Chew, Kui Tat

Emmerline Shelda, Siaw

Ibrahim, Bin Bohari

Norhunaini, Binti Mohd Shaipullah

Nur Fazliana, Binti Rahim

Tan, George Geok Shim [editor].

Unimas Institutional Repository

for Asasi UN/MAS Uni QA 39.2 M425 20 16 IBRAHIM BOHARI NORHDNAINI MOHO SHAlPDllAH ClEW KIDI TAT Pt' -: • •,..... .1 ...... " • .. I~·"Y- .. t ft '-'""'d~'"":\. 1""" il,.n -,"'-, ... \"4"'''' ,1\ P .KHJOhAT MAKLUMAT AK ~D ~MIK UHIMAS 1111 11 1111111111111111111 1000272845 athematics for Asasi UN/MAS SEMESTER 2 .... I'u,,! Klti nil! "".. tJu",,' AJI.a:!,-l 1 '\'TYF.1lC;m "tALAY3lA SARAWM\. a ematies for Asasi UN/MAS SEMESTER 2 t I r I IBRAHIM BOHARI NORHUNAINI MOHO SHAIPULLAH CHEW KHUI TAT NUR FAZLIANA RAHIM EMMERLINE SHELOA SlAW GEORGE TAN GEOK SHIM FARAH LlYANA AZIZAN Universiti Malaysia Sarawak Rota Sarnarahan (';I Ibra him Bohari Norhunaini Mohd Shaipullah Chew Khui Tat Nur Fazliana Rahim Em merline S helda Siaw George Tan Geok Shim Farah Uyana Azizan, 2016 All rights reserved . No part of this publication may be reproduced, s tored in l-etrieval sys tem, or transmitted, in a ny form or by a ny mea ns, electronic, mechanical, photocopying, recording or otherw ise, without the prior permiss ion of t he publisher. Published in Malaysia by UNIMAS Publisher , Universiti Malaysia Sarawak , 94300 Kota Sama rahan, Sarawak, MaIaysla . Printed in Malaysia by InfoGrafik Press Sdn. Dhd. Lot 20 1, No. 265, Section 50, Jalan Nipah, o lr J a lan Abell Utara , 93100 Kuching, Sarawak, Malaysia. Catalogui ng· in · Publication DataPerpustakaan Negar a Mala'ys ia Ibra him Bohari MATHEMATICS for As.,i UNIMAS . SEMESETER 2/ 1BRAH IM BOHARI, NORHUNANI MOHO SHAIPULLAH , CHEW KHUI TAT, NUR FAZLIANA RAHIM, EMMERLINE SHELOA SlAW, GEORGE TAN GEOK SHIM, FARAH LlYANA AZIZAN. includes index Bibliography: pages 333 ISBN 978-967-2008-04-0 1. Mathematics..Problems, exercises, etc. I. Norhunaini Mohd. Shaipullah. II. Chew Khui Ta t. ., 111. NUT Faz li ana Rahim . IV. Emmel'line, Shelda Siaw. -' .,. V. Tan. George Geok Shim. Vi. Fara h Liyana Aziz3n. VII. Tit.le, 810.76 ~'!~" 1(>.:" ..... !t'.l·"lmqt .kademlk , " 1,..,'1 "-'41_."M.& S"R.AV . CONTENT List of table ....... .................. .. ... ... .................................................................... viii List of figures .......... ... ... .. .... .......................................... .. .... ... .. ......... .. .............. ix Preface .......................................... ... ............................................... .. ................. xi 1 Different iat ion ................................................................... ...... .. ... ... .......... .. 1 2 Applications of D ifferentiation ...................................... ... ... .... ..... ......... 33 3 Integration .................................................................. ............. ...... .. ..... .. ... . 67 4 Data Description ....................................... .. . , ..... ... .. ......... .. .... . ...... .. , .... ... 129 5 Permutations and Com binations .................................. .......... .. .......... 