MAT 17A Week 1 of Notes
MAT 17A Week 1 of Notes MAT 17A
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This 8 page Class Notes was uploaded by CarolineTylerSmith on Friday October 2, 2015. The Class Notes belongs to MAT 17A at University of California - Davis taught by A. Dad-Del in Fall 2015. Since its upload, it has received 267 views. For similar materials see Calculus for Biosci in Math at University of California - Davis.
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Date Created: 10/02/15
MAT 17A Week 1 Hi my name is Tyler Smith and I am in MAT 17A with Dr Daddel I will be taking notes during his lectures I will be uploading each week s notes on Thurs of Friday for Study Soup These notes will include lecture information with examples From Lectures 925 and 928 and 930 Review of Real s Real s R 00oo Rational s Whole or fractional numbers and O as long as the denominator is not 0 o Integers Whole s and O 32101234 I Natural s N Whole s 1 2 3 4 Irrational s that cannot be simpli ed like 2 or 15 e 7 Venn Diagram Real Numbers consist of all the other categories listed above All natural s are Integers but not all Integers are Natural s a All Integers are Rational s but not all Rational s are Integers etc Imaginary s i is not a real thus it is imaginary To remove a from under the radical you replace it with an i I 43 N3 42i Ii We get the common expression iI from x210 Subtract 1 from both sides x21 Square root both sides x1i Complex s consist of a real part and an imaginary part a b i real part imaginary part I 4 3139 When adding or subtracting complex numbers add subtract the real parts together and add and subtract the imaginary parts together 42i 13i 4 1 2i 3z 3 a and b are s CH 1 and Review Intervals Open parenthesis ab this means altXltb all numbers between a and b excluding a and b themselves 18 l l In W l T W uses open circles a b Closed brackets ab this means aSXSb all numbers between a and b including a and b themselves used closed circles a Half Open ab this means altXSb all numbers between a and b including only b ab this means a Xltb all numbers between a and b including only a T l l I39 l l l T L l l l l 41 b a b a Unboundedoo aoo 00a aoo 00a XZa XSa Xgta Xlta UnionOR ab U cd this means values that are in a within b and are within c and d U or g E It is OR when D 3 Remember this by a b c d X23 GO LA IntersectionAND this means values that belong Greater LESS to both ab and cd where they both have the It is AND when Than 01 than and same X values Xlta Basically if a number is positive in an YZX absolute value sign it remains positive and if a number is negative in an absolute value sign it becomes X if X20 X X if XS 0 Wcause a would yield a a Absolute Value intersection is at 1 and 5 X23 X23 x5 xl y 3 I Solve for X23 9Must separate into the negative and positive possibility I I I I equalities S 2 b 3X4 lt2 remem er to switch the sign I 3X4S 2 3X4Z 2 It is OR When X S 2 AND XZ23 xgt a 28362 Xza Remgrgbelrghls by I 3X4l Z2 gfleater t ess d X 22 OR It is AND when an or an an 2 00 or 00 23 Xlta XS d Distance between points on a number line 2 units D abba I a3 b5 35 53 3 5 22 22 Distance between points in a grave XV Dlane 2 2 Q J XI X2 YIYZ x 21 Equation of a circle comes from the distance 0 x2 12 formula i I u distance from point 12 and 55 d V1 52 252 169 V25 radius Xa2yb2 square both sides Sunits r2Xa2yb2 bt b34ac Factoring and Quadratic Formula 2a When you cannot factor using the Greatest Common Factors Grouping Cubed Roots or any other typical way you can use the Quadratic Formula X2 5X 29 cannot factor in a typical way so must use Quadratic Formula a b c 5i52412 SiV258 5iV33 the factors 21 2 2 Completing the Square X26x92 if you subtracted 2 from both sides and got X26X119that is not easily factorable so instead complete the square9 then like using geometry and the Area of a square side2 you will get a factor that is squared and easy to use I X26x50 X26X 5 take middle X term and divide it by 2 then square it9 62 squared equals 99 this value will be added to both sides to complete the square X26X 9 5 2 x3249 now you can go further to square root both sides and then you would get x3 2 so X 5 