MAA 101 Calculus for Science Students I [Kalkulus untuk Pelajar Sains I]
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1 UNIVERSITI SAINS MALAYSIA Peperiksaan Kursus Semasa Cuti Panjang Sidang Akademik 9/ Jun 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 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 all ten [] questions [Arahan : Jawab semua sepuluh [] 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 ] If f show that f ( f ( )) f ( f ( )) [7 marks] Let f a product of two functions defined by f ( ) ln(5 ) Find the domain of f Find the interval of that satisfies + 6 [9 marks] 3 Evaluate the following limit The L Hospital rule can be applied whenever applicable lim lim ln(3 ) (c) lim 3 5 [ marks] Given f ( ) a, b, Find a and b so that f is differentiable at Show that the equation solution on the interval [,] has at least one [ marks] 3/-
3 3 [MAA ] Jika f, tunjukkan bahawa f ( f ( )) f ( f ( )) [7 markah] Biarkan f hasildarab dua fungsi yang ditakrif sebagai f ( ) ln(5 ) Cari domain f Dapatkan selang bagi yang memenuhi + 6 [9 markah] 3 Nilaikan had berikut Hukum L Hospital boleh digunakan di tempat yang sesuai had ln(3 ) had (c) had 3 5 [ markah] Diberi f ( ) a, b, Dapatkan a dan b sedemikian f terbezakan pada Tunjukkan bahawa persamaan sekurang-kurangnya satu penyelesaian pada selang [,] mempunyai [ markah] /-
4 [MAA ] 5 Find the derivative dy of the following equations d y (tan3 e) 5 sin( y) y [8 marks] 3 6 Find the constants, and that guarantee the graph of f () will have a local maimum at ( 3,8) and a local minimum at (, ) Find the equation of tangent line to the curve y at the point A(,) [ marks] 7 Find the absolute etrema value of f() on the interval [,] Sketch a graph of a function g that is continuous and differentiable on interval [,] and satisfies the following conditions: (i) the function g is decreasing on (,3) and increasing [,) and (3,] (ii) the largest value of g is 5 and the smallest value of g is (iii) the graph of g has local etrema at (,5) and (3,) [ marks] 8 Evaluate the following integral / sin cos d tan d [ marks] 5/-
5 5 [MAA ] 5 Dapatkan pembezaan dy persamaan-persamaan berikut d y (tan3 e) 5 sin( y) y [8 markah] 6 Dapatkan pemalar, dan yang menjaminkan bahawa graf 3 f () akan mempunyai maksimum tempatan pada ( 3,8) dan minimum tempatan pada (, ) Dapatkan persamaan suatu garis tangen kepada lengkung pada titik A(,) y [ markah] 7 Dapatkan nilai ekstrema mutlak bagi f() pada selang [,] Lakarkan graf untuk fungsi g yang selanjar dan terbezakan pada selang [,] dan memenuhi keadaan berikut: (i) fungsi g menyusut pada (,3) dan menokok pada[,) dan (3,] (ii) nilai terbesar bagi g ialah 5 dan nilai terkecil bagi g ialah (iii) graf g mempunyai ekstrema tempatan pada (,5) and (3,) [ markah] 8 Nilaikan kamiran berikut / tan sin cos d d [ markah] 6/-
6 6 [MAA ] 9 Show that Hence, evaluate d Given F is a continuous function such that F( ) t dt t What is F ()? [ marks] Let R be the region bounded by the graphs of equations y and y Sketch the region R and find its area Set up without evaluating the integral for the volume of the solid obtained by rotating the region R about the specified line using both cylindrical shell and washer method (i) y-ais (ii) y (iii) [3 marks] 7/-
7 7 [MAA ] 9 Tunjukkan bahawa Seterusnya nilaikan d Diberi F suatu fungsi yang selanjar yang mana F( ) t dt t Apakah F ()? [ markah] Biarkan R rantau yang di batasi oleh persamaan y dan y Lakarkan rantau R dan cari luas rantau tersebut Nyatakan tanpa menilaikan, kamiran untuk isipadu bongkah kisaran yang terhasil apabila rantau R dikisarkan terhadap garis yang dispesikasikan menggunakan kedua-kedua kaedah petala selinder dan washer (i) y-ais (ii) y (iii) [3 markah] -oooooo-
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