الاجوبة النموذجية لامتحان الشهر الاول / الفصل الثاني

اجوبة شعبة A
(1) Find f_xx ,f_xy , f_yy for f(x,y)=xy^3+( y^2)/x^2 =xy^3+y^2 x^(-2) ?5?
f_x=y^3-2y^2 x^(-3) ? f_xx=6y^2 x^(-4)
f_xy=3y^2-4yx^(-3)
f_y=3xy^2+2yx^(-2) ? f_yy=6xy+2x^(-2)
(2) Find ?×F for F(x,y,z)=x^2 yi+xy^2 j+ye^xz k .
?×F=| ?(i&j&k@?/?x&?/?y&?/?z @x^2 y&xy^2&ye^xz )|
=((?ye^xz)/?y-(?xy^2)/?z)i-((?ye^xz)/?x-(?x^2 y)/?z)j+((?xy^2)/?x-(?x^2 y)/?y)k
=e^xz i-yze^xz j+(y^2-x^2 )k
(3) ?_0^2??_0^(y^2)??(y^3+1) dxdy=? ?_0^2?x?|_0^(y^2 ) ?(y^3+1) dy=?_0^2??y^2 ?(y^3+1)? dy
=? ( 1 )/( 3 )×( 2 )/( 3 ) (y^3+1)^(3?2) ?|_0^2=( 2 )/( 9 ) (27-1)=( 52 )/( 9 )
(4) ?_D??(x+3y) ? dA=?_0^(??4)??_0^2?(r cos??+3r sin?? )rdrd?=?_0^(??4)??_0^2??(cos??+3sin?? ) r^2 drd??
=?_0^(??4)?(cos??+3 sin?? ) ? ( r^3)/( 3 )?|_0^2 d?
=?_0^(??4)??( 8 )/3 (cos??+3 sin?? ) ? d?=? ( 8 )/3 (?sin??-3cos??? )?|_0^(??4)
=( 8 )/3 (( 1 )/?(2 )-( 3 )/?(2 )-0+3)=( 8 )/3 (3-?(2 ))



(1) Find f_xx ,f_xy , f_yy for f(x,y)=x^2 y^2+( y^3)/x^2 =x^2 y^2+y^3 x^(-2) ?6?
f_x=2xy^2-2y^3 x^(-3) ? f_xx=2y^2+6y^3 x^(-4)
f_xy=4xy-6y^2 x^(-3)
f_y=2x^2 y+3y^2 x^(-2) ? f_yy=2x^2+6yx^(-2)
(2) Find ?×F for F(x,y,z)=xy^2 i+x^2 yj+xe^yz k .

?×F=| ?(i&j&k@?/?x&?/?y&?/?z @xy^2&x^2 y&xe^yz )|
=((?xe^yz)/?y-(?x^2 y)/?z)i-((?xe^yz)/?x-(?xy^2)/?z)j+((?x^2 y)/?x-(?xy^2)/?y)k
=xze^yz i-e^yz j+(2xy-2xy)k=xze^yz i-e^yz j
اجوبة شعبة B
(1) Find f_xx ,f_xy , f_yy for f(x,y)=( x^2)/y^2 +x^3 y^2=x^2 y^(-2)+x^3 y^2 ?7?
f_x=2xy^(-2)+3x^2 y^2 ? f_xx=2y^(-2)+6xy^2
f_xy=-4xy^(-3)+6x^2 y
f_y=-2x^2 y^(-3)+2x^3 y ? f_yy=6x^2 y^(-4)+2x^3
(2) Find ?×F for F(x,y,z)=z sin?(xy)i+xy^2 j+xyz^2 k .
?×F=| ?(i&j&k@?/?x&?/?y&?/?z @z sin?(xy)&xy^2&xyz^2 )|
=((?xyz^2)/?y-(?xy^2)/?z)i-((?xyz^2)/?x-(?z sin?(xy))/?z)j+((?xy^2)/?x-(?z sin?(xy))/?y)k
=xz^2 i-(yz^2-sin?(xy) )j+(y^2-xz cos?(xy) )k
(3) ?_0^2??_0^(x^2)??(x^3+1) dydx=? ?_0^2?y?|_0^(x^2 ) ?(x^3+1) dx=?_0^2??x^2 ?(x^3+1)? dx
=? ( 1 )/( 3 )×( 2 )/( 3 ) (x^3+1)^(3?2) ?|_0^2=( 2 )/( 9 ) (27-1)=( 52 )/( 9 )
(4) ?_D??(x+3y) ? dA=?_0^???_0^1?(r cos??+3r sin?? )rdrd?=?_0^???_0^1??(cos??+3sin?? ) r^2 drd??
=?_0^??(cos??+3 sin?? ) ? ( r^3)/( 3 )?|_0^1 d?
=?_0^???( 1 )/3 (cos??+3 sin?? ) ? d?=? ( 1 )/3 (?sin??-3cos??? )?|_0^?
=( 1 )/3 (0+3-0+3)=2
(1) Find f_xx ,f_xy , f_yy for f(x,y)=( x^3)/y^2 +x^2 y^3=x^3 y^(-2)+x^2 y^3 ?8?
f_x=3x^2 y^(-2)+2xy^3 ? f_xx=6xy^(-2)+2y^3
f_xy=-6x^2 y^(-3)+6xy^2
f_y=-2x^3 y^(-3)+3x^2 y^2 ? f_yy=6x^3 y^(-4)+6x^2 y

(2) Find ?×F for F(x,y,z)=x sin?(yz)i+x^2 yzj+yz^2 k .
?×F=| ?(i&j&k@?/?x&?/?y&?/?z @x sin?(yz)&x^2 yz&yz^2 )|
=((?yz^2)/?y-(?x^2 yz)/?z)i-((?yz^2)/?x-(?x sin?(yz))/?z)j+((?x^2 yz)/?x-(?x sin?(yz))/?y)k
=(z^2-x^2 y)i-(-xy cos?(yz) )j+(2xyz-xz cos?(yz) )k
=(z^2-x^2 y)i+xy cos?(yz)j+(2xyz-xz cos?(yz) )k