Mathmatics Compl numbr Functions: sinusoids Sin function, cosin function Diffrntiation Intgration Quadratic quation Quadratic quations: a b c 0 Solution: b b 4ac a Eampl: 1 0 a= b=- c=1 4 1 1or 1 1 Quadratic quation Ral solutions if b 4ac 0 What about this cas? b Eampl: 1 0 4ac 0 Compl Numbr rctangular form Compl numbr: Rctangular form: Imaginary, j z a Ral part jb (a,b) Imaginary part R z = a, Im z = b b b 4ac a j 1 j b a Ral 4
Calculations in rctangular form z = a + jb w = c + jd Addition/subtraction z+w = (a+c) + j(b+d) z-w = (a-c) + j(b-d) Multiplication zw = (a+jb)(c+jd) = ac + jad + jbc + j*j*bd = ac + jad + jbc - bd = (ac-bd) + j (ad+bc) Compl numbrs A = 1 + j B = j4 A+B = AB = 5 6 Calculations in rctangular form Ercis 1 z = a + jb w = c + jd Division a jbc jd z a jb w c jd c jd c jd z ac bd j bc ad w c d Solution: z w j j? 7 8
Compl Numbr polar form Rctangular form: z a jb Eampl z=-1-j Polar form: Imaginary, j b=sin (a,b) Ral a=cos z z z z a z tan z j b 1 z cos b a j j sin 9 z z z j 4 10 Radian angl Radian Standard unit of angular masur Equal to th lngth of th arc of a unit circl /60 Ercis Eprss th following numbrs in polar form: z 1 j z z 1 1 1 j j 11 1
Ercis Eprss th following numbrs in polar form: z z z 1 j j 5 Ercis 4 Givn: z 1 =+j z =--j Calculat (z 1 ) / z 1 14 Ercis 5 9 j Simplify j 1 answr 1 j j 9 j Conjugat Compl numbr: z a jb Conjugat: * z a jb Imaginary, j b j (a,b) b a Ra l 15 16
Eulr s Rlation j j cos cos cos sin j j j j sin j sin j j Functions and Graphs f : input For ach input actly on output =, y= 11 : indpndnt variabl f(): function output / dpndnt variabl 17 18 Diffrnt kinds of functions Constant function Diffrnt kinds of functions Quadratic function Linar functions Polynomial functions 19 0
Diffrnt kinds of functions Eponntial function Diffrnt kinds of functions Trigonomtric function sin: cosin: 1 Sinusoids Sinusoids Basis of all signals A sinusoid is a signal that has its magnitud changs in tim according to a sin function sin() sin Radian Standard unit of angular masur Equal to th lngth of th arc of a unit circl /60 (in dgr) 4
Sinusoids Radian: /60 5 6 Chang of Amplitud sin() A sin () Chang of Amplitud How larg th sin wav is 5 sin() sin() 7 8
Chang of Frquncy sin() Tim shift and phas shift sin() sin(b) How fast th sin wav is changing (t) is th rcivd signal If this signal is rcivd t 0 sconds latr (t+ t 0 ) Sin() 9 0 Ercis 6 Sktch: y=sin() Sktch: y=sin(+1) Ercis 7 Sktch: y=sin(+pi/6) Sktch: y=sin(+pi/4) 1
Diffrntiation Rviw of Calculus - Diffrntiation Rlatd to find vlocity of an objct at a particular instant Objct movs along th -ais and its displacmnt s at tim is: Diffrntiation is th mathmatical procss to valuat th drivativ of a function (signal) Drivativ rfrs to th instantanous rat of chang of a function (signal) For a function f() to b continuous at a point, th function must ist at th point, and a small chang in producs only a small chang in f() y y = f() Vlocity at t=? y Small chang in, i.. will caus a small chang in y, i.. y, sinc y = f() 4 Drivativ and th slop of a curv Th drivativ y Th slop of th lin through P and y Q is y1 m 1 Th slop at any point of a curv (say point P) is th limiting valu of th slop of th PQ as Q approachs P, i.., whn is vry small P ( 1, y 1 ) Q (, y ) 5 Givn y=f(), th drivativ of y with rspct to is givn by dy d f f d d f h f lim h0 h Th drivativ of a curv y=f() at a point (.g., 1 ) is th slop of th curv at that point 6
