Lecture 2 Notes (MATH M-119; Brief Survey of Calculus I, Staff)

# Lecture 2 Notes (MATH M-119; Brief Survey of Calculus I, Staff)

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Unformatted text preview: M119 Notes, Lecture 2 1.1 Functions Math functions a lot like other languages. We have constants (numbers) and variabies that work like nouns and that ﬁmctions work like verbs. Just like in language, this notation can sometimes be ambiguous and we need to be careful and sometimes get the meaning from the content. In language, we often see words act in both ways: “Start your engines”werb, “Line up at the start ” — noun- In math, we might see: Mr) : x2 where h acts as a function, while we also might see the area of a triangle written as Pugh) = A where h is the variable height and the parentheses represent multiplication. What letters are typically used for variables? _‘f_j“_ C‘ ‘_3_‘_ ' _ _ i k‘) i 4. I; M} II. 3.1-.» XI- ,“ - "I J I I r I. \K '7" I. l. '- i— i ul‘r’l'i What letters are typically used for functions? {7 _ I in. ‘3 The minus sign (—) can be even more confusing. Even though it always means opposite or inverse, there are lots of inverses in mathematics. In addition to meaning subtraction and negative, we also have: _. 1 x =— x l f (x) BUT: f 'l (x) at in general (this is the notation for the inverse function) Here is a list of commonly used ﬁmctions and their inverses. Multiplication 3 _ 2 \d >< a l w h.) U3 “ergHwaikﬁ [\J H'E‘t: m 23% H O p: O M ._t 0 GO “1' I a i ‘s: Note that we normally notate y as the output of ﬁll) so that y = f (x) , but we can just as easily write x = f ( y) as x, y arejust variable names. 0/. .- - I.. . ' I.-‘ , _ . Ex. Proﬁt, H, in millions of dollars, is a ﬁmction of q, the quantity of items sold in thousands of items. What does H(25) = 100 mean? Lamp: 't‘r'Lrt‘ -"' 2 Ex. Let y=f(x)=x—+7 3 A) What is the value of y whenxis zero? 3 " 'L " . II B) What is f(3)? to —_ 3, s, f H I f .f 3 C) What values of x, if any, give y a value of 34?, D) What values of x, if any, give y a value of 5‘? 1: " - 5: 7%! ’} -:- j ‘I‘F-Jc "“-. . (Do Qaufﬁ m J 7" "r\'.:'|i;.‘ll L'I".-.:"I'I In I |-_u..n-_- s. 4 hr-ﬁ Illil'lL'il'vi“ am“ all iila.‘ 1": Inna-int- '—.|-.-'-|LI."‘"- VIE: sly rah-‘4'} [hr [in :ctmnnmrr ;.',1'.L;1'I1. MI- I had in»? La.::.‘. unim- nth-:14 ! r‘cnII..u-a.l I "Lu: Z'.I.1‘-_".‘l*:“.‘l :m “Hm-Iva. LIN-.1 1-.“ | avg-iv Im-cL. It: '9‘”le |l‘l~."lH up. (h: T'In:|_-__r-- m cm 111w man] I in»; .l 11ml mu. |{"l l xmru'd win mlmiy Hu'. «qua; ILU mimi I regimen-.1 ] an.“ gum; In E'Ir.‘ Lua- . ... -...II|—‘ Figure 1.4 x—N ‘1 "9% Ex. For the functions listed below, ﬁnd f(5) and f "1 (S) r 5.. 1)f(x)=3x“4 - \5 A = \\ 375” Fm 5-3341"; :3 QCs‘Jcaté‘J'W ‘ f— 7 ﬁnial-1% .1: 1.; “Mix; 1.2 Linear Functions How does slope of a line affect the graph? A»; ‘ ‘Erwl How does the intercept of a line affect the graph. Mwé V': "f GTaph: 6x — 2y = 10.? What is the slope?';5 What is the intercept? = *‘5 ___2\\I{_1; N .- w -- t? , .2 , _:__- V 5 " .«o f, i I. What slope does a horizontal line have? What is the equation for a horizontal line? A, a aciwhﬂeo i: What slope does a vertical line have? What is the equation for a vertical line? \F Iii-pig“ X 5 (:7 _- '-." \ Find an equation for the line going through (2, 4) and (4, 8). Double check by graphing. - " zc _ ‘ Line forms: Slope, intercept y = mx + b Slope, point (y — yl) = m(x — a) 2 points Find the slope and then either form Ex. (12) A cell phone company charges a monthly fee of \$25 plus 5 cents per minute. Find a formula for the monthly charge, C, in dollars, as a ﬁmction of the number of <. H) minutes, In, the phone is used each month. C A rm ti an“: dint. ~'—‘-'/I - v r~ r —’r ‘ . . ' ' I. ..j -j' =' ' C; 0511/] 25 may we} C0145) I; 5 C 2.5.053 <- :1 [I ‘ Fét‘ﬂ‘} ...,. ’l’ "l 5'71”: j /‘ (11/. 2 Ex. Which of the following tables could represent a linear ction? : .93 t A) 2A 2/4. 2/1 27, Z . M ma 9 1 2 3 4 5 (l'ﬂg'/ q 3 5 7 9 11 (71 “L B) P 1 2 3 i’ 4 C1 5 13‘1" LIAM/HF" q 1 7145/! 9" 16 [Has/U C) p 2 4 6 a 9 5 12 M1“ Uﬂfiﬁﬁz q 5 5/2, 10 5/95 20 If; 25 D) w p 1 3 2 Q, 4 3% 5 6 KIM/gm q 4 A 7 13 ' 16 ' 19 K‘ For those functions that are linear, ﬁnd a formula for: A) q as a linear function of g P B) p as a linear function of ﬁg {H1- ? r , 2 ...
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## This note was uploaded on 12/21/2011 for the course BUS 100 taught by Professor Intro during the Fall '11 term at UCSB.

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Lecture 2 Notes (MATH M-119; Brief Survey of Calculus I, Staff)

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