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# ALGEBRA & TRIG FOR CALCULUS MA 110

UK

GPA 3.54

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This 7 page Class Notes was uploaded by Kennith Herman on Friday October 23, 2015. The Class Notes belongs to MA 110 at University of Kentucky taught by Staff in Fall. Since its upload, it has received 14 views. For similar materials see /class/228155/ma-110-university-of-kentucky in Mathematics (M) at University of Kentucky.

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Date Created: 10/23/15

1 Formulas to remember Complex numbers Let 2 z W be a complex number 1 E0 00 H F 9 4 The direction from 0 for a non zero 2 The conjugate E z 7 z y 7 7 J The real part Rez 7 z 7 Q The imaginary part 7712 y 2i The norm V23 dz yz m7iy 1 7 The rec1procal i muff Zl The Euler representation 2 rexpz 6 where r and t9 Argz arotanyz The angle 6 can be replaced by any angle in the set argz Argz 2n7rln E Finite series formulas Arithmetic series n2a n71d aadan71dZaz71d 2 l 1 Geometric series formula aararn71Zari 1alir 1 177 Telescoping series formula lf g f 7 fz 7 17 then Sums of powers a H F 9 Power series Geometric series Zaziaazax2 0 Exponential series n1 7n 2 n 12n 1 n 27 212 7 6 n2n12 393 4 24 7 nn 12n 13n2 3n 71 1 7 30 39 a 17 where lt1 00 2 expxin1z 2 for allregk 0 Basic Trigonometric series sinx ZO 5 r 2i1 3 5 z z 7forallzegt 4 00 i 2 2 cosx071 W17 17 forallzegk Generalized Binomial series n l nn1n 1i where lt 1 or n is a non negative integer and z is any complex number 3 5 Log series 2 3 n1 log1x x73 71 n1 wherez isrealwith lt1 6 Arctan series arctanz z 7 3 j 7 n zzn 3 2n 1 where z is real with lt 1 7 Arcsine series l m mowgm arosin z g3 24 5 246 7 where z is real with lt 1 Geometry H The determinant of a 2 gtlt 2 matrix a b c d adibc E0 Cramer7s Rule To solve azbyeandczdyf de ne AiabidbA76bid dAiaei i C d 7a 70 17 f d 76 ifcan y C f iafiec lfAy Othen the answer is We g If A 0 and at least one of Am Ay is non zero then there is no solution If all A A1 Ay are zero7 then there are in nitely many solutions provided each of the two equations has at least one solution 3 The distance between two points A and B on the real line is dA7 B lA 7 Bl 7 7 U 03 0051 3 For two points P1a17b1 and 132127 b2 in the xy plane The distance between the points is dltP7 12 7 112 32 7 b12 The parametric two point form of the line containing the points 131117 b1 and P2127 2 is i 11 b12 7 117y 1 7 The compact parametric two point form of the line is 9671 P1 tP2 7 P1 or Ly 17tP1 tPZ The midpoint of two points AB is A4213 The two point form of the equation through the points P1117b17 132127 b2 is 12 7 a1y 7 b2 7 b1x 12b1 7 a1b2 The slope of the line through two points 131117 b17 P2a27b2 is 52751 02701 where the slope is in nite and the line is vertical if 11 12 The slope intercept form of the line joining P1117b17 132127 b2 is y mx c where b 7 b m g is the slope and 12 7 11 a b 7 a b C M is the y intercept 12 7 11 lf 0 31 p is the x intercept and 0 31 q is the y intercept7 then the line is 10 q A line parallel to ax by c is ax by k for some k A line perpendicular to ax by c is bx 7 ay k for some k The quadratic function Qx axz bx c has extremum value at x 7271 The value is if It is a maximum if a lt 0 and minimum if a gt 0 The equation axz bx c 0 with a 31 0 has solutions 7bixb2 7406 Quadratic Formula 2a 16 17 The equation of a circle with center 17 k and radius r is z 7 h y 7 k For the circle 2y2usvyw the center is 71127 7112 and radius is iw A parametric form of the circle 2 y2 r2 is 2 2m T1mzgt The trigonometry parameterization for the same circle is z rexpz 6 or x7 y 7quot cos0 7 r sin6 17m 1mz7 MI 7 where m is the parameter with parameter 0 A circle with diameter 1311171917 P2a27b2 is 95 i 0195 i 12 y 51W 52 0 The distance from a point 1071 to the line ax by c 0 is lap bq cl The expression up bq c 0 if and only if the point lies on the line Moreover7 the sign of the expression up bq 0 changes if the point moves from one side of the line to the other Trigonometry ma 00Msecgcsc tanz i may tlt Sinwy cosz7 SinW sin2z cos2 17sec2z 1 tan2x7csc2z 1 cot2 sinx y sinz cosy cosx siny cosz y cosx cosy 7 sinz siny 5 tan tany t Ema y 1 7 tan tany 6 cos7x coszsin7 7sinx 7 cos2x 00322 7 sin2z 2 0032z 7 1 1 7 2 sin2x 8 sin2 2 sinx cosx 9 1 mi i m 2 2 sin i 1 7 COSW 2 2 The signs need to be xed by the position of the locator point 10 Sine Law s1nA s1nB s1nC a b c 11 Cosine Law a2 b2 02 7 2bc cosA 6 Derivative Formulas 1 If c is a constant7 then DAG 0 Moreover7 Dwcy cDmy 2 Sum Rule Dmy 2 Dzy Dz2 3 Power Rule For any real n7 Dzyn nynilDAyl As a special case Dmz nznil r U 03 I 00 Product Rule Quotient Rule Chain Rule Linear approximation formula Linear approximation to fx at z a is M96 fa WWW 0 Newton7s Method for nding a root 0 Start with some convenient guess 0 This is the current guess o If xn is the current guess then the new current guess shall be zn1 zn 7 f n 0 Stop when successive current guesses are close to each other within the set accuracy level or when the value of the function is suf ciently close to zero within the set accuracy

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