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Midterm Math2

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    M IDTERM M ATHEMATICS 2

    COURSE OVERVIEW

    www.facebook.com/B.EngSocial1

    !"#$%&'()*+,)-./0-12

    345%6

    !$"718!$"718!$%$-2

    !$"718743%*

    Midterm Examination

    '9&2'3!12/ :8++3!1)$

    !$"718;

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    MIDTERM MATHEMATICS 2

    1 !"#$%&'()*+,-.(/-01

    Introduction >(? >$7/1..........................................................................................................................-%@!&A/*'(A$

    1.

    2. 3. 4. B/15?@& x-y 3*-%@

    '23456'

    C?!+@11< ?:B3%77/1C!*!"#$%&'()*+,)-./0-128= D!'5=&128/E/79FGH=I58)I$A >B?$?!*JK)E/1,/9FGH=69C?/*6$:3?@C!*%*'7-1B? P(n) :9$C?!+@/89= N'7= N&@C?!*7%6 n :3M-?!*7/1EMK)0

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    MIDTERM MATHEMATICS 2

    $!7E/7$% T$ B3%77/194IE9&23)8)-C!*345%6+3?/&7/1B/3)8)-'C?/>73?!$%$-2C!*;U*72(%$+P! -./#01 n 23 453'"!$167168#9(12$ :1;!?6@A)(('5B*$#'$)>!/A>2)C@B

    E/7$% T$>(?086%-)5%*$= T 1.

    2.

    Tool2: 'BE.B!F5F*9(1GH1'+I$)23 453E7J>%%K5@'".)# (Indeterminate forms)+P!;U*72(%$9= N'8P N!:9$+A/-%@:#1-/8:3?@!&< A >$1$1(?7HI3#X-/3 (LHopitals rule)

    I5&@)Y=7/1:7?IE9&21

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    MIDTERM MATHEMATICS 2

    Tool3: 0

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    MIDTERM MATHEMATICS 2

    789%6'()>95&B

    @)Y=9= N1 @)Y=Z3-A/*

    @)Y=9= N2 @)Y=Z3B/1

    @)Y=9= N3 5);

    =(0:(

    1) B/7'#O$345%6($)5'K) N8CW T$ a1 EM'#O$C!6'C-3A/* 2) B/7'#O$345%6($)5353* a1 EM'#O$C!6'C-6$

    Step3: B/C!6'C-6$C!*345%69= N'K) N8CW T$ B1P! C!6'C-3A/*C!*345%69= N353* I5&K)E/1,/5%*$= T I5&9/* takeVW N*B/7B/+A/D8A D5? +P!D8A8=C!6'C- lim

    n !"

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    9FGH=: EM'1=&7 {an}@A/...

    345%6'K) N8 ->345%6D8A35 ->345%635 ->345%6D8A'K) N8 ->

    a 1 < a 2 < a 3 < ... < a n < ...

    a 1 ! a 2 ! a 3 ! ... ! a n ! ...

    a1

    > a2

    > a3

    > ... > an

    > ...

    a1

    ! a2

    ! a3

    ! ... ! an

    ! ...

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    MIDTERM MATHEMATICS 2

    4. (Mid. 2/2554)74B$5345%6 E*B/ 7) 345%69= N74B$5>B?'#O$345%6($)5>5 C) C!6'C-6$$?!&0"5:3MC!6'C-3A/*8/70"5 +) 345%69= N74B$5>B?'#O$345%63< A'C?/B1P!3< A!!7

    3 !$"=0;!$%$-1

    $,&(; !$"718!$%$-2(VW N*-A!D#EM'1=&70% T$J@A/!$"718) 0/8/1Q'C=&$!&< A >$1

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    MIDTERM MATHEMATICS 2

    5. (Mid. 2/2554)E*:05*@A/!$"718 '#O$!$"7183< A'C?/B1P!3< A!!7 Q?/3< A'C?/Z36@7C!*

    !$"718'#O$'9A/>5

    !$"=0;!$%$-1 2 C$,9'5 D*B00@ E 1. !$"718'1C/+,)-

    I5&EM 3< A'C?/ Q?/ | r | < 1, 0/8/1Q+4$@,Z36@7!$%$-2 D5?E/7 3< A!!7 Q?/ | r | >/= 1

    2. !$"718K=

    I5&EM 3< A'C?/ Q?/ p >13< A!!7 Q?/ p

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    MIDTERM MATHEMATICS 2

    B,F5=(0'9/!6=(07@ D>

    !!7

    n

    n 2 + 2n=1

    !

