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

836 Chapter 25 Functions of several variables
Example25.10 Afunction, f, issuch that
f(3,1)=2
∂f
∂f
(3,1) = −1
∂x
∂y (3,1)=4
(a) State the first-order Taylor polynomial generated by f about (3, 1).
(b) Estimate the values of f(3.5,1.2) and f(3.2,0.7).
(c) Verifythatthe Taylor polynomial and the function have identical values at(3, 1).
(d) VerifythatthefirstpartialderivativesoftheTaylorpolynomialandthefunctionare
identical at(3, 1).
Solution (a) Inthisexamplea = 3 andb= 1.Hence
p 1
(x,y) = f(3,1)+(x−3) ∂f
∂x (3,1)+(y−1)∂f ∂y (3,1)
=2+(x−3)(−1)+(y−1)4
=4y−x+1
The first-order Taylor polynomial generated by f about (3, 1) is
p 1
(x,y)=4y−x+1
(b) We use p 1
(x,y) toestimate f(3.5,1.2) and f(3.2,0.7).
p 1
(3.5,1.2) = 4(1.2) −3.5 +1 = 2.3
p 1
(3.2,0.7) = 4(0.7) −3.2 +1 = 0.6
Hence 2.3 isan estimateof f(3.5,1.2) and 0.6 isanestimate of f(3.2,0.7).
(c) We aregiven f (3,1) = 2.Also
p 1
(3,1)=4(1)−3+1=2
andso f(3,1) = p 1
(3,1).
(d) We aregiven ∂f
∂f
(3,1) = −1and (3,1) =4.Now
∂x ∂y
p 1
(x,y)=4y−x+1
and so
∂p 1
∂x = −1 ∂p 1
∂y = 4
Hence
∂p 1
∂f
(3,1) =
∂x ∂x (3,1) = −1 ∂p 1
∂f
(3,1) =
∂y ∂y (3,1)=4
Example25.11 Afunction, f (x,y), isdefinedby
f(x,y)=x 2 +xy−y 3
(a) State the first-order Taylor polynomialgeneratedby f about(1, 2).
(b) Verify that the Taylor polynomial in (a) and the function f have identical values at
(1, 2).