IUK 191 MATHEMATICS I [MATEMATIK I]
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1 UNIVERSITI SAINS MALAYSIA First Semester Eamination 00/0 Academic Session November 00 IUK 9 MATHEMATICS I [MATEMATIK I] Duration: hours Masa: [ jam] Please check that this eamination paper consists of SEVEN pages of printed material before you begin the eamination. [Sila pastikan bahawa kertas peperiksaan ini mengandungi TUJUH muka surat yang bercetak sebelum anda memulakan peperiksaan ini.] Instructions: Answer FIVE questions. You may answer the questions either in Bahasa Malaysia or in English. [Arahan: Jawab LIMA soalan. Anda dibenarkan menjawab soalan sama ada dalam Bahasa Malaysia atau Bahasa Inggeris.] In the event of any discrepancies, the English version shall be used. [Sekiranya terdapat sebarang percanggahan pada soalan peperiksaan, versi Bahasa Inggeris hendaklah diguna pakai.] /-
2 - - [IUK 9]. (a) Suppose that 5 A if f ( ) if, where A and B are constants. B if (i) If f( ) is continuous at, show that A B 0. (ii) Find lim f( ). (6 marks) If f( ), (i) find f ' ( ). Simplify your answer. (ii) find equations of the tangent line and normal line to the curve of f( ) at the point. (0 marks) Evaluate lim 0 6 tan 6 sinh. (Hint: lim ) h 0 h ( marks). (a) If f ( ) g( ), where g (4) 8 and (i) f ' (4). g ' (4) 7, find (ii) f g (4). ( marks) Use definition of limit to prove that lim( 5) 7. 4 Give the eact value for cos sin cos 5 5. (Hint: cos( y) cos cos y sin sin y, sin sin cos ) cos cos sin, (4 marks) /-
3 - - [IUK 9] (d) ' Suppose a company has a marginal cost function y C ( ) 9, where is the number of thousands of items sold and the cost C is in thousands of ringgit. If fied costs are RM thousand, what is the total cost of manufacturing the first 4 thousand items?. (a) According to a certain model, the world population (in billions) t years 40 after 960 is modelled by the function pt (). 0.08t e (i) If this model is correct, at what rate will the world population be increasing with respect to time in the year 00? (ii) At what percentage rate will it be increasing at this time? (iii) What is the highest world population epected to be? (8 marks) Solve the initial value problem of the differential equations (i) dv t tv t e dt with v (0) 5 (ii) dy d sin e y cos with y (0) (0 marks) Two ships leave the same port at p.m. The first ship heads due north and at p.m. is observed to be 6 miles due north of port. The ship is going at.5 miles per hour at that instant. The second ship leaves the port at the same time as the first ship, and heads due east. One hour later, the second ship is observed to be 8 miles due east of port and going at 4.5 miles per hour. At what rate is the distance between the two ships changing at p.m.? Hint: Let at () and bt () be the distances that the first and second ships are from port at time t (hours) after leaving. 4. (a) Find the critical points of f ( ) ( 5) 4. Find the area of the region bounded by the curve y 4 and the -ais. 4/-
4 - 4 - [IUK 9] A manufacturer needs to make a cylindrical can that will hold.5 L of liquid. Determine the dimensions that will minimize the cost of the material to manufacture the can. (Hint: Volume, V r h and surface area, A r rh, L 000 cm ) 5. (a) Evaluate (i) 4 (5 ) d (ii) ( ) e d 0 (iii) d (iv) 4 sin cos d (0 marks) Find the length of the curve y between the points (,) and (4,8). (4 marks) Solve by Cramer s Rule. y z y z 0 y z 0 (4 marks) (d) Sketch the graph of f ( ) 5. (8 marks)...5/-
5 - 5 - [IUK 9]. (a) Andaikan 5 A if f ( ) if, di mana A dan B adalah pemalar. B if (i) Jika f( ) selanjar pada, tunjukkan bahawa A B 0. (ii) Cari lim f( ). (6 markah) Jika f( ), (i) cari f ' ( ). Permudahkan jawapan anda. (ii) cari persamaan bagi garis tangen dan garis normal pada lengkung f( ) pada titik. (0 markah) Kirakan lim 0 6 tan 6 sinh. (Petunjuk: lim ) h 0 h ( markah). (a) Jika f ( ) g( ), di mana g (4) 8 dan (i) f ' (4). g ' (4) 7, cari (ii) f g (4). Gunakan definisi had untuk membuktikan bahawa lim( 5) 7. 4 ( markah) Berikan nilai yang tepat untuk cos sin cos 5 5. (Petunjuk: cos( y) cos cos y sin sin y, sin sin cos ) cos cos sin, (4 markah) 6/-
6 - 6 [IUK 9] (d) Andaikan sebuah syarikat mempunyai fungsi maginal kos ' y C ( ) 9, di mana ialah jumlah ribuan item yang terjual dan kos C adalah dalam ribuan ringgit. Jika kos tetap ialah RM ribu, apakah jumlah kos pengeluaran 4 ribu item pertama?. (a) Berdasarkan kepada suatu model, populasi dunia (dalam billion) t tahun selepas 960 dimodelkan oleh fungsi pt () t e. (i) Jika model ini betul, pada kadar berapakah populasi dunia akan meningkat terhadap masa dalam tahun 00? (ii) Pada kadar peratusan berapakah populasi dunia akan meningkat ketika ini? (iii) Apakah jumlah penduduk dunia tertinggi yang dijangka? (8 markah) Selesaikan masalah nilai awal persamaan pembezaan (i) dv t tv t e dt dengan v (0) 5 (ii) dy d sin e y cos dengan y (0) (0 markah) Dua kapal meninggalkan pelabuhan yang sama pada pukul petang. Kapal pertama menuju ke utara dan pada pukul petang dilihat berada 6 batu ke utara pelabuhan. Kapal ini bergerak pada.5 batu per jam pada saat itu. Kapal kedua meninggalkan pelabuhan pada masa yang sama dengan kapal pertama dan menuju ke timur. Satu jam kemudian, kapal kedua dilihat berada 8 batu ke timur pelabuhan dan bergerak pada 4.5 batu per jam. Pada kadar apakah, jarak di antara kedua-dua kapal berubah pada pukul petang? Petunjuk: Biarkan at () dan bt () sebagai jarak kapal pertama dan kedua dari pelabuhan pada masa t (jam) selepas meninggalkan pelabuhan....7/-
7 4. (a) Cari titik-titik genting bagi f ( ) ( 5) 4. [IUK 9] Cari luas rantau yang dibatasi oleh lengkung y 4 dan paksi-. Seorang pengeluar perlu membuat satu tin selinder yang akan menampung.5 L cecair. Tentukan dimensi yang akan meminimumkan kos bahan untuk menghasilkan tin tersebut. (Petunjuk: Isipadu, V r h dan luas permukaan, A r rh, L 000 cm ) 5. (a) Nilaikan (i) 4 (5 ) d (ii) ( ) e d 0 (iii) d (iv) 4 sin cos d (0 markah) Cari panjang lengkung y di antara titik-titik (,) dan (4,8). (4 markah) Selesaikan dengan Petua Cramer y z y z 0 y z 0 (4 markah) (d) Lakarkan graf bagi f ( ) 5. (8 markah) -ooo0ooo-
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