    "

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    '#3= N&$ an '#O$ f(x) 04B1%6 x \ 1 B/7 f(x) '#O$;U*72(%$35 :3M-A!'$P N!* EM947/1950!6D5?I5&947/1!)$9)71%3

    B/7B/+A/D5?:05*@A/3< A'C?/ f(x)

    1

    + !

    " dx

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    MIDTERM MATHEMATICS 2

    AFR323 42 'BE2#6(%N#O;JE3O%;23O% (Comparison Test)

    7. E*:05*@A/!$"7189= N74B$5>B? '#O$!$"7183< A'C?/1P!3< A!!7 1) 2)

    @)Y=9= N3 7/1950!6I5&7/1'#1=&6'9=&63)8)- (Limit Comparison Test)

    1

    n(n+1)(n+ 2)n = 1

    !

    " 1

    n 2 nn = 1

    !

    "

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    >B? ]an :3M ]bn '#O$!$"7186@7 EMD5?@A/ 1. Q?/ an bn :3M ]bn 3< A'C?/:3?@ EMD5?@A/ ]an 3< A'C?/5?@& 2. Q?/ an \ bn :3M ]bn 3< A!!7:3?@ EMD5?@A/ ]an 3< A!!75?@& B8/&'B-": $!7'B$P!E/7$= T01"#D8A D5?

    "#$ ! a n %&' ! bn ()*+,+-./012. 3'45$267

    1. 8$7 :3?@ ]an :3M ]bn EM3< A!!7B1P!3< A'C?/'B8P!$7%$ 2. 8$7 :3?@ ]bn '#O$!$"7183< A'C?/ :3?@EMD5?@A/ ]an 3< A'C?/5?@& 3. 8$7 :3?@ ]bn '#O$!$"7183< A!!7 :3?@EMD5?@A/ ]an 3< A!!75?@&

    limn !"

    a nb

    n

    = c > 0

    limn !"

    an

    bn

    = 0

    limn !"

    an

    bn

    = + "

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    MIDTERM MATHEMATICS 2

    8. (Mid. 2/2554)E*>(?7/1950!6:66'#1=&6'9=&63)8)-950!6!$"718 @A/'#O$!$"7183< A

    'C?/B1P!3< A!!7

    AFR323 44 'BE2#6(%N#O($*EB6@A) (Ratio Test)

    9. (Mid. 2/2554)E*>(?7/1950!6:66!%-1/0A@$(Ratio test) 950!6!$"718 @A/'#O$!$"7183< A'C?/

    B1P!3< A!!7

    1

    3 n ! cos nn=1

    "

    #

    n 3

    3 nn=1

    !

    "

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    >B? ]an '#O$!$"7189= N-?!*7/1950!6 Q?/ :3?@ EMD5?@A/

    1. Q?/ L < 1 :3?@ ]an '#O$!$"7183< A'C?/ 2. Q?/ L > 1 :3?@ ]an '#O$!$"7183< A!!7 3. Q?/ L = 1 :3?@01"#D8A D5?@A/ ]an 3< A'C?/B1P!3< A!!7

    limn !"

    an+1

    an

    = L

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    MIDTERM MATHEMATICS 2

    AFR323 45 'BE2#6(%N#O-I/EB'23 4 n (n th Root Test)

    10. E*:05*@A/!$"718 '#O$!$"7183< A'C?/B1P!3< A!!7

    !$"=0;/7%6

    9:2,;67 0

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    MIDTERM MATHEMATICS 2

    7/1950!67/13< A'C?/C!*!$"71803%6

    =(07@ D>)H D!$I<

    11. (Mid. 2/2554)E*950!6!$"71803%6 @A/'#O$:66>5:66B$W N*>$0/8:665%*-A!D#$= T +P!

