x, x, e are not periodic. Properties of periodic function: 1. For any integer n,

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1 Chpr Fourir Sri, Igrl, d Tror. Fourir Sri A uio i lld priodi i hr i o poiiv ur p uh h p, p i lld priod o R i,, r priodi uio.,, r o priodi. Propri o priodi uio:. For y igr, p. I d g hv priod p, h h g lo h h priod p.. I priodi uio h ll priod p>, h i i o lld h udl priod o. Th udl priod o Th udl priod o i, i i, i wihou udl priod. i h oo priod o,,i,,i,,,i, A rigoori ri h h or i i ud o rpr y priodi uio wih h priod i W ll Fourir ri. Au h i priodi uio o priod h rprd y rigoori ri, 5 i W w o dri h oii o h orrpodig ri 5. d d d d Siilrly, w hv d d d d i d i d

2 d, d d i d i d d i d Thu w oi h rul Filly, w go d,,, i d i d i d i i d i d, i i d d i d i d d i d Thu w oi h rul ulr orul: 6 i d,,, d d i d,,,, Fourir ri o wih 6 Soluio. i odd uio, o w hv i. Fid h Fourir oii o 7 d d d i d i d odd v

3 Thu h Fourir ri o i 8 i i i 5 5 Furhror, w i i i y iiz, Th rigoori y,,i,,i,,,i, i orhogol o h irvl I orul, or y igr d, i i d d i d iludig Giv piwi oiuou uio o priod, w g h oii ordig o 6 d h oi Fourir ri o, dod ~ i Udr wh odiio w g o ow i? Thor I priodi uio i piwi oiuou i h irvl d h l-hd driviv d righ-hd driviv h poi o h irvl, h i Priulrly, i i oiuou i

4 . I pl, h jup,. So w oi i 5* 5 i * i. Fuio o Ay Priod p W w o id rigoori ri o rpr uio o priod p, whr., h g h priod. Thu, w hv g i ~ g whr,, i,, d g d g d g W rpl y, h i ~ g 5 wih Fourir oii,, i,, d d d 6 hr w oly vriy d d g d g y ig. Fid h Fourir ri o, p Soluio. I i v uio.

5 5,,, i d d d d H h rul i,,5, 5 5 ±. Fid h Fourir ri o, p Soluio. i odd uio, o w hv d d d d v odd i i Thu h Fourir ri o i i 5 5 i i. Fid h Fourir ri o p u, i, Soluio. By orul 6 w hv d d i i > i ] i [i ] i [i i d d d d >

6 i i d [ ] d [ ] d [ ] d i i > > Thu u i 5. v d Odd Fuio Hl-Rg pio Propri o h v or odd uio:.i g i v uio, h g d g d.i h i odd uio, h h d Thor Th Fourir ri o v uio o priod i i ri wih Fourir oii d, d,,, Th Fourir ri o odd uio o priod i i ri wih Fourir oii i i d,,, I h o priod uio iply * wih oii *, w giv or v d, d,,, 6

7 I h o priod uio iply * wih oii, w giv or odd i Thor Th Fourir oii o u r h u o h orrpodig Fourir oii o d Th Fourir oii o r i h orrpodig Fourir oii o * i d,,,.. Fid h Fourir ri o Soluio. g whr g i odd uio. For g, w hv g i d i d odd v g ~ i i i 5 5 i i i 5,, ± 5. Fid h Fourir ri o d Soluio. i odd. i d i d i i i, ± 7

8 8 Hl-Rg pio Giv uio, w o h v io;, h odd io;, W y h oh d r hl-rg pio.. Fid h wo hl-rg pio o,, Soluio. v priodi io. 8 8 d d [ ] ydy y ydy y ydy y ydy y ydy y d d Thu or h odd odd v y ydy y y ydy y i 8 i i 8 8 h v hl-rg pio i 5 6 Odd priodi io.

9 i d Thu or h v y i ydy [ ] y i ydy i d y i ydy For h odd 8 i y ydy 8 y y ydy 8 8 i y i H h odd hl-rg pio i 8 5 i i i 5. Copl Fourir Sri Th Fourir ri i wri i opl or y h ulr orul ± i ± i i Thu w oi i i i i, i i i i i i i i i i i i W wri, i, i Th h Fourir ri o 5 i i d, i i i, ±, ±, For h uio o priod, h opl Fourir ri giv 6 whr 7 i i d,, ±, ±, 9

10 . Fid h opl Fourir ri o priodi uio Soluio. W o h, ± i i d i i i i i ih i H h opl Fourir ri i ih i i, Fro hi l u driv h rl Fourir ri i i i ii i i i Oiig h igiry pr, w hv ih i ih i.6 Approiio y Trigoori Polyoil i I good pproiio o? rigoori polyoil o dgr F A A B i i pproiio o, Wh i F? Or wh i A, A, B i, uh h F hiv h iiu or ll rigoori polyoil o dgr. i lld h ol qur rror o F rliv o h uio o h irvl F d d F d d F d whr F d A A A A B i d A B i d A B F d A A A B i d A B

11 Thu w hv B A A d B A A B A A d ] [ I i lr h rh h iiu i * d 6 i d oly i B B A A,,,, H w prov h hor low. Thor Th ol qur rror o i rliv o h uio o h irvl i iiu i d oly i h oii o i r h Fourir oii o. Th iiu vlu i giv y 6. F F Th Bl iquliy. or y, d 7 Th Prvl idiy d 8. Copu h ol qur rror o wih rliv o F Soluio. W hv d d i i F i i i.567 ] [ ] [ * d

12 . Copu h vlu o Soluio., W hv i d i d ~ i i i Aordig o h Prvl idiy d So w hv 6.7 Fourir Igrl Coidr priodi uio o priod h rprd y Fourir ri w i w, w Aordig o h ulr orul, w hv i w d w vi w vdv v w vdv W ow w w w i w d w w w i w w v wvdv v w vdv vi wvdv vi wvdvdw W do A w v wvdv, B w vi wvdv Th w rwri hi i h or 5 [ A w w B w i w]dw 5 i lld rprio o y Fourir igrl. Thor I i piwi oiuou i vry ii irvl d h righ-hd driviv d l-hd driviv vry poi d i h i wll did, h rprd y Fourir igrl 5. A poi whr i dioiuou h vlu o h Fourir igrl qul h vrg o h l- d righ-hd lii o h poi.

