1.1 Solutions of Linear Equations and Inequalities in One Variable

Transcription

1 P Chapter 1 Linear Equations and Functions 11 Solutions of Linear Equations and Inequalities in One Variable Solve A P + Prt : a) for r A P + Prt b) for P A Ptprt P AIRE Ap P p Ap r or rap A p ( ltrt ) fipch# # tfrt P or P a BreakEven Point: The level(s) of production where Profit 0 In other words, R C Ex: #48, Page 61 Breakeven Dish Systems manufactures satellite systems and has its monthly profit P in dollars related to the number of satellite systems, x, by 4P 81x 29,970 Find the number of systems that Dish Systems needs to produce and sell in order to break even , ) 81 29, ,970 29, systems

2 008 D 77,600 Ex: #56, Page 61 Investment Yields One safe investment pays 10% per year, and a riskier investment pays 18% per year A woman who has $145,600 to invest wants to have an income of $20,000 per year from her investments How much should she invest at each rate? To ffteefooeeenofimixns@ioo0lox ( 145,600 20, , , ,208, good 20,000 26,208 It , 10% : $77,600 18% : 145,600 $68,000

3 12 Functions Independent variable: x Dependent variable: y Function: Each x value gnfdue Vertical Line Test: III <#En * Ex: Is this graph the graph of a function? Why or why not? Y y X x Function Notation Ex: Evaluate, given that f(x) 2x 3 3x and g(x) 3x a) f(3) b) f(0) c) f(t) ( o ) 3 2t3 3 :(3) 3 (3) two 3 ( d) g(12) 3 ( 1 2) 36 gcx ) 3 e) g(x+h) 3 ( xth) 3 +3 h

4 f G) 4 f ( i ) 4 14 f Ct ) 4t f ( xth ) 4 ( xth ) 4 +4 h 2 1 or f (D) 201 f ( xth ) 2 ( xth ) 1 g#2x2t2 gcxth) 2 ( Xthj ' + 2

5 tttaumieent Ex: Given f(x), find a) f(x) x + 3! "#$ %!(") $ (h¹0) f( th) ( xth ) +3 f (D) D + 3 ( X+ht3)x3) h #h+3f*#h b) f(x) 2x 2 4x + 5 f(x+h)z( xthj ' zxhth ' ) 4 4h xhtzh ' f ( * f h ) (2x' ) ( ) #t4 ht2hhxxh##*# 4xhiLK#h(4I2h4) h f ( )

6 (x+h)2 ( xthkxth) x2+ xhtxhtk ' + Zxhth ' ; A + B) 2 AZZAB +132

7 12 Domain and Range Domain: X, Input, set of x 's Range: y, Output, Ex: Find the domain and range of each of the following functions a) y 3x 2 b) y! c) f(x) ) "%* D : R : R R D :X >_ 0 R : y 20 set of all Y 's Ex: Mortgage Payment This table shows the approximate number of years that it will take a couple to pay off a $100,000 mortgage at several different interest rates if they pay $800 per month If r denotes the rate and f(r ) denotes the number of years: r f(r ) FA a) What is f(63) and what does it mean in English? }) 17 at 63% it will take, 17 b) If f(r ) 30, what is r? years to pay f 9 the c) Does 2 * f(26) f(2*26)? 2 i If ( s y 24 / 15 No off mortgage

8 13 Linear Functions The math is review, but the finance is new Slope: me ' i free # y mx + b slope form intercept Ex: Find an equation of the line that passes through (2, 3) and (4, 6) Stepl : Find slope m3fip? E or 6 I m E ( 4, 6) ( 2 3) m 4,3 ) or y y, MG x,) ymxtb slope int Y 3 3f( x 2) (2) + b Y " I n?i t / y

9 Rate of Change: This measure tells you how fast something is changing, and in what direction it is changing Rate of Change ROC +,/ 1$234 56,72,! $ 9: ;::4< Ex: #56 Gross Domestic Product The US gross domestic product (GDP), y, for the years from 2011 and projected to 2021 is given by y 9778 x m where x is the number of years past 1990 and y is in billions of dollars a) What is the slope of the graph of this function? m $9778billions Hear b) Interpret the slope as a rate of change Each year, GDP increases by $9778 billion

10 16 Applications of Functions in Business and Economics Marginal Cost: The cost to produce one more unit at the current level of production Also: Marginal Revenue, Marginal Profit 6 < Ex: A linear cost function is C(x) 2755 x a) What are the slope and Cintercept? m 2755 variable cost C int int y variable 5180 fixed cost b) What is the marginal cost, and what does it mean? µ } more unit fixed Maginaffgpte Supply and Demand Supply: The number of units that a company/market is willing to produce at a given price Demand: The number of units that consumers are willing to buy at a given price Market Equilibrium: A price and quantity where supply demand Surplus: There is a surplus when demand is less than supply (There are units left over) Shortage: There is a shortage when demand is greater than supply (There aren t enough units to go around)

11 Ex: If the demand function for Zbrand phones is p + 2q 100, and the supply function is 35p 20q 350, what happens when p 14? Are there surplus phones, or is there a phone shortage? fprieuqvanktl D: 100 p+2q S : 35p 20g 350 p$l4 D: q 43 S : 3544 ) q7 D: 43 > S :7 Shortage

12 #40 on page , 1340 EpI Demand down End Supply ' sup a Label these as Demand and Supply and then answer these: a) If the price is $30, what quantity is demanded? 20 units b) If the price is $30, what quantity is supplied? 40 units c) Is there a surplus or shortage when the price is $30? By how many units? Surplus Price should of units 20 go # to get to market equilibrium

13 Ex: Find the equilibrium point for the following supply and demand functions: p price Demand: p 480 3q Supply: p 17q 80 g quantity Substitution or Elimination? Substitution is easier for this problem S : D p ( 480 g) 3 17g q 480 3g 17g g p q q 20 units 3g ) Equilibrium p $420 q 20