Transcription
PRE-CALCULUSFORMULABOOKLET
UNIT 1CHAPTER 1RELATIONS, FUNCTIONS,AND GRAPHSSLOPE: m y 2 y1x 2 x1SLOPE-INTERCEPT FORM OF A LINE: y mx bPOINT-SLOPE FORM OF A LINE: ( y y1 ) m( x x1 )STANDARD FORM OF A LINE: Ax By C 0 or Ax By CCHAPTER 2SYSTEMSOF LINEAREQUATIONS AND INEQUALITIESREFLECTION MATRICES 1 0 1 0 ry axis rx axis 0 1 0 1 0 1 ry x 1 0 ROTATION MATRICES 0 1 1 0 R90 R 0 1 180 1 0 0 1 R270 1 0 DETERMINANTS a b a b ad bc2X2: det c d c d a13x3: det a 2 a3b1b2b3c1 a1c3 a 2c3 a3b1b2b3c1bc2 a1 2b3c3c2a b1 2c3a3c2a c1 2c3a3b2b3
CHAPTER 3THE NATURE OF GRAPHSEVEN FUNCTIONS: f ( x) f ( x); symmetric with respect to the y-axisODD FUNCTIONS: f ( x) f ( x); symmetric with respect to the originDIRECT VARIATION: y kx n , n 0INVERSE VARIATION: y k,n 0xnJOINT VARIATION: y kx n z n , where x 0, z 0 and n 0CHAPTER 4POLYNOMIALS AND RATIONAL FUNCTIONSQUADRATIC FORMULA: x b b 2 4ac2aUNIT 2CHAPTER 5THE TRIGONOMETRIC FUNCTIONSLAW OF SINES:abcsin A sin B sin Cor sin A sin B sin CabcAREA OF A TRIANGLE11 sin A sin BK ab sin CK c222sin CHero’s Formula: K s ( s a)( s b)( s c) , where s LAW OF COSINES: c 2 a 2 b 2 2ab cosCa b c2
CHAPTER 6GRAPHS OF TRIGONOMETRIC FUNCTIONSLENGTH OF AN ARC: s r AREA OF A CIRCULAR SECTOR: A ANGULAR VELOCITY: LINEAR VELOCITY: v r1 2r 2 t tCHAPTER 7TRIGONOMETRIC IDENTITIES AND EQUATIONSPYTHAGOREAN IDENTITIES:sin 2 cos2 1tan 2 1 sec 2 1 cot 2 csc2 SUM AND DIFFERENCE IDENTITIES:cos( ) cos cos sin sin sin( ) sin cos cos sin tan tan tan( ) 1 tan tan DOUBLE-ANGLE IDENTITIES:sin 2 2 sin cos cos 2 cos2 sin 2 cos 2 2 cos2 1cos 2 1 2 sin 2 2 tan tan 2 1 tan 2
CHAPTER 7 CONTINUEDHALF-ANGLE IDENTITES: 1 cos sin 22costan 2 2 1 cos 2 1 cos , cos 11 cos NORMAL FORM OF A LINEAR EQUATION:x cos y sin p 0DISTANCE FROM A POINT TO A LINE:Ax By1 Cd 1 A2 B 2CHAPTER 8VECTORS AND PARAMETRIC EQUATIONS VECTORS IN A PLANE (2-D) a a1 , a 2 and b b1 ,b2 INNER (DOT) PRODUCT: a b a1b1 a2 b2 VECTORS IN SPACE (3-D) a a1 , a 2 , a3 and b b1 , b2 , b3 INNER (DOT) PRODUCT: a b a1b1 a 2 b2 a3 b3 a2CROSS PRODUCT: a b b2a3 a1i b3b1a3 a1j b3b1a2 kb2PARAMETRIC EQUATIONS FOR THE PATH OF A PROJECTILE x t v cos 1y t v sin gt 22
UNIT 3CHAPTER 9POLAR COORDINATES AND COMPLEX NUMBERSMULTIPLE REPRESENTATIONS OF (r , ).(r , 2 k ) or ( r , (2k 1) )DISTANCE FORMULA IN POLAR PLANEIf P1 (r1 , 1 ) and P2 (r2 , 2 ) , then P1 P2 r1 r2 2r1r2 cos( 2 1 )22CONVERTINGPOLAR TO RECTANGULAR (r , ) ( x, y)x r cos y r sin RECTANGULAR TO POLAR ( x, y) (r , )r x2 y2 y x y tan 1 , when x 0 x POLAR FORM OF A LINEAR EQUATIONp r cos( )POLAR FORM OF A COMPLEX NUMBERr (cos i sin )PRODUCT OF A COMPLEX NUMBER IN POLAR FORMr1 (cos 1 i sin 1 ) r2 (cos 2 i sin 2 ) r1r2 (cos( 1 2 ) i sin( 1 2 ))QUOTIENT OF A COMPLEX NUMBER IN POLAR FORMr1 (cos 1 i sin 1 ) r1 [cos( 1 2 ) i sin( 1 2 )]r2 (cos 2 i sin 2 ) r2De MOIVRE’S THEOREM[r (cos i sin )]n r n (cos n i sin n ) tan 1 , when x 0THE p DISTINCT pth ROOTS OF A COMPLEX NUMBER1p1p[r (cos i sin )] r (cos 2n where n 0, 1, 2, 3, , p – 1p i sin 2n p)
CHAPTER 10INTRODUCTION TO ANALYTIC GEOMETRYDISTANCE FORMULA FOR TWO POINTSd ( x2 x1 ) 2 ( y2 y1 ) 2MIDPOINT OF A LINE SEGMENTx x y y2midpt ( 1 2 , 1)22STANDARD FORM OF THE EQUATION OF A CIRCLE( x h) 2 ( y k ) 2 r 2GENERAL FORM OF THE EQUATION OF A CIRCLEx 2 y 2 Dx Ey F 0 , where D, E and F are constantsSTANDARD FORM OF THE EQUATION OF AN ELLIPSE( x h) 2 ( y k ) 2( y k ) 2 ( x h) 2or 11a2a2b2b2where c 2 a 2 b 2GENERAL FORM OF THE EQUATION OF AN ELLIPSEAx 2 Cy 2 Dx Ey F 0 , where A 0 and C 0 and A and C have the samesigns.STANDARD FORM OF THE EQUATION OF A HYPERBOLA( x h) 2 ( y k ) 2( y k ) 2 ( x h) 2Form 1:orForm2: 11a2a2b2b2where a 2 b 2 c 2EQUATIONS OF THE ASYMPTOTES OF A HYPERBOLAbaForm 1: y k ( x h) or Form 2: y k ( x h)abRECTANGULAR HYPERBOLAxy c where c is a nonzero constantGENERAL FORM OF THE EQUATION OF A HYPERBOLAAx 2 Cy 2 Dx Ey F 0 where A 0, C 0 and A and C have differentsigns.