181 6 Probability ............. ..................... .. ... ............. .. ............... .............. .... .... .. ... 197 . 7 Random Variables ............................................ ...... ... ... ....... ..... .. .. .. .. ...... . 245 8 Special Probability Distributions .......... ......... ... ... .. ...... ...... .. .. .. .. .... .. .. 277 Glossary ......... .. ......................................... .. .... .. .. ... ... ... .. , .. ........... ........ ..... ..... 329 Bibliography ........................ ............... .. ... .. ...... ........ .. ............... ... ..... ... .. ........ 333 Index .. .......................................................... ....... ..... .... .... .... ... ... ... ... ... ... .......... 33 5 vn ILLUSTRATION LIST LIST OF TABLES Table 1.1 10 Table l.2 10 Table 2.1 36 Table 2.2 37 Table 2.3 40 Table 2.4 41 Table 3. 1 88 Table 4.1 162 Table 5. 1 181 Table 7. 1 262 Table 8.1 278 Table 8.2 309 Vlll PREFACE Mathematics for Asasi UNIMAS Semester 2 is written for student at Centre for Pre­University Universiti Malaysia Sarawak and all local matriculation centr e. This book conta in all the required sub-topic for pre-university for Mathematics Semester 2. The objective of this book is to develop the understanding of mathematical concepts and their application. At the same time, to build proble m solving skill s and to formulate problems into mathematical terms and try to solve the problems. All mathematical concepts are presented in simple English for easy understanding. This books contains eight chapter that had been planned and arran ged carefully based on the university syllabus. In each chapter, all concepts a nd skills presented for each subtopic are accompanied by detailed explaination. Question and solution for each chapter provide a wide range of examination type question based on the concepts and theories learnt. The topics are arra nged in a systematic manner to ma ke learning easier and more effective. The mathe matical concepts of each topic presented in a comprehensive for easy understanding. Answers are also provided for a ll qu estion on each chapter. The book is a lso suitable for first year undergraduate Statistics of a degree or diploma programme. We hope this book will assist Asasi students in mastering mathematical concepts as well as preparing them for their examination. Ibrahim Bohari Norhunaini Moh d Shaipullah Chew Khui Tat Nur Fazliana Rahim Emmerline Shelda Siaw George Ta n Geok Shim Farah Liyana Azizan Xl DIFFERENTIATIONS 1.1 Derivative of a Function The derivative of a function f(x) with respec t to x from first principles is f'(x) = dy = Iim f(x+ h) - f(x) dx ,, ~O h Example 1.1 Find the derivatives of the fo llowing functions from the fir st principles (a) f(x) = 3x + 2 (b) f(x) =5x ' - 4 Solu t ion (a) f(x) = 3x + 2 , f(x+h ) = 3(x + h)+ 2 rex) = lim (3(x + h)+ 2) - (3x + 2) h-4 0 h ,. 3x + 3h + 2 - 3x - 2 = lin h--+ O h 3h = lim­h--+ O h = lim 3 h O = 3 1 Mathematics for Asasi UNIMAS, Semester 2 (b) j(x)=Sx' -4, j(x+h) = S(x +h)' -4 = lim S(x' + 2x" + h ' )- 4- Sx' + 4 h-tO h , Sx ' + IOxh+Sh ' - Sx' = IIm - - --'- ---­h~O h , Sh' + IOxh = IIm--- ­11-.0 h = lim h(Sh + lOx) 11 ..... 