# MAT 101 Calculus [ Kalkulus] Duration : 3 hours [Masa : 3 jam]

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1 UNIVERSITI SAINS MALAYSIA Peperiksaan Semester Pertama Sidang Akademik 011/01 Januari 01 MAT 101 Calculus [ Kalkulus] Duration : 3 hours [Masa : 3 jam] Please check that this eamination paper consists of SIX pages of printed material before you begin the eamination. [Sila pastikan bahawa kertas peperiksaan ini mengandungi ENAM muka surat yang bercetak sebelum anda memulakan peperiksaan ini.] Instructions: Answer all eight [8] questions. [Arahan: Jawab semua 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 - - [MAT 101] 1 (a) Let A = R or [ a, a]. Give the definition of an even function f : A B. Also give the definition of an odd function f : A B. (b) Determine if each of the following functions is odd, even or neither, when f is even and g is odd. (i) f g; (ii) f g. [1 marks] 1 (a) Andaikan A = R atau [ a, a]. Beri takrif untuk fungsi genap f : A B. Juga berikan takrif untuk fungsi ganjil f : A B. (b) Tentukan sama setiap fungsi berikut ganjil, genap atau bukan kedua-duanya, jika f genap dan g ganjil. (i) f g; (ii) f g. [1 markah] (a) Give the definition of a one to one function f : A B. Give two eamples of one to one function f : R R. (b) A function f : A B is onto if f(a) = B. A student shows that the function f : R R, f() = + 3, is onto as follows: Consider any b in R. Then and So f is onto. f(a) = b a + 3 = b a = b 3 ( ) b 3 f(a) = + 3 = b. Eplain why this solution is sufficient to show that f is onto. [7 marks]...3/-

3 - 3 - [MAT 101] (a) Beri takrif untuk fungsi satu ke satu f : A B. Beri dua contoh fungsi satu ke satu f : R R. (b) Fungsi f : A B adalah keseluruh jika f(a) = B. Seorang pelajar menunjukkan bahawa fungsi f : R R, f() = + 3, adalah keseluruh seperti berikut: Pertimbangkan sebarang b dalam R. Maka f(a) = b a + 3 = b a = b 3 dan ( ) b 3 f(a) = + 3 = b. Jadi f adalah keseluruh. Terangkan mengapa penyelesaian ini cukup untuk menunjukkan f keseluruh. [7 markah] 3 (a) Find the nonzero value for the constant k that makes continuous at = 0. f() = { tan k, < 0; 3 + k, 0, (b) If f and g are continuous on [a, b], and f(a) > g(a), f(b) < g(b), show that there is at least one solution in (a, b) for the equation f() = g(). [16 marks]...4/-

4 - 4 - [MAT 101] 3 (a) Cari nilai tak sifar untuk pemalar k yang menjadikan selanjar pada = 0. f() = { tan k, < 0; 3 + k, 0, (b) Jika f dan g selanjar pada [a, b] dan f(a) > g(a), f(b) < g(b), tunjukkan bahawa terdapat sekurang-kurangnya satu penyelesaian dalam (a, b) untuk persamaan f() = g(). [16 markah] 4 (a) Differentiate the function, WITHOUT simplfying it: (i) f() = e tan 1 (3), (ii) g() = +, (iii) h() = ln(cos(sin )). (b) (i) If f and g are differentiable at = a, give the formula of the derivative of f + g at = a. (ii) If f + g is differentiable at = a, are f and g necessarily differentiable at = a? Prove it or give a countereample to disprove it. [13 marks] 4 (a) Bezakan fungsi berikut, TANPA meringkaskannya: (i) f() = e tan 1 (3), (ii) g() = +, (iii) h() = ln(cos(sin )). (b) (i) Jika f dan g terbezakan pada = a, beri rumus untuk terbitan bagi f + g pada = a. (ii) Jika f + g terbezakan pada = a, adakah f dan g perlu terbezakan pada = a? Buktikannya atau beri satu contoh lawan untuk menyangkalnya. [13 markah]...5/-

5 - 5 - [MAT 101] 5 (a) Suppose that f is twice differentiable on an interval (a, b) that contains c and f has an inflection point at c. By considering g = f, prove that f (c) = 0. (b) Find all the asymptotes (vertical, horizontal, slant) of the graph of y =. +1 [16 marks] 5 (a) Andaikan f terbezakan dua kali pada selang (a, b) yang mengandungi c dan f mempunyai titk lengkuk balas pada c. Dengan mempertimbangkan g = f, buktikan bahawa f (c) = 0. (b) Cari semua asimptot (mencancang, mengufuk, sendeng) untuk graf untuk y = +1. [16 markah] 6 (a) Find the following integrals/antiderivatives. (i) cos 3 d, 4 (ii) d, 4 (iii) (b) Evaluate ln 4 [18 marks] 6 (a) Cari kamiran/antiterbitan berikut. (i) cos 3 d, 4 (ii) d, 4 (iii) (b) Nilaikan ln 4 [18 markah]...6/-

6 - 6 - [MAT 101] 7 (a) Find a function g such that 0 tg(t) dt = sin 3, for > 0. (b) If f is continuous and 3 f() d = 6, show that f takes on the value at least 0 once on the interval [0, 3]. [8 marks] 7 (a) Cari fungsi g supaya 0 tg(t) dt = sin 3, untuk > 0. (b) Jika f selanjar dan 3 f() d = 6, tunjukkan bahawa f mempunyai nilai 0 sekurang-kurangnya sekali pada selang [0, 3]. [8 markah] 8 (a) Find the intersection point(s) of the curves = y(4 y) and y + 3 = 0. Then sketch both curves on a same coordinate plane. (b) Setup, without evaluating, the integral for the volume of the solid generated by rotating about y = 1 the region bounded by = y(4 y), y + 3 = 0 and y = 1. [14 marks] 8 (a) Cari titik(-titik) persilangan untuk lengkung = y(4 y) dan y + 3 = 0. Kemudian lakarkan kedua-dua lengkung pada satu satah koordinat yang sama. (b) Bentukkan, tanpa menilai, kamiran untuk isipadu bongkah yang terjana dengan memutar sekitar y = 1 rantau yang dibatasi oleh = y(4 y), y + 3 = 0 dan y = 1. [14 markah] - ooo O ooo -

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