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الكلية كلية العلوم     القسم قسم الكيمياء     المرحلة 1
أستاذ المادة فؤاد حمزة عبد الشريفي       06/04/2017 17:55:52
(1)-A
1.Show that w_?=2 cot?2? if w=ln?(uv) ,u=cos?? sin?? ,v=sin?? cos??
2. Evaluate ?_0^(2??3)??_0^3??r?(9-r^2 )? drd?
3. Determine whether the sequence converges or diverges.
(a){ ?(?4n?^2+3n) -2n } (b){ln??((n^2-3n)/n^2 )^n ? }
4. Determine whether the series converges or diverges.
(a) ?_(n=1)^??1/(n^2-2n-15) (b) ?_(n=1)^??(3n/( 5n+2))^(-n)




(2)- A
1.Show that w_?=2 cot??2?? if w=ln?(uv) ,u=cos?? sin?? ,v=sin?? cos??
2. Evaluate ?_0^(??4)??_0^2??r?(4-r^2 )? drd?
3. Determine whether the sequence converges or diverges.
(a){ ?(?9n?^2+2n) -3n } (b){ln??((n^2-2n)/n^2 )^n ? }
4. Determine whether the series converges or diverges.
(a) ?_(n=1)^??1/(n^2-n-12) (b) ?_(n=1)^??(2n/( 3n+5 ))^(-n)


(1)-A
1.Show that w_?=2 cot?2? if w=ln?(uv) ,u=cos?? sin?? ,v=sin?? cos??
?w/??=?w/?u×?u/??+?w/?v×?v/??
?w/?u=( v )/( uv )=( 1 )/( u )=( 1 )/( cos?? sin?? ) and ?w/?v=( u )/( uv )=( 1 )/( v ) =( 1 )/( sin?? cos?? )
?u/??=cos?? cos?? and ?v/??=-sin?? sin??
?w/??=( 1 )/( cos?? sin?? )×cos?? cos??+( 1 )/( sin?? cos?? )×(-sin?? sin?? )
=( cos?? )/( sin?? )-( sin?? )/( cos?? )=(cos^2??-sin^2??)/sin??? cos?? ? =cos?2?/(sin?2?/2)=2 cot?2?
2. Evaluate ?_0^(2??3)??_0^3??r?(9-r^2 )? drd? =? ?_0^(2??3)??-?( 1 )/( 3 ) (9-r^2 )?^(3?2) ??|_0^3 d?
=?_0^2??(-?( 1 )/( 3 ) (9-9)?^(3?2)+?( 1 )/( 3 ) (9)?^(3?2) ) d?
=? ?_0^(2??3)?9 d?=9??|_0^(2??3)=6?
3. Determine whether the sequence converges or diverges.
(a) lim?(n??)??(?(?4n?^2+3n) -2n)×(?(?4n?^2+3n)+2n)/(?(?4n?^2+3n)+2n)?=lim?(n??)??(?4n?^2+3n-4n^2)/(?(?4n?^2+3n)+2n)?
=lim?(n??)??3n/(?(?4n?^2+3n)+2n)?=lim?(n??)??3/(?(4+3/( n ))+2)?
=3/( 4 ) converges to 3/( 4 )

(b) lim?(n??)??ln??((n^2-3n)/n^2 )^n ?=ln??lim?(n??) (1+(-3)/n)^n ? ?=ln??e^(-3) ?=-3 converges to -3


4. Determine whether the series converges or diverges.
(a) 1/(x^2-2x-15)=1/(x-5)(x+3) =A/(x-5)+B/(x+3)
x=5 ?A=( 1 )/8 and x=-3 ?B=-1/8
?_1^??dx/(x^2-2x-15)=( 1 )/( 8 ) ? ?_1^??(1/(x-5)- 1/(x+3)) dx=( 1 )/( 8 ) ln?|(x-5)/(x+3)| ?|_1^?
=? ( 1 )/( 8 ) ln?|(1-( 5 )/x)/(1+( 3 )/x)| ?|_1^?=( 1 )/( 8 ) (ln?1-ln??4/4? )=0 converge
(b) lim?(n??)??(n&a_n )=lim?(n??)??(3n/( 5n+2 ))^(-1) ?=lim?(n??)??(5n+2)/( 3n )?=( 5 )/3>1 diverge


