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# Obj 11c-14a MAC1105

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## Popular in Calculus and Pre Calculus

This 10 page Class Notes was uploaded by Michaela Maynard on Saturday October 17, 2015. The Class Notes belongs to MAC1105 at Florida State University taught by Pennington LeNoir in Summer 2015. Since its upload, it has received 73 views. For similar materials see College Algebra in Calculus and Pre Calculus at Florida State University.

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Date Created: 10/17/15

Obj 11c Average Rate of Change O 9 5 The average rate of change 011 between c and a is the of the secant line between c fE and 51 fz r 3 Cant me Comn66t5 a Average Rate of Change 2 1 two 370mb 0 L C a h MCMofVLC a 9 Formum Ty WRL DW The average rate of change the slope of the graph at x c In calculus the derivative gives the slope of the graph at a c which is the slope of the tangent line at a z c Plan of Attack Set up the two ordered pairs 0 f and x f ll in the values then just nd the slope Always REDUCES A1 Obj 110 Example Find the average rate of change of f between 3 and a for f as 3a2 2x 5 o 1803 3 3333 3 5 R39I W5 33 5 33 W 9 239va Rate Bl Gwaan 003ch 0 0 3239 2 5j 6 C 30quot in 5quot 3933 3x 2a f j 5 an 33 quot x3 X 3 x3 2 33 Bic 0643 31o e 139 7 v 3 2 Obj llc Example Find the average rate of change of f between 4 and 1 for 04 if t 433 HI aggt 5 quot39 quot quot39 202 23 Z39 2 4 ivy Rafe M Chan b 1 T A p K gt 3 b C L quota 2K quotquot2 L K 39l l 3042 f 39 d 2W zq 3n39l M 39 Ubj 11c Example Find the average rate of change of f between 1 and a for f I2 2x R r 2 1 03 26 M 5 54 Jc 2x a a 3 a 3 L32K3 Hv Katt 0Q O M V T 66Q Qquot 47 K 3 F f K H L C KZ R U Q Viv 3 nl m quot Obj 11c Example Find the average rate of chan e of f between 4 and a for f 13 21 mpg Hra we2 no 80 09 Oman 39 H0462 94 21 a 1 at a all 104 1 0 339EI N aaw M l L Ll m w 464 v i quot aw JV 27 L gt Mule A C howe thCC39irmc rationalize numeracowr k WW Obj llc Example Find the average rate of change of f between 5 and 2 for fa 2 2J5 The square root function examples will quiz only not test 5 86 am to W M c Me It to Avg Ratc olquot Chanj 3 4Q 11 a qgg an NS 3 qu 30 40425 L C 71 quot5 EVEng I 6 53 3 539 r Do Noe mutt ow MGM 25 q 2 ML jarfrail 1 Z 17445 Checc Obj 11c Example Find the average rate of change of f etween 3 and 1 for f at xx 1 The square root function examples will quiz only not test 93 WW a 35 mm Irma Rate 0 Cmngu 1900190 27 2 WET r22 K C 39 L g mi9 7 10quot 39Ll M Do 40 malt ouel ZD f 3gt W 39 W Mult Chance Obj 11d Slope of the Secant Line This is the basis of the de nition of the derivative in calculus The Ob 11d problems gill quiz only not test 5 9 LA 7quot Find the slope of the secant line between 1 f x5 and a h f a h for fc x2 21 3 3 m 3 r w h My mm 4 n quot NW0 MD 3 W3 3 x h so h 242 3 KRmhhaah fK w m 26 1 n h snowman s MC3 Rgt a h9L crfs h K x No Cayman Hamilquot Find the slope of the secant line between 13 fx and a h f w h for fa a2 3 New 39uvb za x 11 73 km a 3xh39na393 3 39 war so V 3 a 3 WC jggzmm 33 2 LaRlbh Ma 3 m x I A uhx RW V BW h dbl 7 T T mr h thamp irhh 93 Obj 119 Symmetry Properties for Functions 3 5 3 Qio Er 1 y M 0 VODMCt I n 1 KI L x IL x39 w ODD No rail 15 Graph is symmetric Graph is symmetric 38 mm c Pbr with respect to l39 0 5 with respect to Q 5l Qu nchov flt agt 19L fxgt LEE Plan of Attack To test for Symmetry nd f 50 If f x then function is m If f 513 f then function is Q If f v 2W then function is Mix CVCn of Odd Obj 1 le Even Odd Neither fxx4 3cc25 fxx7 