Midterm Math2
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Transcript of Midterm Math2
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8/13/2019 Midterm Math2
1/20
M IDTERM M ATHEMATICS 2
COURSE OVERVIEW
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Midterm Examination
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MIDTERM MATHEMATICS 2
1 !"#$%&'()*+,-.(/-01
Introduction >(? >$7/1..........................................................................................................................-%@!&A/*'(A$
1.
2. 3. 4. B/15?@& x-y 3*-%@
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MIDTERM MATHEMATICS 2
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MIDTERM MATHEMATICS 2
Tool3: 0
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MIDTERM MATHEMATICS 2
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a 1 < a 2 < a 3 < ... < a n < ...
a 1 ! a 2 ! a 3 ! ... ! a n ! ...
a1
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> ... > an
> ...
a1
! a2
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MIDTERM MATHEMATICS 2
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MIDTERM MATHEMATICS 2
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2. !$"718K=
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MIDTERM MATHEMATICS 2
B,F5=(0'9/!6=(07@ D>
!!7
n
n 2 + 2n=1
!
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1
+ !
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MIDTERM MATHEMATICS 2
AFR323 42 'BE2#6(%N#O;JE3O%;23O% (Comparison Test)
7. E*:05*@A/!$"7189= N74B$5>B? '#O$!$"7183< A'C?/1P!3< A!!7 1) 2)
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1
n(n+1)(n+ 2)n = 1
!
" 1
n 2 nn = 1
!
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>B? ]an :3M ]bn '#O$!$"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$= T01"#D8A D5?
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limn !"
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an
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MIDTERM MATHEMATICS 2
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AFR323 44 'BE2#6(%N#O($*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
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#
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/!$"718 '#O$!$"7183< A'C?/B1P!3< A!!7
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MIDTERM MATHEMATICS 2
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MIDTERM MATHEMATICS 2
4 !$"=0;=87%)
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MIDTERM MATHEMATICS 2
12. (Mid. 2/2554)74B$5!$"718743%*
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nn=1
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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
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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)
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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
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MIDTERM MATHEMATICS 2
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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(%$+= N)
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MIDTERM MATHEMATICS 2
6. Q?/ f(x) '#O$;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;