(2)- A
1.Show that w_?=2 cot??2?? if w=ln?(uv) ,u=cos?? sin?? ,v=sin?? cos??
?w/(??)=?w/?u×?u/(??)+?w/?v×?v/(??)
?w/?u=( v )/( uv )=( 1 )/( u )=( 1 )/( cos?? sin?? ) and ?w/?v=( u )/( uv )=( 1 )/( v ) =( 1 )/( sin?? cos?? )
?u/(??)=-sin?? sin?? and ?v/(??)=cos?? cos??
?w/??=( 1 )/( cos?? sin?? )×(-sin?? sin?? )+( 1 )/( sin?? cos?? )×cos?? cos??
=-( sin?? )/( cos?? )+( cos?? )/( sin?? )=(cos^2??-sin^2??)/sin???cos?? ?
=cos??2??/(sin??2??/2)=2 cot??2??


2. Evaluate ?_0^(??4)??_0^2??r?(4-r^2 )? drd?=? ?_0^(??4)??-?( 1 )/( 3 ) (4-r^2 )?^(3?2) ??|_0^2 d?
=?_0^2??(-?( 1 )/( 3 ) (4-4)?^(3?2)+?( 1 )/( 3 ) (4)?^(3?2) ) d?=? ?_0^(??4)?( 8 )/3 d?=8??|_0^(??4)=( 2 )/3 ?
3. Determine whether the sequence converges or diverges.
(a) lim?(n??)??(?(?9n?^2+2n) -3n )×(?(?9n?^2+2n)+3n )/(?(?9n?^2+2n)+3n )?=lim?(n??)??(?9n?^2+2n-9n^2)/(?(?9n?^2+2n)+3n)?
=lim?(n??)??2n/(?(?9n?^2+2n)+3n)?=lim?(n??)??2/(?(9+2/( n ))+3)?
=( 2)/( 6 ) converges to ( 1)/( 3 )
(b) lim?(n??)??ln??((n^2-2n)/n^2 )^n ?=ln??lim?(n??) (1+(-2)/n)^n ? ?=ln??e^(-2) ?=-2 converges to -2
4. Determine whether the series converges or diverges.
(a) 1/(x^2-x-12)=1/(x-4)(x+3) =A/(x-4)+B/(x+3)
x=4 ?A=( 1 )/7 and x=-3 ?B=-( 1 )/( 7 )
?_1^??dx/(x^2-x-12)=( 1 )/7 ? ?_1^??(1/(x-4)- 1/(x+3)) dx=( 1 )/7 ln?|(x-4)/(x+3)| ?|_1^?
=? ( 1 )/7 ln?|(1-( 4 )/x)/(1+( 3 )/x)| ?|_1^?=( 1 )/7 (ln?1-ln?|(-3)/4| )=-( 1 )/7 ln?(( 3 )/4) converge
(b) lim?(n??)??(n&a_n )=lim?(n??)??(2n/( 3n+5 ))^(-1) ?=lim?(n??)??(3n+5)/( 2n )?=( 3 )/2>1 diverge




(1)-B
1. Show that w(u,v)=ln??(u^2+v^2 ) satisfy Laplace s equation.
2. Evaluate ?_0^2??_0^(?(4-y^2 ))?ydxdy
3. Determine whether the sequence converges or diverges.
(a){ ?(?4n?^2-n) -2n } (b){ln??(?2n?^2/(?2n?^2-n))^(-n) ? }
4. Determine whether the series converges or diverges.
(a) ?_(n=1)^??1/(n^2+2n-15) (b) ?_(n=1)^??(( 2n+1 )/( 3n ))^n