75m53a 5 39 1560 5 Oq3C z7 r5 Pquot quotquot7 73971quot 36quot 39 0quot 3 6qu2 5 2quot 5x5quot3 75H ev n s M 7 Invcoww mhan Qwe 39OM fZ l L74 5b53p gt quotFCQ fxx5 5x2 fc 2x22i3 3 cm 405 5916 w am 31 Ma a t wag 3 026074 2V3 not CVC n 3 we 1w w L rcaraf Mt 1 00 nob We wooing odd 39 9 quot0396 Udd 55k fx35 fa14 3 L nzi 1 t amp zx not 73 we 3 X quot x 21quot amp2q quot even o hO39t QVCY AA a I 39 Nanak ed ltxgt xq NW quot F Z gt 23 Tiquot 9quot 04 odd Obj 12 Collection of 7 basic functions Plan of Attack You DO NOT need to memorize the properties Learn the GRAPH and the FUNCTION rule You should be able to answer any question if you know the GRAPH 1 Special case of a linear function f as 10 y 9160400 2 30 4 0 odd 9on055mm wrt Ofl lh thYEOShij on 39wlw 2 Squaring function f 513 932 wquot D OOIODgt 0 w aCvx 53mm wrt 9 4quot 5 DECICOSn3 IMHO Manama 01 00 3 Cubing function f x3 9 4 91 00 R of Odo ngm Wt 006er intvCaSnrx on 3900 0 AK RV gt39Hr Z quotl o l 4 o x t4 3 395 7 3 0 o J L a 7 M eve v63 4 Square Root function f cc NO 38mm0 am 3 nexmev odd nor even N on 000 020 065 f 5 Cube Root function f x Nab 175 wew odd lune R 4 3 mm m 09V nC On oo 0gt 6 Absolute value function f as a l 39039 quot 5 RI 0 co Even PLAN 3de W f o y aw w ok DEC 0 MIO C on 7 Reciprocal function f 4 K 0 91zw0uo 0gt a wow wow 7 tuft ng decr 000 MO b AR Obj 12 example Enter the equation of the function whose graph is shown Enter both sides of the equation not just the right hand side You must use 3 on the left side of the equation f a is not owed in this 103639 V sMust MCMOI IZC Fume quot Mora for thwi arr 2 W WCs on came Your Answer c 9 l l Dcm It 90 EaventhCMS Ob j 12 example Select all of the following that are symmetric with respect to the yaxis There may be more than one 311 a 3PM x3 PM 3 fx 35 ms 5 lt35 Obj 12 example 7 5 Select all of the following that are true for the function f x ac There may be more than one ltI Domain is 0 00 Domain is 00 O U 000 Range is 00 00 Range is 00 0 U 0 oo raph is symmetric with respect to theyg Graph is symmetric with respect to the origin 763 Obj 13 Functions de ned piecewise A function de ned piecewise has a different function rule for different parts of the domain Obj 13a Evaluate a functions de ned piecewise Plan of Attack Compare the xvalue with the restrictions on the right that tells you which function rule to use DO NOT plug as into each A7 x2 ifxlt3 fc 112 if3ltx5 O ifasgt5 Find each of the following f1 l r R f301 095 90mm f5001 O 1 if x S 1 fx 0 if 1ltalt1 1 if a 21 Find each of the following f 1 x a q H933 1 0 f3 3 5 f35 3 5 R 15 mm O f05 o Obj 13b Graph a functions de ned piecewise A warmup question Graphfxc 2ifx2 1 A g quot74 x T a 1 ng Value hot 0 Pm Pfr AunChon f5 53 Graph faa Qifagt1 II 4y K 2 AR quot l 0 01 Rf Graph f52x ifxlt3 K 7 O 5 1 x43 th 95 Ax 3x 1 ifxlt 1 Graphfx 4 ifxgt 1 Ay Ma a r X H CWCIQ ovt OPB 6 U 4 own male 4391 3 b Ix 0 39 lLl x4quot twn 6 i man 8 3X 5 3 4 4 39 ifxgt0 3 Graph m it i x u i 3Q twat ovc a 2 4 f lt0 v60 oven cwue x X 0 LI 0 R H x 20 I k 40 lme MWEVCW GOUNd I 9 5 th j 4 a 44quot behawm Hum 3 5 AG 333 5 ifxlt2 Graphfm 2 ifxgt2 2 R 393 b H 399 l gt643 M Li 46va 33Lv5 U g Obj 14 Graphing with Re ections Compressions Stretching Translations Obj 14a Graphing with Re ections How do these compare y fx y fx 3 fv We will consider a speci c example to justify the general case y5 czo yE 530 y x LZO 1c f O y across y Krams Aycztxoss i 030 grams g R1 0001 1 j x I x 5 D39 00 O y y R 0 w 011000 X IZ fl 6 o 0 Rquot 0 w 0 4 x l H 3 2 O O quotI l q a m

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