Drivativ of a constant y=f()=c, c is a constant dy d d c d c c lim 0 h0 h Drivativ of a straight lin y=f()= dy d d d h lim h0 h 7 8 Drivativ of t^ Ercis 8 of finding th drivativ ds t d t dt dt y f th t lim limtht h0 h h0 Slop at = is 1 9 40
Basic Diffrntiation Formulas f() Constant, c 0 n a ln n n 1 a ln a 1 df d 41 Diffrntiation Diffrntiation is th mathmatical procss to obtain th drivativ of a function Givn y=f()=-, diffrntiat y with rspct to (i.., drivativ of y with rspct to ) dy d d d d d d d d 0 d 4 Ercis 9 Givn y f Diffrntiat y with rspct to : Drivativ of commonly usd functions Givn y f Diffrntiat y with rspct to 4 44
Drivativ of commonly usd functions Drivativ of commonly usd functions 45 46 Ercis 10 y f dy d d d Finding th maimum/minimum On important applications of diffrntiation is to find th maimum/minimum point(s) of a curv 47 48
Find th maimum/minimum dy d dy d 5 10 100 10 10 d d dy 0 d Stp 1: Find Stp : St 10 10 0 10 100 4 10 4.187 or 0.8054 49 Ercis 11 Find th drivativs of th following functions: 1. y. y. y 50 Ercis 1 Rviw of Calculus - Intgration A rctangular bo without lid is to b mad from a squar cardboard of sids 18cm by cutting qual squars from ach cornr and thn folding up th sids. Find th lngth of th sid of th squar that must b cut off if th volum of th bo is to b maimizd. 51 Intgration is a mathmatical opration that allows th valuation of th total sum of a function within a crtain valuation window Eampl application It is known that, on avrag, th Intrnt traffic y in normal wkdays is givn by y=f(), whr is th tim of a day (00:00 to :59) For planning th ntworking systm, nd to know th total traffic during th offic hour from 9:00 am to 5:00pm 5
Sum of a function If th traffic is constant in a day, th total traffic can b obtaind as, y 17 9 c Sum of a function y f 9:00 1hr f 10 : 00 1hr... f 17 : 00 1hr Traffic is not constant??? 5 54 Anti-diffrntiation Intgration: rvrs opration of diffrntiation By diffrntiating a function y=f(), w gt th chang of y, dy for a small chang in By intgrating dy/d, w gt back th original function y 55 56
Eampl y=, dy/d=1 57 58 Th indfinit intgral If d F f f d F c d F() is known as th indfinit intgral of f() Th constant c is ndd sinc th drivativ of a constant is zro Th rsult of an anti-drivativ is not uniqu ( c can b any valu) Eampl d d 8 8 d F c Find th indfinit intgral of Solution: n n n1 d d 8 4 8 d F d 4 d c 8 59 60
Ercis 1 9 9 d Fc d F Find th indfinit intgral of Solution: d 9 A tabl of Intgrals n 1 function n1 c n 1 ln c c Intgral sin cos c cos sin c 61 6 Ercis 14 Th dfinit intgral Givn f d F c Th total sum of th function from =a to =b: b b F c F c F F a a 6 64
Th dfinit intgral Th dfinit intgral of a function f() is dfind as, a: lowr limit b: uppr limit b a f d F F b a Th dfinit intgral: ara undr th curv of y=f() from =a to =b (summation) 65 Proprtis of dfinit intgral Linarity b b b f g d f d g d a a a Inquality If m f() M, thn b m ba f d M ba a 66 Proprtis of dfinit intgral Additivity of intgration on intrval If a c b, thn b c b f d f d f d a a c Application of Intgration Rsrv limit of dfinit intgral b a a b f d f d ara A b a f d 67 68
Eampl Sktch th function Shad in th ara dfind by th intgral Comput th intgral f() = + 1 A A 1 1 0 f f d d Eampl A A 1 1 0 f f d d 1 0 1 4 1 1 0 69 70 Ercis 15 Comput th intgral Shad in th ara dfind by th intgral F dt Solution 1 Ercis 16 Find th ara of th rgion that is boundd by th lin =1, th -ais and th curv >1 71 7
Ara btwn graphs of function Eampl Find out th ara lis blow y=+, abov that of y=^, and btwn =- 1 and = 7 74 Ercis 17 Find th ara of th combind rgion boundd by th -ais and th curv 75