MAT Calculus [Kalkulus]

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1 UNIVERSITI SAINS MALAYSIA Firs Semeser Eaminaion 3/ Academic Session Jan MAT - Calculus [Kalkulus] Duraion : 3 hours [Masa : 3 jam] Please check ha his eaminaion paper consiss of SEVEN pages of prined maerial before you begin he eaminaion [Sila pasikan bahawa keras peperiksaan ini mengandungi TUJUH muka sura yang berceak sebelum anda memulakan peperiksaan ini] Insrucions: [Arahan: Answer all si [6] quesions Jawab semua enam [6] soalan] In he even of any discrepancies, he English version shall be used [Sekiranya erdapa sebarang percanggahan pada soalan peperiksaan, versi Bahasa Inggeris hendaklah diguna pakai] /-

2 -- [MAT ], > (a) Suppose f( ) = +, + a, < Find lim f( ) Why is f coninuous a? (iii) Wha should be he value of a if f is coninuous a? [9 marks] (b) Find he following limi if i eiss sin lim sin lim( + ) (a) Show ha Suppose = lim sin wheher f () eiss sin, f( ) =, = 3 [6 marks] Using he definiion of derivaive, deermine [5 marks] (b) Suppose f is coninuous on [ ab, ] and differeniable on ( ab, ) If f( a ) = and f ( ) > for all ( ab, ), prove ha f( ) > for all ( ab, ] [ marks] 3 (a) Prove lim(3 + ) = using he ε -δ -definiion [9 marks] (b) Find he derivaive of he funcion Do no simplify your answer y = ( + ) sin y = + 5 an (ln( e )) [6 marks] 3/-

3 -3- [MAT ], > (a) Andaikan f( ) = +, + a, < Cari lim f( ) Kenapa f selanjar pada? (iii) Apakah nilai a yang sepaunya jika f adalah selanjar pada? (b) Cari had yang beriku jika ia wujud [9 markah] sin lim sin lim( + ) (a) Tunjukkan bahawa = lim sin [6 markah] Andaikan bahawa enukan sama ada f () wujud sin, f( ) =, = 3 Dengan menggunakan akrif erbian, [5 markah] (b) Andaikan f adalah selanjar pada [ ab, ] dan erbezakan pada ( ab, ) Jika f( a ) = dan f ( ) > bagi semua ( ab, ), bukikan bahawa f( ) > bagi semua ( ab, ] [ markah] 3 (a) Bukikan bahawa lim(3 + ) = dengan menggunakan akrif ε -δ [9 markah] (b) Cari erbian bagi fungsi yang beriku Jangan permudahkan jawapan anda y = + sin y = + 5 an (ln( e )) [6 markah] /-

4 -- [MAT ] (c) Wrie down he correc unique answer Suppose f : R R If f( ) f(), hen f is no an odd funcion True or false? Ans: True / False Suppose neiher lim f ( ) nor lim g ( ) eiss Can lim[ f( ) + g ( )] eis? a a a Ans: Can / Canno f( a+ h) f( a) (iii) If lim eiss, hen lim ( ) h h f eiss True or false? a Ans: True / False (iv) The Ferma s Theorem says ha if f has a local maimum or local minimum a c, hen f () c = True or false? Ans: True / False (v) Are he following wo saemens equivalen? For every ε >, here eiss δ > such ha δ < ε There eiss δ > such ha for every ε >, we have δ < ε Ans: Yes / No [ marks] (a) If If f ( ) d =, f ( ) d =, and cos F( ) = (cos ) d, find F '( π ) f ( ) d = Compue f ( ) d [3 marks] (b) Inegrae d e 3 Le f be a coninuous funcion on he inerval [,5], and i saisfies informaion evaluae f ( + ) d 5 f ( ) d = 6 Wih his [ marks] 5/-

5 -5- [MAT ] (c) Tulis jawapan unik yang beul Andaikan f : R R Jika f( ) f(), maka f bukan fungsi ganjil Benar aau palsu? Jawapan: Benar / Palsu Andaikan kedua-dua had f ( ) dan had g( ) idak wujud Bolehkah had[ f ( ) + g( )] a wujud? Jawapan: Boleh / Tidak Boleh f( a+ h) f( a) (iii) Jika had wujud, maka had f ( ) wujud Benar aau palsu? h h a Jawapan: Benar / Palsu a (iv) Teorem Ferma menyaakan bahawa jika f mempunyai maksimum seempa aau a minimum seempa pada c, maka f () c = Benar aau palsu? Jawapan: Benar / Palsu (v) Adakah dua kenyaaan beriku bersamaan? Jawapan: Ya / Tidak Bagi seiap ε >, wujud δ > di mana δ < ε Wujud δ > di mana bagi seipa ε >, kia ada δ < ε [ markah] (a) Jika Jika f ( ) d =, f ( ) d =, dan cos F( ) = (cos ) d, cari F '( π ) f ( ) d = Hiung f ( ) d [3 markah] (b) Kamirkan d e 3 Biar f suau fungsi selanjar pada selang [,5], dan ia memenuhi makluma ini nilaikan f ( + ) d 5 f ( ) d = 6 Dengan [ markah] 6/-

6 -6- [MAT ] 5 (a) The region bounded by y = and y = is roaed hrough ais = Compue he volume of roaion by using he shell mehod (b) Find he arc lengh of he curve Evaluae 3 d 3 y = which lies above he ais 36 abou he verical [3 marks] [ marks] 6 (a) Evaluae he following inegrals π ( + an + sin ) d 3 d [ marks] (b) For > and f( ) = +, using Mean Value Theorem show ha here is + c (, ) such ha = + c From par b deduce ha + + for > (iii) Using par b, show ha 6 + d [3 marks] /-

7 -- [MAT ] 5 (a) Ranau yang dibendung oleh y = dan y = Hiung isipadu kisaran denagn menguna kaedah silinder (b) Cari panjang lengkuk graf 3 dikisar melalui y = yang erleak sebelah aas paksi 36 pada paksi egak = [3 markah] [ markah] 6 (a) Nilaikan kamiran beriku π ( + an + sin ) d 3 d [ markah] (b ) Unuk > dan f( ) = +, dengan Teorem Nilai Min unjukkan erdapa c (, ) supaya + = + c Dari bahagian b simpulkan + + unuk > (iii) Dari bahagian b unjukkan 6 + d [3 markah] -oooooo-

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