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

Review exercises 25 847
4 Find allfirstand second partialderivatives of
f(x,y) = (ax +by) n
wherea,bandnare constants.
5 Find the firstpartial derivatives of
f(x,y) = (ax 2 +by 2 +cxy) n
wherea,b,candnare constants.
6 Verifythat
z(x,y)=3x+2y+1
isasolution of
∂z
∂x − ∂z
∂y = 1
7 Verifythat
z(x,y)=2xy−x+y
isasolution of
∂z
∂x + ∂z
∂y =2(x+y)
8 Verifythat
f(x,y)=x 2 +y 2 −2xy
isasolution of
∂f
∂x + ∂f
∂y = 0
9 Verifythat
z(x,y) =sinx +cosy
isasolution of
∂ 2 z
∂x 2 + ∂2 z
∂y 2 +z=0
10 Verify that
z(x,y) =xye x
is asolution of
∂ 2 z
∂x 2 + ∂2 z
∂y 2 −y ∂2 z
∂x∂y =yex
11 (a) Writedown the second-order Taylor polynomial
generatedby f (x,y) aboutx = 2,y = 3 given
f(x,y) =3x 3 y−x 2 y 3
(b) Estimate f(2.1,2.9) usingyourpolynomial from
(a)andcompare thiswith the exact answer.
12 Writedown the second-order Taylor polynomial
generated byz(x,y) aboutx = 1,y = 1 given
z(x,y) = x +y
x
13 Calculate the second-order Taylor polynomial
generated by
√
f(x,y)= x 2 +y 2
aboutx=1,y=0.
14 Locateandidentify all the stationary pointsofthe
following functions:
(a) f(x,y)=x 2 +y 3 −3y
(b) f(x,y) =4xy−x 2 y
(c) f(x,y) =x 3 +2y 2 −12x
(d) f(x,y)=xy−y 2 −x 3
15 Locateandidentify the stationary pointsof
f(x,y)= y x −x2 +y 2
Solutions
1 (a)
∂z ∂z
= 18x,
∂x ∂y = 4y
(b) 9x 2 y 6 ,18x 3 y 5
(c) 12x 2 , −15y 2
(d) y+2xy 2 ,x+2x 2 y
(e) 2ycos(2xy),2xcos(2xy)
(f) 6ye 3xy ,6xe 3xy
∂f ∂f
2 (a) (−1,2) =1, (−1,2) = −8
∂x ∂y
(b) 2,1 (c) 0.5403, −1.8186
3 (a)
(d) 25, −12 (e) − 4 9 ,−2 9
(f) 5.4366,5.4366
∂ 2 z
∂x 2 = 36x2 +6xy 3 ,
∂ 2 z
∂x∂y =9x2 y 2 ,
∂ 2 z
∂y 2 = −108y2 +6x 3 y
(b) 12x 2 +4y 2 ,8xy,4x 2 +12y 2
(c) 32e 4x−3y , −24e 4x−3y ,18e 4x−3y