and xl Equations of a Line Equation to nd slope rise in y direction over run in the X direction when graphing thus m VIY2X1X2 yintercept form ymxb point slope form yy1mXX1 Standard Form AxByCO 9m this form A must be a whole number and positive I 2 3 1 5 nd the equation of the line slopem 2l35 l2 use slope and one of the points in ymxb 3122b therefore b4 so 1 1 y intercept equation y12 4 S ope slope intercept form y312X29 y12 4 Parallel lines have the same slope m1m2 Perpendicular lines have opposite reciprocal slopes m2lm1 Trigonometry sin9yr csc9lsin9 Slope 12 Slope 2 cos9Xr sec9lcos9 tanGyx sinGcosG cotGltan9 cosGsinG cos29 sin2 91 if you divide everything by sin29 then you will get cot2 Gcsc2 91 if you divide everything by cos29 then you will get tan2 Gsec2 91 12 Special Triangles Unit Circle 5 2 45 32 From radians to degrees replace 7 With 1800 I 2n39degrees921803 120 O From degrees to radians multiple by 751800 and reduce I 4509 radians9 45 075 1800754 Double Angle Formulas can use one to get another easier to memorize sin AB sinAcosBcosAsinB sin AAsin2AsinAcosAcosAsinA 2sinAcosA cosABcosAcosBsinAsinB cosAAcos2AcosAcosAsinAsinAcos2Asin2A 9 sin2A1cos2A so cos2Alcos2A cos2A cos2A 12cos2A1 sin2Al2cos2Al Exponents things to remember r 1 a arxasars arlar any9ayn 0 a1s the base arasars a 1 1 r is the power ats ars arm n am n or a nm LogLn things to remember logo is the one in the calculator most common base lnxlogeX lnABlnAlnB lnAB lnAlnB lnAn nlnA logAB logA log B log AB logA log B log An nlogA ylnX9 ey 722Mquot 9 xey logbxlnXlxb log4 2 42 xl6 J COSX e3x391291n both sides 321lneln2 3Xlln2 xln2l3 Functions is a relation between two sets A and B such that for every element of A there is exactly one element of B function range subset of codomain domain codomain A B 0 L each 1 S 2 B input does to exactly one r D output 2 C 3 Although it is not a oneto A one function each input has only 1 unique output To be able to have an inverse must be a one to one function 5 or graphically you can reverse the direction of EX 1 and still have each input have one output Input C has 2 outputs9NOT ALLOWED Also D and A are inputs with 0 outputs9 NOT ALLOWED I Is a function passes the vertical line test for every X value there is only one y value 15 not one to one does not pass the horizontal line test no inverse I You can also test the inverse by nding x2 s inverse the inVel SC You 1 exchange x and y xy2 find is not a 2 solve for y y x function so there is no inverse NOTE yx top part and yx bottom part are both functions if treated separately Onto Function or Not Rzreal 3 OO OO I x yzx2 x y2X2 eliminates from codomain fX R9R Where the domains and codomains are de ned fX R9 Ooo A9B domame COdomam NOT the same function to be two alike function must have fXfx and have same codomains NOT ONTO FUNCTION all the y IS AN ONTO FUNCTION all values that are possible 00 00 are the y values that are possible not used like all the values of y 0 00 are used Domain values that X can be is a rational function I Find the domain ofyfX yogi 1 gt polynomialpolynomial X210 zero can never be in the denominator so there are some X values X lemust be excluded that must be excluded in the domain Domain 00 1 U 41 U 1900 Zero cannot be in the denominator and there cannot be negative values under the radical sign I Find the domain of yfX Vx11V4 x2 1 x10 9 X2 1 red 2 4X2ZO 9 XS 2 or X229 2 2 blue the domain mUSt fOllOW all 3 conditions gt I I I Domain 1 V3 U V3 2 Composition of Functions like a plane either a nonstop ight or a ight with connecting ights domain ofg fogfgX g00gfX I fltxgtx2 gltxgt1ltx1gt f o gX f gX 9use gX in fX so fgX9 lXl 2 9 lXl 2X2Xl 2XlXl g o fX g fX 9 use fX in gX so gfX 9 lX2l 1X1 Inverse y f391X inverse function f391 s21ltfx g is the inverse of f if and only if g o fX X for all X in domain of f f o g X X for all X in domain g I y 2 10g2y yfX2X f391Xlog2X I y fXaX f391X logaX fo 1 xx f391 x o f x Therefore 1 a X X W X X Even or Odd Functions Even symmetric about the y aXis Odd symmetric about the origin YZX2 9 yZX47 YZX6 YZX yZX37 yZXS fXfX fXfX YZCOSX ysinX cosXCOSX sinXsinX I COSX COSX
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