13 . Fid h Fourir igrl rprio o h uio > Soluio. Fro B w i w A w v wvdv wvdv w d 5 giv h wr w i w dw w w i w dw w > W h i w dw w I i v, h Bw, d h A w v wvdv 5 h rdu o h Fourir i igrl. A w wdw I i odd, h Aw, d h B w vi wvdv 5 h rdu o h Fourir i igrl. B w i wdw I i did o hl rg, w id h v or odd rprio, rpivly. vluio o igrl. Fid h Fourir i d i igrl o, >, > Soluio. ro w hv v A w wvdv v w i wv wv w w Th Fourir i igrl rprio w dw, w >, > w dw, w >, > ro w hv v B w i wvdv w v w i wv wv w w w Th Fourir i igrl rprio i w w dw, w >, > wi w dw, w >, >

14 . vluio o Igrl or h ollow w wi w dw w > Soluio. Fro w hv or h rh uio A w v wvdv B w vi wvdv wvdv i wvdv w w w So w hv w wi w dw w >.8 Fourir Coi d Si Tror For v uio, w hv wih A w wdw A w ˆ w v wvdv wd ˆ w wdw Th d ˆ w i lld h Fourir i ror o, d i lld h ivr Fourir i ror o ˆ w For odd uio, w hv wih B w i wdw B w vi wvdv Th 5 6 ˆ w i wd d ˆ wi wdw ˆ w i lld h Fourir i ror o, d i lld h ivr Fourir i ror o ˆ w W lo do F { } ˆ, F { } ˆ, F ˆ { }, F { ˆ }. Fid h Fourir i d Fourir i ror o h uio > Soluio. Fro d 5 w hv ˆ w i w wd w w ˆ w i wd w

15 . Fid F { } Soluio. By w g F { } wd w wi w w w I i y o how h h Fourir i d i ror r lir oprio, 7 F { g} F { } F { g} g} F { } F { g} Thor oiuou d oluly igrl o h -i, l piwi oiuou o h ii irvl, d l,, h F { } w } 8 F { } wf { } Proo. Thi ollow ro h diiio F { } wd w w i wd w }; F { } i wd i w w wf { } wd 9 I i y o how y hor h F { } wf { } w F { } F { } wf { ' } w F { } w.. Fid h Fourir i ror o whr > Soluio. By diriio, F { } w F { } F { } w },.9 Fourir Tror W idr h Fourir igrl [ A w w B w i w]dw A w v wvdv, B w v i wvdv Coiig oghr, w hv v [ wv w i wvi w] v [ w v ] dvdw dvdw 5

16 Ad h v[ w v ] dvdw W lo g h i[ ] v w v dvdw -i d rulig v v { [ w v ] i i[ w v ] } iw v dvdw dvdw Hr rwri iwv iw 5 v dv dw Furhror iw 6 ˆ w d i lld h Fourir ror o, d iw 7 ˆ w dw i lld h ivr Fourir ror o ˆ w F { } ˆ w F { ˆ w} Suii or h i o h Fourir ror 6 r h ollowig wo odiio:. i piwi oiuou.. i oluly igrl o h -i.. Fid h Fourir ror o i d ohrwi. Soluio. For 6 y igrio, iw iw iw ˆ w d iw iw. Fid h Fourir ror o Soluio. }, whr >. p p iw d iw iw d iw p w p w p p p v iw dv w p d Thor Th Fourir ror i lir oprio. Th F { g } } g } Proo. I i y o g ordig o h diiio o Fourir ror. Thor oiuou o h -i d Furhror, l oluly igrl o h -i. Th 9 F { } iw } 6

17 Proo. Fro h diiio, w hv iw } d iw iw d iw iw d iw } } iw } w }.. Fid h Fourir ror o Soluio. g g } y., w hv } w oi h g iw iw } g } g } w Th ovoluio *g o d g i did y * g v g v dv v g v dv Thor. Suppo h d g r piwi oiuou, oudd, d oluly igrl o h -i. Th * g} } g} Proo. By h diiio iw * g} v g v dvd iw v g v ddv iw v p v g p dpdv iwv iwp v dv g p dp g} By ig h ivr Fourir ror, * iw g ˆ w gˆ w dw 7

(A) 1 (B) 1 + (sin 1) (C) 1 (sin 1) (D) (sin 1) 1 (C) and g be the inverse of f. Then the value of g'(0) is. (C) a. dx (a > 0) is

(A) 1 (B) 1 + (sin 1) (C) 1 (sin 1) (D) (sin 1) 1 (C) and g be the inverse of f. Then the value of g'(0) is. (C) a. dx (a > 0) is [STRAIGHT OBJECTIVE TYPE] l Q. Th vlu of h dfii igrl, cos d is + (si ) (si ) (si ) Q. Th vlu of h dfii igrl si d whr [, ] cos cos Q. Vlu of h dfii igrl ( si Q. L f () = d ( ) cos 7 ( ) )d d g b h ivrs