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

25.7 Maximum and minimum points of a function of two variables 841
7 Afunction, f (x,y),is defined by
f(x,y) =x 3 y+xy 3
(a) Calculate the first-orderTaylor polynomial
generated by f about (1,1).
(b) Calculate the second-order Taylor polynomial
generated by f about (1,1).
(c) Estimate f(1.2, 1.2) usingthe first-orderTaylor
polynomial.
(d) Estimate f(1.2, 1.2) usingthe second-order
Taylor polynomial.
(e) Compare your answers in (c) and (d)with the
true value of f(1.2, 1.2).
8 Afunctiong(x,y) is defined by
g(x,y)=xsiny+ x y
(a) Calculate the first-orderTaylor polynomial
generated bygabout (0,1).
(b) Calculate the second-order Taylor polynomial
generated bygabout (0,1).
(c) Estimateg(0.2, 0.9) usingthe polynomial in (a).
(d) Estimateg(0.2, 0.9) usingthe polynomial in (b).
(e) Compare your estimateswith the exact value of
g(0.2, 0.9).
9 A functionh(x,y) isdefined by
h(x,y) =e x y+x 2 e y
(a) Calculate the first-orderTaylor polynomial
generated byhabout (0,0).
(b) Calculate the second-order Taylor polynomial
generated byhabout (0,0).
(c) Estimateh(0.2, 0.15)usingthe polynomial
from (a).
(d) Estimateh(0.2, 0.15)usingthe polynomial
from (b).
(e) Compare youranswers in (c)and(d)with the
exact value ofh(0.2,0.15).
10 A function, f (x,y,z),is defined by
f(x,y,z)=x 2 +xyz+yz 2
(a) Writedown the first-orderTaylor polynomial
generated by f about (0,1,2).
(b) Use the polynomial from (a)to estimate
f(0.1,1.2, 1.9).
(c) Compare youranswerin (b)with the exact value
of f(0.1,1.2, 1.9).
Solutions
1 4.4
2 5.6
3 3.8
4 1.465
5 1.305
6 3.625
7 (a) 4x+4y−6
(b) 3x 2 +3y 2 +6xy−8x−8y+6
(c) 3.6 (d) 4.08 (e) 4.1472
8 (a) 1.8415x (b) 2.3012x −0.4597xy
(c) 0.3683 (d) 0.3775
(e) 0.3789
9 (a) y (b) x 2 +xy +y (c) 0.15
(d) 0.22 (e) 0.2297
10 (a) 2x+4y+4z−8 (b) 4.6
(c) 4.57
25.7 MAXIMUMANDMINIMUMPOINTSOFAFUNCTION
OFTWOVARIABLES
We saw inChapter12 thattofind the turningpoints ofy(x) wesolve
dy
dx = 0