    3< A'C?/:66086

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    MIDTERM MATHEMATICS 2

    4 !$"=0;=87%)

    $,&(; !$"718743%*>$ x (a=0) '(A$

    !$"718743%*>$ x-a (a _0) '(A$

    CDB)

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    MIDTERM MATHEMATICS 2

    12. (Mid. 2/2554)74B$5!$"718743%*

    7) E*:05*7/1B/1%.8=:3M(A@*C!*7/13< A'C?/I5&K)E/1,/E"5#3/&9% T*0!*C!*(A@*5?@& C) x 8=+A/'#O$'9A/>59= N94>B?!$"7183< A'C?/:6608659= N94>B?!$"7183< A'C?/:668='*P N!$DC

    !$"=0;>'&1>7!01 A7G!$"=0;A;**7!0,$

    (2 x ! 1) n

    nn=1

    "

    #

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    9FGH=69 !$"718'9&2'3!12: >B? f '#O$;U*72(%$9= NB/!$"K%$Y2 D5?9"7!%$5%6 '9&2'3!121!6E"5 x=a 'C=&$D5?5%*$= T

    !$"718:8++3!1)$: B1P!'1=&7!=7!&A/*@A/!$"718'9&2'3!121!6E"5 a=0 'C=&$D5?5%*$= T

    f ( x) = f (a ) + f '(a )(x - a) + f ''(a )(x - a) 2

    2!+ ... + f (n ) (a)

    (x - a) n

    n!+ ...

    f ( x) = f (0 ) + f '(0 )(x) + f ''(0 )(x) 2

    2!+ ... + f ( n ) (0)

    (x)n

    n!+ ...

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    MIDTERM MATHEMATICS 2

    13. (Mid. 2/2554)E*B/!$"718:8++3!1)$C!*;U*72(%$

    5 !$"=0;J@>05&01

    !$"718;

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    MIDTERM MATHEMATICS 2

    14. E*'C=&$71/;;U*72(%$+/69= N74B$5>B?-A!D#$= T K1?!81M6"+/6C!*;U*72(%$$% T$J 1) 2) 3)

    4) 5)

    !$"=0;J@>05&$1 ()8'E5G7;E3OE+23 453CB%;JT) 2 V

    f ( x) = sin x f ( x) = cos x f ( x) = x ; x ! ( " # , # )

    f ( x) =! ;" ! # x < 0 x ; 0 # x < !

    $%&

    f ( x) =! x ;! L " x < 0 x ; 0 " x < L

    #$%

    , f ( x + 2 L) = f ( x)

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    f(x) = a0

    2+ (a n cos nx+b n sin nx)n=1

    !

    "

    a 0 =1!

    f ( x) dx ;n = 0,1,2,.." !

    !

    # .

    a n =

    1! f ( x)cos nxdx ;n

    =

    0,1,2,.." !

    !

    # .

    bn =1!

    f ( x)sin nxdx ;n = 1,2,3,..." !

    !

    #

    f(x0- )+ f(x 0

    + )2

    Q?/ f(x) $)&/8'#O$;U*72(%$+/68=+/6'9A/7%6 2a :3?@!$"718;

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    MIDTERM MATHEMATICS 2

    ()8'E5G7;E3OE+23 453CB%;JT) 2L

    GH1'+I$)C7 @ - GH1'+I$)C3 4

    086%-)6/*#1M7/1C!*;U*72(%$+< A:3M;U*72(%$+= N 1. (;U*72(%$+< A)x(;U*72(%$+< A) = ( ;U*72(%$+< A)2. (;U*72(%$+< A)x(;U*72(%$+= N) = ( ;U*72(%$+= N)3. (;U*72(%$+= N)x(;U*72(%$+= N) = ( ;U*72(%$+< A)4. (;U*72(%$+< A)(;U*72(%$+< A) = ( ;U*72(%$+< A)

    5. (;U*72(%$+= N) ( ;U*72(%$+= N) = ( ;U*72(%$+= N)

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    Q?/ f(x) $)&/8'#O$;U*72(%$+/68=+/6'9A/7%6 2L :3?@!$"718;

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    MIDTERM MATHEMATICS 2

    6. Q?/ f(x) '#O$;U*72(%$+< A:3?@

    7. Q?/ f(x) '#O$;U*72(%$+= N:3?@

    15. (Mid. 2/2554)74B$5;U*72(%$+/6 f(x) $)&/8I5&

    7) E*@/571/;C!* f(x) >$(A@*+@/8&/@ 1 +/6 C) f(x) '#O$;U*72(%$+< AB1P!+= N B1P!D8A'#O$9% T*;U*72(%$+< A:3M+= N E*>B?'B-"Z3#1M7!6 +) E*B/!$"718;

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    MIDTERM MATHEMATICS 2

    15. (Mid. 2/2554)74B$5 8/+1W N*(A@* E*B/!$"718;


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