CHAPTER 10 CONTINUEDSTANDARD FORM OF THE EQUATION OF A PARABOLA( y k ) 2 4 p( x h) or ( x h) 2 4 p( y k )GENERAL FORM OF THE EQUATION OF A PARABOLAy 2 Dx Ey F 0 when the directrix is parallel to the y-axisx 2 Dx Ey F 0 when the directrix is parallel to the x-axisGENERAL EQUATION FOR CONIC SECTIONSAx 2 Bxy Cy 2 Dx Ey F 0 where A, B and C are not all zeroCHAPTER 11EXPONENTIAL AND LOGARITHMIC FUNCTIONSDEFINITION OF b1n1nb n bRATIONAL EXPONENTSmb n n b m (n b ) mNEGATIVE EXPONENTS1b n nbEXPONENTIAL GROWTH OR DECAYN N 0 (1 r ) tCOMPOUND INTEREST r A(t ) P 1 n IN TERMS OF eN N 0 e ktCONTINUOUS COMPOUNDED INTERESTntLOGARITHMIC FUNCTIONSy log b x b y xA(t ) Pe rt
CHAPTER 11 CONTINUEDLOGARITHMIC PROPERTIES (work for both log and ln)log b mn log b m log b nm log b m log b nnlog b m p p log b mlog blog b m log b n m nCHANGE OF BASE FORMULAlog b nlog a n log b aCHAPTER 12SEQUENCES AND SERIESTHE nth TERM OF AN ARITHMETIC SEQUENCEa n a1 (n 1)dSUM OF A FINITE ARITHMETIC SERIESnS n (a1 a n )2THE nth TERM OF A GEOMETRIC SEQUENCEa n a1 r n 1SUM OF A FINITE GEOMETRIC SERIESa a1 r nSn 11 rTHEOREMS FOR LIMITSLIMIT OF A SUMlim (an bn ) lim a n lim bnlim (an bn ) lim a n lim bnn LIMIT OF A DIFFERENCEn LIMIT OF A PRODUCTlim an nn n n n bn lim a n lim bnn n
CHAPTER 12 CONTINUEDLIMIT OF A QUOTIENTLIMIT OF A CONSTANTan limn bnlim cn nlim alim bn n nn, wherelimbn n 0 c , where cn c for each nSUM OF AN INFINITE GEOMETRIC SERIESaS 1 , when r 11 rn FACTORIALn! n(n 1)(n 2)(n 3) 3 2 1BINOMIAL THEOREMn r 0nC r x n r y r n C 0 x n y 0 n C1 x n 1 y 1 n C 2 x n 2 y 2 n C n 1 x1 y n 1 n C n x 0 y nEXPONENTIAL SERIES xnx 2 x3 x4 x5ex 1 x 2! 3! 4! 5!n 0 n!TRIGONOMETRIC SERIES ( 1) n x 2 nx 2 x 4 x 6 x8cos x 1 (2n)!2! 4! 6! 8!n 0( 1) n x 2 n 1x3 x5 x7 x9 x 3! 5! 7! 9!n 0 ( 2n 1)! sin x EULER’S FORMULAe i cos i sin
DISTANCE FORMULA FOR TWO POINTS 2 2 1 2 d (x 2 x 1) (y y) MIDPOINT OF A LINE SEGMENT ) 2, 2 (1 2 x x y y t STANDARD FORM OF THE EQUATION OF A CIRCLE (x h)2 (y k)2 r2 GENERAL FORM OF THE EQUATION OF A CIRCLE x2 y2 Dx Ey F 0, where D, E and F are constants STANDARD FORM OF THE EQUATION OF AN ELLIPSE 1 ( ) ( ) 2 2 2 2 b y k a x h or 1 ( ) ( ) 2 2 2 2 b x h a y k where