(2)-B

1. Show that z(x,y)=ln??(x^2+y^2 ) satisfy Laplace s equation.
2. Evaluate ?_0^3??_0^(?(9-x^2 ))?xdydx
3. Determine whether the sequence converges or diverges.
(a){ ?(?9n?^2-n) -3n } (b){ln??(n^2/(n^2-n))^(-n) ? }
4. Determine whether the series converges or diverges.
(a) ?_(n=1)^??1/(n^2+n-12) (b) ?_(n=1)^??(( 3n+2 )/( 5n ))^n



(1)-B
1. Show that w(u,v)=ln??(u^2+v^2 ) satisfy Laplace s equation.
w(u,v)=( 1 )/( 2 ) ln?(u^2+v^2 )
?w/?u=2u/2(u^2+v^2 ) = u/((u^2+v^2 ) ) ? (?^2 w)/(?u^2 )=((u^2+v^2 )-u×2u)/(u^2+v^2 )^2
(?^2 w)/(?u^2 )=(u^2+v^2-2u^2)/(u^2+v^2 )^2 =(v^2-u^2)/(u^2+v^2 )^2
?w/?v=2v/2(u^2+v^2 ) = v/((u^2+v^2 ) ) ? (?^2 w)/(?v^2 )=((u^2+v^2 )-v×2v)/(u^2+v^2 )^2
(?^2 w)/(?v^2 )=(u^2+v^2-2v^2)/(u^2+v^2 )^2 =(u^2-v^2)/(u^2+v^2 )^2

(?^2 w)/(?u^2 )+(?^2 w)/(?v^2 )=(v^2-u^2)/(u^2+v^2 )^2 +(u^2-v^2)/(u^2+v^2 )^2 =0
2. Evaluate ?_0^2??_0^(?(4-y^2 ))?ydxdy
?_0^2??_0^(?(4-y^2 ))?ydxdy=? ?_0^2?yx?|_0^?(4-y^2 ) dy=? ?_0^2??y?(4-y^2 ) dy?=-( 1 )/( 2 )×( 2 )/( 3 ) (4-y^2 )^(3?2) ?|_0^2
=-( 1 )/( 3 ) [(4-4)^(3?2)-(4-0)^(3?2) ]=( 8 )/( 3 )
3. Determine whether the sequence converges or diverges.
(a) lim?(n??)??(?(?4n?^2-n) -2n )×(?(?4n?^2-n)+2n )/(?(?4n?^2-n)+2n )?=lim?(n??)??(?4n?^2-n-4n^2)/(?(?4n?^2-n)+2n)?
=lim?(n??)??(-n)/(?(?4n?^2-n)+2n)?=lim?(n??)??(-1)/(?(4-1/( n ))+2)?
=(-1)/( 4 ) converges to (-1)/( 4 )
(b) lim?(n??) ln??(?2n?^2/(?2n?^2-n))^(-n) ?=lim?(n??)??ln??((?2n?^2-n)/?2n?^2 )^n ?=ln??lim?(n??) (1+(-1)/2n)^n ? ?=ln??e^((-1)?2) ?=-( 1 )/( 2 ) converges to -( 1 )/( 2 )
4. Determine whether the series converges or diverges.
(a) 1/(x^2+2x-15)=1/(x+5)(x-3) =A/(x+5)+B/(x-3)
x=-5 ?A=-( 1 )/8 and x=3 ?B=( 1 )/8
?_1^??dx/(x^2+2x-15)=( 1 )/( 8 ) ? ?_1^??(1/(x-3)- 1/(x+5)) dx=( 1 )/( 8 ) ln?|(x-3)/(x+5)| ?|_1^?
=? ( 1 )/( 8 ) ln?|(1-( 3 )/x)/(1+( 5 )/x)| ?|_1^?=( 1 )/( 8 ) (ln?1-ln??( 2 )/( 6 )? )=( 1 )/( 8 ) ln?3 converge
(b) lim?(n??)??(n&a_n )=lim?(n??)?(( 2n+1 )/( 3n ))=( 2 )/( 3 )<1converge
(2)-B
1. Show that z(x,y)=ln??(x^2+y^2 ) satisfy Laplace s equation.
z(x,y)=( 1 )/( 2 ) ln?(x^2+y^2 )
?w/?x=2x/2(x^2+y^2 ) = x/((x^2+y^2 ) ) ? (?^2 w)/(?x^2 )=((x^2+y^2 )-x×2x)/(x^2+y^2 )^2
(?^2 w)/(?x^2 )=(x^2+y^2-2x^2)/(x^2+y^2 )^2 =(y^2-x^2)/(x^2+y^2 )^2
?w/?y=2y/2(x^2+y^2 ) = y/((x^2+y^2 ) ) ? (?^2 w)/(?y^2 )=((x^2+y^2 )-y×2y)/(x^2+y^2 )^2
(?^2 w)/(?y^2 )=(x^2+y^2-2y^2)/(x^2+y^2 )^2 =(x^2-y^2)/(x^2+y^2 )^2