0 h = 1imSh+ IOx HO = S(O) + lOx = IOx 1.2 Rule s of Differentiation 1. The constant rule d - (c)= 0 , where c is real number. dx 2. The power rule ~(xn) = nx,,·1 dx 3. The constant multiple rule ~ [cj(x)] = c~ [j(x)] = cj ' (x) dx dx 2 Differentiatio n 4. The sum and difference rule d 	 d r r d- [f(x) ± g(x)]= - LJ (x )] ± - [g(x)]dx 	 dx dx 5. 	 The product rule ~[J(x) g(x)]= f(x)~ [g(x)]+ g(x )~ [J (x )]dx 	 dx dx 6. The quotient rule ~[f(X) ] _ g(x)! [f(x)] - f(x) ! [g (x)] - [ ( )] ' where g(x);< Odx gx() 	 gx 7. 	 The chain rule dy = dyx du where y = flu ) and u = f(x ).dx du dx A simpler version of the chain rule: ~[g(x)]" =neg (x)r' g '(x)dx Example 	1.2 Find dy of the following functions . dx (a) y = 5 (b) y =Xl (c) 	 Y =8x ' 10(d) y =­.r; (e) y = X '+10x -· 8XJ 3 Mathematics for Asasi UNIMAS, Semester 2 (f) y = (8 X4 + 12X5x' - JO) x 3 + I(g) y =-­x- 4 (h) y = ( X 10 + 17 ) ' Solut ion (a) y = 5 dY = ~(5) dx dx = 0 (b) y = x' = 3x 2 (e) y =8x' = 8(S)x' = 40x ' 4 Differentiation P,,~at Khldma' M_k1UUlBl Ak.adrmlk If''J'\'''Ic!>m ~~AJ."'YSI... ~~ ~""AI( (d) 10 Y=-E I = lOx 2 dy = ~(IOx -~ J dx dxl =10 :lX-~J =10(- ~}-~ 3 =-5x "-2 5 ( ) e y=X ; + 10X- 8x 3 dy = ~ (X5 + lOx _ 8x 3 ) dx dx = ~(X5)+ ~(lOx)- ~(8X3)dx dx dx = ~C~5)+ 10~(x)- 8~(x])dx dx ctt =5x' +10-8(3)x2 = 5x' + 10 - 24x2 5 ---- - -Mathematics for Asasi UNIMAS, Semester 2 dy = ~(8x' + 12XSx ' - 10) dx dx = (8x' 	+ 12)~ (Sx ' - J 0)+ (Sx ' - J0)~(8x' + J 2)dr dx =(8x ' + 12XlOx )+ (Sx ' - JOX32x ' ) = 80x; + J20x + 160x ' - 320x 1 =240x' - 320x 3 + 120x Xl + 1 (g) y =-­x - 4 dy _ 	d (Xl+ l) dx 	 dx x - 4 (X - 4)~ (X3 + 1)- (x 3+ 1 )~ (x - 4) _ dx 	 dx - (x - 4)' _ (x- 4X3x ' )- (x 3 +lXI) - (x- 4)' 3x3 - 12x' - x' - I = (x - 4)' 2x3-12x 2 -1 = (x - 4)' 6 Different iation dy = ~ (X'Q + 17 )' dx dx =5(x'O + 17)' ~ (X' O + 17)dx J<;xample 1.3 F· d d 2 y l In - - and d ~ of the fo llowing functions. dx ' d," -(a) y = ) X 10 (b) y = 5x l - 2x Solution 5x 'Q(a) y = dy =~ (5X 'O )dx dt = 5~ (x ' O )dx =5(JOx' ) =SOx' 7 Mathematics for Asas; UNIMAS, Semester 2 d'y _~(dY ) dx ' - dT dx = 50~ (X0 )d, = 50(9x' ) =450x ' 3dy _~(d ' Y J dx 2dx ' dx = 450 ~ (X R )dx = 450(SX7) = 3600x 7 (b) y = 5x ' - 2x dy d ( 3 2) -=- 5x -2x dx dx _d ( ,) d ( 2) = )- X - 2 - X dx dx = 5(3 )x 2 - 2(2x) = J5x ' - 4x 8 Differentiation =~(15x ' - 4x)dx = 15~(x2) - 4 ~(x)dx dx = 15(2}x - 4 = 30x - 4 2yd ' y _ d ( d ) dx 3 - dx dx ' = -d - - (30x - 4)dx d d =30--(x) --- (4)dx dx = 30 1.3 	 Differentiation of Exponential, Logarithm.ic and Trigonometric Functions 1. 	 ~(e') =e' . If u=g(x) , then ~(e")=e" du. dx 	 dx dx 2. 	 ~(a' )=a ' ln a,where a E R. l fu = g(x),then ~(a " ) = a " du l na. dx 	 dx dx dId I ~ 3. 	 - (In x)=- . If u = g(x) , then - (In u)=-- . dx x 	 dx udx 4. 	 -d (log" x) =--I , where a > 0, a * I . If u = g(x), then dx x Ina ~(iog ,,)=_l _ du . dx " "In a dx 9 Mathematics for Asasi UNIMAS, Semester 2 5. Function derivatives Table 1.1 Function Dedvatives smx cos x cos x - SIn x tan x ., sec · x sec x sec x tan x cosec x -cosec x cot x cot x - cosec 2X 6. Function derivatives Table 1.2 Function Der'ivatives sin I(x) cos I(x) ~ I(x)dx cos I(x) . d - sm ICx)-/(x)dx tan I(x) " dsec ICx)-/(x)dx sec I(x) d sec I (x) tan I(x) - /(x)dx cosec I(x) d - cosec I(x) cot I(x) - /(x)dx cot I(x) - cosec 2/(x)~ I(x)dx 10 Differentiation Example 1.4 Find the derivatives of the following functions. e 5O(a) y = . (b) y = 6' 4 r-2 1(c) y = e +­e" (d) y = 7" + e' (e) y = x ' e"+ l Solution (a) y = e" dy = ~(e" ) dx dx (b) y =6' dy = ~(6 ' ) dx dx = 6' In 6 4). -2 1(c) y = e +­e' dy = ~ (e"-' +_1 ) dx dx e' -_ d (e4",' ) d ( -x )- +- e dx dx =4e 4x - 2 __1 e' 11 Mathematics for Asasi UNIMAS, Semester 2 (d) 	 y = 7" +e' dy _ d(7 4 ' , )--- +e de dx = ~(7 " 	 )+~ (e' )dx; dx; = 7" (4)ln 7+e ' 2 11" .. 1(e) y = x e Example 1.5 Find th e derivatives of the following functions. (a) 	 y = ln2x (b) 	 y = In (Sx + 2) (c) 	 y = logSx (d) 	 y =log] .)Sx - 3 12 Differentiation Solution (a) 	 y = ln 2x Let u = 2x dy d ( )- =- ln2x dx dx J d = - -- (2x)2x d, =- -I (2)2x = x (b) 	 y = ln(5x+2) Let u = Sx+ 2 dy. =~(ln(5x + 2))d, dx 1 d_ =----()x +2)5x +2 	 dx = _  1- (S) )x +2 5 = 5x +2 13 Mathemat ics for Asas i UNIMAS, Semester 2 (c) 	 y ~ log5x Let u ~5x dy d 	( _)- ~- log ) x dx dx 1 d _ ~ - ()x)5x In 10 dx ~ I (5) 5x In 10 1 ~--x ln 10 (d) 	 Y = log] -JSx - 3 Let u ~ 5x -3 dy d 	( _ ~)- ~ - 100- () x - 3)2 dx dx 	 0 ] 1 d I ~ I -- (5x - 3)2 (Sx - 3)2 1n 3 dx ~ 	 I - (.!.(SX - 3t~~ (5X - 3))I (5x - 3), ln 3 2 dx ~ 	 I -(.!.(SX-3t~(5))I (5 x - 3)2 In 3 2 5 14 Differentiation 'xamplc 1.6 Find the deriva tives of the fo llowing fun ctions. (a) 	 y =sin3x (b) 	 y =tan 5x' (c) 	 y =sec(2x 3 + 4) (d) 	 y =cot ' 3x Solut ion (a) y = sin 3x dy d . - =- (sm3x)dr dx =3cos3x (b) 	 y = tall5x' dy = ~ (tan5x' )dx dx = sec ' 5x'~ (5x' ) dx = sec' 5x' (J 5x' ) = 15x' sec ' 5x' 15 Mathematics for Asasi UNIMAS, Semester 2 dy = ~ (sec(2x3 + 4))dx dx = sec(2x' + 4)tan(2x1 + 4)~(2x3 + 4) dx = sec(2x) + 4 ) tan (2x .l + 4). (6x' ) =6x' sec(2x; + 4)tan (2x ' + 4) Cd) y = cot' 	3x dy = ~(cot 4 3x) dx de =~ (cot 3X)'dx , , d 	( )= -4 cot·, 3x cosec 3x - 3x dx =--4 cot 1 3x cosec 1 3x . (3) = - 12 cot' 3x cosec 1 3x 1.4 Implic it Differentiation 1. If y is not alone on one side of the equation , the equation defines y im plicitly as fun ction of x. For example y 2 + x ' = y - 2x. 2. 	 To find dYof an implicit function, differentia te each terms wi th dt respect to x . 16 Differentiation Example 1.7 Find dy for the following functions" dx a ' + 3 = - ,() y x ) x - - y (b) 5x + xy' = 5y Solution , 3 - ,(a) y - + x = )x - Y ' -d (y-' +3x )=-d (_ ) X -- y ) dx d( ~ (y' )+ ~ (3x) = ~ (Sx' )- ~ (y) dx dx dx dx d ( ,) dY d ( ) dl"\y - " +3= IOx - - y - " dy dx ely dx dy dy2y - +3 = 10x - l­dx dx ely dv2y - + 1- = IOx - 3 dt dx dv(2y+ l)- =10x- 3 dx ely 10x - 3 = dx 2y + I 17 

MATHEMATICS for Asasi UNIMAS : SEMESTER 2

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MATHEMATICS for Asasi UNIMAS : SEMESTER 2

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