Week 2

# Week 2 - 1 I Ti “1'2(3)‘2.L—t l l l.1 r E l[10 con...

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Unformatted text preview: 1 I Ti “1 '2.(3)‘2__.L—t l l l .1 r: E l \[10] con Average Rate of Change: I Finalin 5‘6??? all) 0" “we Definition: The average rate of change of a function f[x] on [a , b] is the Slope of the line . and (b, f(b)]. If y = f[x], the average rate of change on [a, biggin m) —- fCa) " Ex: The growth rate of a fetus in the mother’s womb [b weight in grams] is modeled by the graph shown below beginning with the 25th week of gestation. a) Calculate the average rate of change [slope of the secant line) between th_e_25£1_week {D and the 29th week. 15 the slope of the secant line positive or negative? Discuss what . . .i (X‘Nme slope means In this context. b ,Is the 7_ - . g a ' ing weight faster between the 25th and 29th week, or between the - 32nd and 36th wee "1 Compare the slopes of both secant 'r(res and discuss. 3900 e .1":- r , _ /—-* .8200 7. lo) '32, moo) \$159 267-0 3M _. a He), ‘ 0‘ , Hm) [9r ) 2,1600) a) (x‘}~j‘):(23)6wo) _ magma“) Aveer tickle (x 24m ; (m H of? 7..“ Qqﬂoo); 1 V j I. 1 wowgiﬂ)ﬂloooﬁve;m¢je (3307 \ £34?" (25,400) I 39’39 "75712906 (-27“. gm _ a I X2 'X\ b “OK. 24 1;, 283082311» 39 38‘ an 92 I; “oﬂ A33 [We-EMS) ““Z'Ei,2g Ex: Find the average rate of change of the function got) = 2x2 —4 on [—1, 3] tn) 2- we WM 3;? E i. i! as *2] W7 ’W‘f " ' ' , D ""0 m met A 6"mle/ ﬁo‘lg Cwe gig.) 6912‘) ‘9 Cl 1. —— Ll —— —2 m 2 i ‘ a _ end points. EX) x _Hq1 'FormulaLDiffere e Quotient of function Hbi "F (a) ' f(x+ h)- f(x) = f(x + 10-. f(x) fOU=W h Graph: b M Ck g3? Definition of Difference Quotient: Average rate of change over an interval with variable “giving EX: f(x) 2 2x2 —4 Observation between the average rate of change and the difference quotient on the same / function: ‘ a 2.. a ‘ Porgy { 0 too if? e a; q X'h +35“- ‘5. 19(1) -: axlku “ cm E13] h _ - \;( ‘ AVWM)? ﬁCJ/towge =4 ' C Hi1be Let’s generalize it to a common rule: y = f[X] any function C- ‘3 Ccmoi-"(m'i‘ . Equation Effect Equation Effect + «a. 3/ ‘Q'? Ex: Investigate the graph ofthe function: y=—(;vc—3)2 1 + _ - “t ’t-st‘H 3 W5 "'" _. Rag-eat accwss 79m: 2,1 199,11: accws§ Fj‘éUdS 95cm W 11‘ ' Methods of Combining Functions: Iterations (3.5) w u Let’s consider two functions: f(x) = 2x —1, g(x) = x2 — 3x + 8 Adding: Jr. v; EVLX+7 ,- , ' l 72 I , : v Subtracting: h" f < Y W 3 X I t: 1;:x;;3e2ii : 'Vﬁibﬂ g} Multiplying: amt "1; + g) 5 1,1 5” (Balm , \U 7‘ ~;@8 it'll-“W (30“) M'/ 9:21_><5«--~jr><2*+l‘l‘7<--8 39(1) ': 1% ~- g ’ SON ":- 17:- 3x+ ‘57 comprising: 0 3X1) : £08613) 7; 763.5% +8) 5 a y€3x+e)—W e * _ (MW) ‘2» Cﬂﬁil) 7: (Up 1) ~_—.7 (zx—-.;)Q;3(m~-u)+% Domain of f+g, f-g, and [f] [g] =set of all inputs x belonging to both the domain offand the