MAA Calculus for Science Students I [Kalkulus untuk Pelajar Sains I]
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1 UNIVERSITI SAINS MALAYSIA First Semester Eamination Academic Session 6/7 December 6 / January 7 MAA - Calculus for Science Students I [Kalkulus untuk Pelajar Sains I] Duration : 3 hours [Masa : 3 jam] Please check that this eamination paper consists of TEN pages of printed materials before you begin the eamination [Sila pastikan bahawa kertas peperiksaan ini mengandungi SEPULUH muka surat yang bercetak sebelum anda memulakan peperiksaan ini] Instructions: Answer EIGHT (8) questions [Arahan: Jawab LAPAN (8) soalan] 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 - - [MAA ] (a) Solve the inequality: 3 [ 5 marks ] Suppose f 6 and g (i) Find f and its domain Find the domain and range of f o g (iii) Evaluate the limit g lim 5 [ 35 marks ] (a) Selesaikan ketaksamaan: 3 [ 5 markah ] Andaikan f 6 and g (i) Cari f dan domainnya Cari domain dan julat bagi f o g (iii) Cari nilai lim g 5 [ 35 markah ] 3/-
3 - 3 - [MAA ] (a) (i) Determine the eistence of sin lim sin Let f m n 6 Find the values m and n so that f is continuous everywhere [ 3 marks ] Consider the function f For what values of is f differentiable? [ marks ] (a) (i) Tentukan kewujudan sin lim sin Andaikan f m n 6 Cari nilai-nilai m dan n supaya f adalah selanjar pada semua titik [ 3 markah ] Pertimbangkan fungsi f Untuk nilai apakah f adalah terbezakan? [ markah ] /-
4 3 (a) Let f Use the definition of the derivative to find f [MAA ] [ 5 marks ] Find the derivative of the following functions: (i) f tan f ln ln (iii) f 9 cos 5 7 [ marks ] (c) Consider the curve with equation y Find the equation of tangent line to curve at the point 3, 6 [ 5 marks ] 3 (a) Andaikan f 6 Gunakan takrif bagi terbitan untuk mencari f [ 5 markah ] Cari terbitan bagi fungsi berikut: (i) f tan f ln ln (iii) f 9 cos 5 7 [ markah ] (c) Pertimbangkan lengkung dengan persamaan y Dapatkan persamaan garis tangen terhadap lengkung pada titik 3, 6 [ 5 markah ] 5/-
5 - 5 - [MAA ] (a) Given f with f and f 3 Find: (i) all the - and y-intercepts and also the asymptotes; the intervals on which f is increasing or decreasing; (iii) all the local maimum and minimum value of f, if any; (iv) the intervals of concavity and the inflection points, if eist; (v) sketch the graph of f [ 5 marks ] 3 Evaluate the following limits: (i) cos lim lim (iii) lim tan 3 [ 5 marks ] 3 (c) Let f e Using Newton s method with initial value,, find the second approimation, for the solution of f [ marks ] (a) Diberi f f (i) (iii) (iv) 3 3 Cari: dengan f semua pintasan- dan -y dan juga asimptot; selang yang mana f menokok atau menyusut; semua nilai maksimum dan minimum tempatan bagi f, jika ada; selang kecekungan dan titik lengkuk balas, jika wujud; (v) lakarkan graf bagi f dan [ 5 markah ] 6/-
6 - 6 - [MAA ] Nilaikan had berikut: (i) cos lim lim (iii) lim tan3 [ 5 markah ] 3 (c) Andaikan f e Gunakan kaedah Newton dengan nilai awal,, cari nilai anggaran kedua, bagi penyelesaian f [ markah ] 7/-
7 - 7 - [MAA ] 5 (a) Figure below shows the graph of a function f Sketch the graph of an antiderivative of f, given that the antiderivative function starts at the origin of coordinate y 3 5 Find the average value of the function [ 5 marks ] f( ) ln on the interval [, e ] [ 5 marks ] 5 (a) Rajah di bawah menunjukkan graf suatu fungsi f Lakarkan graf antiterbitan bagi f, diberikan bahwa fungsi anti-terbitan bermula pada asalan y 3 5 [ 5 markah ] Cari nilai purata fungsi f( ) ln pada selang [, e ] [ 5 markah ] 8/-
8 - 8-6 (a) Evaluate the indefinite integrals (i) ln d [MAA ] [ 5 marks ] d [ 5 marks ] Use the midpoint approimation to calculate the integral 5 subintervals Round your answer to decimal places sin d with [ marks ] 6 (a) Nilaikan kamiran tak tentu (i) ln d [ 5 markah ] d [ 5 markah ] Gunakan penghampiran titik tengah untuk mengira kamiran dengan 5 subselang Bulatkan jawapan anda ke tempat perpuluhan sin d [ markah ] 9/-
9 - 9-7 (a) Sketch the region enclosed by the curves Find its area [MAA ] y and y [ 5 marks ] Find the volume generated by rotating the region shown in the figure below about the -ais y 3 y cos 6 y y 3 [ 5 marks ] 7 (a) Lakarkan rantau yang dibatasi oleh lengkung-lengkung y Cari luas rantau itu y dan [ 5 markah ] Cari isipadu yang terjana dengan memutarkan rantau yang ditunjukkan dalam rajah di bawah sekitar paksi- y 3 y cos 6 y y 3 [ 5 markah ] /-
10 - - [MAA ] 8 (a) Find the coordinates of the leftmost point on the curve sin t sin t, y cos t cos t, where t [ 5 marks ] Find the length of the curve t, y e e, t t t [ 5 marks ] 8 (a) Cari koordinat titik paling kiri pada lengkung sin t sin t, y cost cost, yang mana t [ 5 markah ] Cari panjang lengkok untuk lengkung t, y e e, t t t [ 5 markah ] - ooo ooo -
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