MAT Calculus [Kalkulus]

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1 UNIVERSITI SAINS MALAYSIA First Semester Eamination 9/ Academic Session November 9 MAT - Calculus [Kalkulus] Duration : 3 ours [Masa : 3 jam] Please ceck tat tis eamination paper consists of SEVEN pages of pried materials before you begin te eamination. [Sila pastikan baawa kertas peperiksaan ini mengandungi TUJUH muka surat yang bercetak sebelum anda memulakan peperiksaan ini.] Instructions: [Araan: Answer all ten [] questions. Jawab semua sepulu [] soalan.] In te eve of any discrepancies, te Englis version sall be used. [Sekiranya terdapat sebarang percanggaan pada soalan peperiksaan, versi Baasa Inggeris endakla diguna pakai]. /-

2 [MAT ]. Find f for te following f (i) f sin f. Using te definition of derivative, find f for f 3. (a) Let f. Given, find in terms of a value of suc tat if ten f 5. [3 marks] Using, prove tat lim 3 [5 marks] 4. Find te following limits (i) lim 5 5 lim 3 lim [9 marks] 5. Suppose f : R R is an even function (i) Prove tat for any R, and f f f f If f is differeiable on R, sow tat f : R R is an odd function. [7 marks] 3/-

3 3 [MAT ]. Cari f uuk f yang berikut (i) f sin f. Menggunakan takrif terbitan cari f uuk f 3. (a) Andai f. Diberi, cari dalam sebutan suatu nilai supaya jika maka f 5. [3 marka] Menggunakan, buktikan baawa ad 3 [5 marka] 4. Cari nilai ad berikut (i) ad ad ad [9 marka] 5. Andaikan f : R R adala suatu fungsi genap (i) Buktikan baawa uuk R, dan f f f f Jika f terbezaan pada R, tunjukkan baawa f : R R adala suatu fungsi ganjil. [7 marka]...4/-

4 4 [MAT ]. (a) Compute te first derivative of te following functions wit respect to (i) f sin n g, Te function f on te ierval, is given by (i) Find c so tat f c Sow tat f c is a local minimum. f d 7. (a) Sow tat Hence determine. Determine e d (c) If f : R R is an odd function, sow tat p f d p f d 8. (a) Let f be a twice differeiable function on ab, and g f f c c f c, c a, b (i) Sow tat c is a critical number of g. If f c, sow by first derivative test tat gc is a local maimum. If f dt, sow tat f e e. 5/-

5 5 [MAT ]. (a) Hitung terbitan pertama teradap uuk fungsi yang berikut (i) f sin n g, Fungsi f pada selang, diberi sebagai (i) Cari c supaya f c f Tunjukkan baawa f c minimum setempat. 7. (a) Tunjukan baawa Seterusnya teukan. [4 marka] Teukan e d [4 marka] (c) Jika f : R R suatu fungsi ganjil, tunjukkan baawa p f d f d p [4 marka] 8. (a) Andai f sebagai suatu fungsi yang terbeza dua kali pada ab, dan g f f c c f c, c a, b (i) Tunjukan c suatu nombor geing g. Jika f c, tunjukkan dengan ujian terbitan pertama baawa Jika f gc iala maksimum setempat. dt, tunjukan baawa f e e. [4 marka] /-

6 [MAT ] 9. A region bounded by te curve 3 y e, ais, y aisand te line u wose area is denoted by Au is rotated troug 3 about te ais. (i) Find Au. State in iegral form te volume of te rotated solid. If te area Au is increasing at unit /sec as te line umoves to te rigt ten find te rate at wic volume of te rotated solid is increasing wen u. [ marks]. (a) Given tat f, 4. (i) Find f and f. Sow tat f as one stationary poi and classify it. Find te equations of one orizoal and two vertical asymptotes. (iv) Sketc te grap of f. [ marks] Evaluate te definite iegrals (i) sin d d 7/-

7 7 [MAT ] 9. Suatu raau yang dibatasi ole lengkung 3 y e, paksi, paksi y dan garis u yang luasnya di tanda dengan Au di putar 3 pada paksi. (i) Cari Au. Nyatakan dalam beuk kamiran isipadu bungka putaran. Jika luas Au menokok pada kadar unit /saat apabila garis u bergerak ke kanan, maka cari kadar perubaan isipadu bungka putaran apabila u. [ marka]. (a) Diberi f, 4. (i) Cari f dan f. Tunjukkan f anya mempunyai satu titik pegun dan teukan jenisnya. Cari persamaan satu asimptot mendatar dan dua asimptot tegak. (iv) Lakar graf f. [ marka] Nilaikan kamiran teu (i) sin d d - ooo O ooo -

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