mum.7: :se mafiewaogie Domain of I- = set of all inputs x belonging to both fan g exce t those values of x which _ make g(x) = O ‘ ' ﬂy M,” “(5% 51mg} big/Of,va . f am mam Domaln of f o g = set ofall those inputs X in the domain of g) for hiC g(x] islin the ' 7 domain of f(x) I L Ex: Let f(x) = x2 1— 4 , g(x) = 1/; N ' “A H” fo‘mmﬁm Find the domain of the following expressions: it“ i N I H v ‘i , J? 03\$ it ‘ T \ ‘ n ‘2 I )\$-;?) :E I“ "Dome/m {(90 1 M x- u] o (Kt-1W ~23 2‘ O :7 1} X: ﬁbril 0L6 U2) 'X43‘l,'xt*?—j ‘7" Dex/main KOO : x 20 ' Ema) ' -' ‘ ‘Dommim c))'¥"%ﬂj g’f) E0;_‘7—\)U (2 1%) _ - I I ‘ Wm: Eye—nu?)ng wag [OEMI)U(_V|)°O) (0509131. %x')=r—1x2,%igi§:ik:= .Qﬂomain off[x): 4M¥ZZQ Y1— L} 7 O r:— ‘ _ » f . (K'Jr-zmc—n l i g; m e ._ 4++++ “My, - . Now find the domain of the following expressions: ll ' \$0»an Pk ' 11% 5-45: 1—52,: —— 9 ,_ - l #5 E- >,\ 3 "A ".3 I App ication Problem: Suppose that a manufacturer knows that the daily production cost to build x bicycles is given by the function C, C(90=100+90x—x2,- 05 x 340 Furthermore, suppose that the numberﬂilf bicycles that can be built in t hours is' given by f, x: OStSS a‘ Find(CEl’f)(r)= CGC[t)\ 3, C(5t\:100t‘10(5tlf~(59 b) Find the production cost- on a day that the factory operates for t = 3 hours - .u—r" '2... Ola} :1 \oo +LRSvll33-'- 13(3) my, c] If the factory runs for 6 hours instead of 3 hours, is the cost twice as much? ‘ QM ‘ CM 7.; \oo +—C%903llo\ higf‘gl‘ Cm "ENDED 44/700 “0‘00; ’ No, Cost is wot «bate; Ex: Express F(x) 2 §\/3x + 4 as a composition oftwo functions _ z :213’3U-‘i 00W) 223%”! -W€C/’“ = Hm :2? mm): H22) BMW Ex: Letg(x)='\/;—3, f(x)=x—1\f:mx+'b WEJ a] Sketch a graph of g[x). Specify the domain and range ‘0 1: - . 7 {fog [DiQM‘Lfv‘i [ED/loo) PT ' ' Raj/«1376, I [f5 I w) b) Sketch the g phof f(X). ecify the domain and range. - ‘ "Domm‘mi (“40;”) r (“00100) c) Find (f o g)(x). Graph and specify domain and range of (f 93000. _ (1303).“) swam) 2H6? a) 3-(Rw3)'\ =1 =>' q ,* I :ﬁvi D: E): w) R : 9,003 Inverse Functions (3.6) This new table represents the graph of f-1(x) i.e. the inverse of f[x]. Graphically: ‘ gawk 0W acmss “EM [\$.49 (ff: X Algebraically: If you are given f(x) = —3x + 2, \d‘ 2 up 5 2" Step 1: K ‘8 VNM‘MrﬂQ/g Definition: Two functions f and g are inverses of each -’ i PogXXEﬂgﬁDq for each x in the domain ofg i» _ »_ '7 ; g(f[x)) =X for each x in the dean of . . ‘- \0/ H110“ _ ' '5 Example: Let f(x):(x—3)3—1 (a 3; (X... )__. ] ‘ 3 a] Find f“1(x) 7 Switun xg‘g X: (‘6-3 --\ I SD\M 15w VJ ' "H * w ‘X-H r.- (‘33) Wmeér boﬂm‘deg. M .2 2V3 ' _ +3 +3 3 XH +3 :y =f“(¥) b] Calculate f*1(7) and -f(—7) . Note the difference conceptually. 4(7):: ﬂ +3 :i‘gq +3 :: 2+3’7‘5 _'.ﬁZed6>nm:} ___ | - f. A” Z. J... .-. W 1: ~——*——* -—— _. new i7 3%) (it—“3):! 