Mathematics. Complex Number rectangular form. Quadratic equation. Quadratic equation. Complex number Functions: sinusoids. Differentiation Integration

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1 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= or 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

2 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

3 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 j j 11 1

4 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

5 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 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

6 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

7 Sinusoids Radian: / 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

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

9 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

10 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

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

11 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

12 Drivativ of commonly usd functions Drivativ of commonly usd functions 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

13 Find th maimum/minimum dy d dy d d d dy 0 d Stp 1: Find Stp : St or 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

14 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

15 Eampl y=, dy/d= 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 d F d 4 d c

16 Ercis 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

17 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

18 Eampl Sktch th function Shad in th ara dfind by th intgral Comput th intgral f() = + 1 A A f f d d Eampl A A f f d d 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

19 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

2008 AP Calculus BC Multiple Choice Exam

2008 AP Calculus BC Multiple Choice Exam 008 AP Multipl Choic Eam Nam 008 AP Calculus BC Multipl Choic Eam Sction No Calculator Activ AP Calculus 008 BC Multipl Choic. At tim t 0, a particl moving in th -plan is th acclration vctor of th particl