الاجوبة النموذجية لاسئلة الامتحان النهائي - الفصل الثاني - الدور الثاني

Q1: A. Let w=xy-z , x=sin?t ,y=cos?t and z=t,find ?w/?t. ?10 marks?

B.Find f_xx and f_xy for f(x,y)=(2x-y)/(x+y). ?10 marks?


Q2: A. Find ?f and ?^2 f for f=x^3 e^y+xy^2 z^3 at (1,0,2) ?10 marks?

B.Evaluate ?_0^2??_1^3??_1^2??xy^2 zdxdydz? ?10 marks?


Q3:Answer Two of the following questions ?20 marks?
A.Convert the point (-4,2?/3)into the given coordinate system.
B.Determine whether the sequence { ?(n+4) -?(n ) } converges or diverges.
C.Find the sum of the series ??( 5^n )/? 3?^2n

Q4: Determine whether Five of the series converges or diverges. ?20 marks?
a. ??(4n^2-n)/(5+2n^2 ) b.??(3/( 2 ))^n c.??1/?(n ) d.??n/(n^2+1) e.??2^n/n! f.??(n/( 2n-1 ))^n

Q5:A. Compute the Maclaurin series of f(x)=ln?(x+1). ?10 marks?
B.Evaluate ?_0^???x?(x ) e^(-x) dx? ?10 marks?





Ass. Lecturer Fouad H. A. Alsharify Prof. Dr. Raheem G. Kadhim
Instructor Head of department
??? With our best wishes ???

Q1:A. ?w/?t=?w/?x×?x/?t+?w/?y×?y/?t+?w/?z×?z/?t
=y×cos?t+x×(-sin?t )+(-1)×1=cos^2?t-sin^2?t-1


B. f_x=(2(x+y)-(2x-y))/(x+y)^2 =3y/(x+y)^2
f_xx=(0-6y(x+y))/(x+y)^4 =(-6y)/(x+y)^3
f_xy=(3(x+y)^2-3y×2(x+y))/(x+y)^4 =(3(x+y)-6y)/(x+y)^3 =(3x-3y)/(x+y)^3
Q2: A. f_x=3x^2 e^y+y^2 z^3 ? f_xx=6xe^y
f_y=x^3 e^y+2xyz^3 ? f_yy=x^3 e^y+2xz^3
f_z=3xy^2 z^2 ? f_zz=6xy^2 z
?f= f_x i+f_y j+f_z k=(3x^2 e^y+y^2 z^3 ) i+(x^3 e^y+2xyz^3 ) j+3xy^2 z^2 k
? ?f?|_( at (1,0,2) )=3 i+ j
?^2 f=f_xx+f_yy+f_zz=6xe^y+x^3 e^y+2xz^3+6xy^2 z
? ?^2 f?|_(at (1,0,2) )=6+1=7

B. ?_0^2??_1^3??_1^2??xy^2 zdxdydz?=?_0^2??_1^3??? x^2/2?|_1^2 y^2 zdydz? =?_0^2??_1^3??(4-( 1 )/2) y^2 zdydz?
=?_0^2??_1^3??( 3 )/2 y^2 zdydz?=?_0^2??? ( y^3 )/2?|_1^3 zdz?=?_0^2?(( 27 )/2-( 1 )/2)zdz=13?_0^2?zdz=13? z^2/2?|_0^2=26


Q3: A. x=r cos?? ? x=-4 cos??2?/3? ? x=-4×(-( 1 )/( 2 ))=2
y=r sin?? ? x=-4 sin??2?/3? ? x=-2?3 ?(-4,2?/3)?(2,-2?3)

B. lim?(n??)?(?(n+4) -?(n ))=lim?(n??)?(?(n+4) -?(n ))×((?(n+4)+?(n )))/((?(n+4)+?(n )) )
=lim?(n??)??(n+4-n)/((?(n+4)+?(n )) )?=0 converges to 0

C. ??( 5^n )/? 3?^2n =??( 5^n )/? 9?^n =( 5 )/( 9 )+( 5^2 )/? 9?^2 +( 5^3 )/? 9?^3 +?
a=( 5 )/( 9 ),r=( 5 )/( 9 ) ?s=a/(1-r)=( 5?9 )/(1-5?9 )=( 5 )/( 9 ) ×( 9 )/( 4 )=( 5 )/( 4 )

Q4:a. lim?(n??)??(4n^2-n^3)/(5+2n^3 )?=lim?(n??)??(8n-3n^2)/(6n^2 )?=lim?(n??)??(8-6n)/12n?=lim?(n??)??(-6)/12?=-( 1 )/( 2 )?0
By the Divergence Test the series diverges.
b.Geometric series with r=3/( 2 )>1 ?diverges to ?
c. p-series with p=1/( 2 )<1 ? diverges
d.?_1^??xdx/(x^2+1)=? 1/2 ln?|x^2+1| ?|_1^? =? ? diverges
e.lim?(n??)??a_(n+1)/a_n ?= lim?(n??) 2^(n+1)/(n+1)!×n!/2^n =lim?(n??) (2^n×2)/((n+1)×n!)×n!/2^n =lim?(n??) 2/(n+1)=0<1 con.
f.lim?(n??)??(n&a_n )=lim?(n??) n/( 2n-1 )=1/( 2 )<1converges

Q5:A. Compute the Maclaurin series of f(x)=ln?(?). ?10 marks?
f(x)= ln?(x+1) ? f_1 (0)=ln?(1)=0
f^ (x)=(x+1)^(-1) ? f^ (0)=1
f^ (x)=-(x+1)^(-2) ? f^ (0)=-1
f^((3) ) (x)= 2(x+1)^(-3) ? f^((3) ) (0)=2
f(x)= ln?(x+1)=x-x^2/2+x^3/3-?

B.?_0^???x?(x ) e^(-x) dx? =?_0^???x^(3/2) e^(-x) dx?
=?(( 3 )/( 2 )+1)=( 3 )/( 2 )×( 1 )/( 2 ) ?(? )=( 3 )/( 4 ) ?(? )