(?^2 w)/(?x^2 )+(?^2 w)/(?y^2 )=(y^2-x^2)/(x^2+y^2 )^2 +(x^2-y^2)/(x^2+y^2 )^2 =0
2. Evaluate ?_0^3??_0^(?(9-x^2 ))?xdydx
?_0^3??_0^(?(9-x^2 ))?xdydx=? ?_0^3?xy?|_0^?(9-x^2 ) dx=? ?_0^3??x?(9-x^2 ) dx?=-( 1 )/( 2 )×( 2 )/( 3 ) (9-x^2 )^(3?2) ?|_0^3
=-( 1 )/( 3 ) [(9-9)^(3?2)-(9-0)^(3?2) ]=9
3. Determine whether the sequence converges or diverges.
(a) lim?(n??)??( ?(?9n?^2-n) -3n )×( ?(?9n?^2-n)+3n )/( ?(?9n?^2-n)+3n )?=lim?(n??)??(?9n?^2-n-9n^2)/( ?(?9n?^2-n)+3n)?
=lim?(n??)??(-n)/(?(?9n?^2-n)+3n)?=lim?(n??)??(-1)/(?(9-1/( n ))+3)?
=(-1)/( 6 ) converges to (-1)/( 6 )
(b) lim?(n??) ln??(n^2/(n^2-n))^(-n) ?=lim?(n??)??ln??((n^2-n)/n^2 )^n ?=ln??lim?(n??) (1+(-1)/n)^n ? ?=ln??e^(-1) ?=-1 converges to -1
4. Determine whether the series converges or diverges.
(a) 1/(x^2+x-12)=1/(x-3)(x+4) =A/(x-3)+B/(x+4)
x=3 ?A=( 1 )/7 and x=-4 ?B=-( 1 )/( 7 )
?_1^??dx/(x^2+x-12)=( 1 )/7 ? ?_1^??(1/(x-3)- 1/(x+4)) dx=( 1 )/7 ln?|(x-3)/(x+4)| ?|_1^?
=? ( 1 )/7 ln?|(1-( 3 )/x)/(1+( 4 )/x)| ?|_1^?=( 1 )/7 (ln?1-ln?|(-2)/5| )=-( 1 )/7 ln?(( 2 )/5) converge
(b) lim?(n??)??(n&a_n )=lim?(n??)?(( 3n+2 )/( 5n ))=( 3 )/( 5 )<1converge


المادة المعروضة اعلاه هي مدخل الى المحاضرة المرفوعة بواسطة استاذ(ة) المادة . وقد تبدو لك غير متكاملة . حيث يضع استاذ المادة في بعض الاحيان فقط الجزء الاول من المحاضرة من اجل الاطلاع على ما ستقوم بتحميله لاحقا . في نظام التعليم الالكتروني نوفر هذه الخدمة لكي نبقيك على اطلاع حول محتوى الملف الذي ستقوم بتحميله .
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