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

840 Chapter 25 Functions of several variables
(c) The partialderivatives of p 2
(x,y) arefound.
∂p 2
∂x =8x+2y−5 ∂p 2
∂y =12y+2x−9
∂ 2 p 2
∂x 2 = 8
∂ 2 p 2
∂x∂y = 2 ∂ 2 p 2
∂y 2 = 12
Thesecondpartialderivativesofp 2
areidenticaltothesecondpartialderivativesof
f evaluated at(1, 1).
25.6.3 Taylorseriesintwovariables
The third-order Taylor polynomial involves all the third-order partial derivatives of f,
the fourth-order Taylor polynomial involves all the fourth-order partial derivatives of f
and so on. Taylor polynomials approximate more and more closely to the generating
function as more and more terms are included. As more and more terms are included,
weobtainaninfiniteseriesknownasaTaylorseriesintwovariables.Thegeneralform
ofthis series isbeyond the scope ofthis book.
EXERCISES25.6
1 Useafirst-orderTaylor polynomial to estimate
f(2.1, 3.2) given
f(2,3)=4
∂f
∂y (2,3)=3
∂f
(2,3) = −2
∂x
2 Useafirst-orderTaylor polynomial to estimate
g(−1.1,0.2) given
g(−1,0) =6
∂g
(−1,0) = −1
∂y
∂g
(−1,0) =2
∂x
3 Useafirst-orderTaylor polynomial to estimate
h(−1.2, −0.7) given
h(−1.3, −0.6) = 4
∂h
(−1.3, −0.6) = −1
∂x
∂h
(−1.3, −0.6) = 1
∂y
4 Useasecond-order Taylor polynomial to estimate
f(3.1, 4.2) given
f(3,4)=1
∂f
∂x (3,4)=0
∂f
∂y (3,4)=2
∂ 2 f
∂x∂y (3,4)=3
∂ 2 f
(3,4) = −1
∂x2 ∂ 2 f
(3,4) =0.5
∂y2 5 Useasecond-order Taylor polynomial to estimate
g(−2.9,3.1) given
g(−3,3) =1
∂g
∂
(−3,3) =4
∂y
∂ 2 g
∂
(−3,3) = −2
∂x∂y
∂g
(−3,3) = −1
∂x
2 g
(−3,3) =3
∂x2 2 g
(−3,3) =2
∂y2 6 Useasecond-order Taylor polynomial to estimate
h(0.1,0.1) given
h(0,0) =4
∂h
∂
(0,0) = −3
∂y
∂ 2 h
∂x∂y (0,0)=2
∂h
(0,0) = −1
∂x
2 h
∂x 2 (0,0)=2
∂ 2 h
(0,0) = −1
∂y2