43-4 ‘0 ﬁx) a. (Jr-3311 c] On the same set of axes graph f[x] and f». 1( x) ivmgmetry ab t the y=x line. 2 ) X3: 3% t , Hana-an \$th 3- MB my | lwl- v "Mum... Question: When does a function have or not have an inverse? Ansmn A too have om ire/arse, {g owl 1m {‘1} it "is one 40 owe- Slmllm’luk)0\ g twelve Sax) does ml Vim/Q 0M lmVe/rSQ iﬁ Wis an not one +0 one. ' Definition: A function f(x) is one to one if f(a) = f[b] then a = b for all a, b in the domain of f[X] Graphically a function is one to one if and only if each horizontal line intersects the graph of 2 ﬁx) in at most one point. .Sifﬂr'l‘ ‘ i /\ Examples: i? Theorem: Arfun tion f(x) has an inverse if and only if fix! is 92g to one. A ’FUMUL‘JA "To 3km) Not one ‘lvm i A We +73 ‘ is Omelnm ,- Alﬁe/bowl ; WC,th Mack/MEN. 800 :— 'XL owx . {VtVQ/(sf. 30Aé§££ tin, ' WWW“.-.- ‘ROO: Ollie 9— b or “Hm-stirs) 0\ L" -_Mo9r OWE lam ail—"1‘0 Example: Let f(x) = Vx+1 —-2 a] Find the domain off[x) Ab“ :DOMAM : 7<+[ 20- I ' .c) Find f—1[x) if it egists/ Y": Ma .-?— 803w xév ' SCAN "GM Y ‘ . - e) Graph ﬁx) on the same Set of axes 3: m 2 Chapter 4: Linear Functions (4.1] Definition: ' unction (always degree 1) is defined by an equation of the form where A and B are constants is called the Standard form. For a Line passing thrgfigh [x50 and [X2,y2); Siope= m z: 82 ~— a ’ 1) Slope-intercept form of Line: I \ UT—m'x-l—b, ."YYl: Shire} 30» "m 2] Point-slope Formi .. - S) I I ' ~— - —-. m 7“? 7 “m: 0?? Lﬁ ta? _ ( ’) (XIJ913:G>DIT6( W . vow/1% “UM Um I 3] Standard Form: . \ri . “We - :2. not + 733 9 A? f) «we, msfangg ngliﬁi}: (Am) - ' If f[3) = 2 and f[-3) = -4, find the linear function defining ffx) in the slope- intercept form. ‘3: ' l < I. (Kw): C3) 51-}- in: Share: ‘82—‘61 3""17‘ “52 :; (“axing I. I = '. - __._.._L fixed cost (the cost before starting production) 0M9, mean iwma . Ex: Letix denote a temperature on the Celsius scale, and let y denote the ‘ '- Q corresponding temperature on the Fahrenheit scale. Y a) Find the linear function relating X and y; use the facts that 3201: corresponds I to 0°C and 212°F corresponds to 1000C. Write the function in standard form. <53} “’5; (0)31] “m: V5145 I; ﬁrm—2 "5?" ._ ' 7c ex 100 ——'o -CX2.)YZ)"—<lo)9—er9—> m: l : l‘g term : mob . ‘lOO . (ﬁgsa: “(3000) :7 6—322ldgK _ I _ b) What Celsius temperature corresponds to 98.60F? F + qégan-Tr \agx +3L rec :- eeo Sgg Tn swam «Cam c) Find a number 2 for which 20F=z°C.- - I "—u a :— F Z iaaaw‘F 1&2 ._—— - b r— F: lasQ +32. “VF—:02 ...
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## This note was uploaded on 02/08/2011 for the course MATH 3 taught by Professor Staff during the Winter '08 term at UCSC.

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Week 2 - 1 I Ti “1'2(3)‘2.L—t l l l.1 r E l[10 con...

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