
x→0 x
7 Given thatysatisfiesthe equation
cos(kx) −1
(b) lim x→0 x
Solutions
1
t 2 2 −4t+19 2
2 (a) −t +7 (b) 4.1, 3.8 (c) 4.105, 3.82
3 (4a+3b+2c+d)t−3a−2b−c+e
4 6x 2 −12x+8
5 (a) et (b) −2.97t +3.11
(c) −0.25t +3.11
6 (a) 4 (b) y ′′′ +xy ′′ −2y ′ = 2x
(c) 4 (d) 1+2x+2x 2 + 2 3 x3
(e) 2.5833