Section 1.1, Linear FunctionsReviewFood for thoughtSection 1.2, Linear Functions and ApplicationsSupply and DemandFood for thoughtSupply and Demand ExampleRevenue vs. CostRevenue vs. Cost ExampleMath 234, Calculus for BusinessLecture 1, Textbook Sections 1.1, 1.2Lines, Supply and Demand, Revenue vs. CostUniversity of Illinois, Urbana-ChampaignJanuary 18th, 2017Lines, Supply and Demand, Revenue vs. Cost (University of Illinois, Urbana-Champaign)Math 234, Calculus for Business January 18th, 2017 1 / 15AnnouncementsRead the syllabus, and textbook Sections 1.1, 1.2.Register on MyMathLab through the main Compass site.A Pearson representative will be available onFriday, January 20, 9am to 4pm in Noyes 216to help with student registration issues. Their name is Scott Brodkin.Contact me and your TA for all DRES accommodations immediately.Contact me for all scheduling and exam conflicts immediately.Lines, Supply and Demand, Revenue vs. Cost (University of Illinois, Urbana-Champaign)Math 234, Calculus for Business January 18th, 2017 2 / 15Section 1.1, Linear Functions ReviewDefinitionA linear function is a function whose graph is a (non -ve rt ic al ) line .Forms Equation InterpretationSlope-intercept y = mx + bm= slope of line,b = y-intercept of linePoint-slope y  y0= m (x  x0) m = slope of line,line goes through (x0, y0)Standard Ax + By = CAB= slope of lineCA= y-intercept of lineNote: The equation Ax = C (i.e., standard form with B = 0) with A 6=0corresponds to a vertical line. Th i s is not a function.Lines, Supply and Demand, Revenue vs. Cost (University of Illinois, Urbana-Champaign)Math 234, Calculus for Business January 18th, 2017 3 / 15Section 1.1, Linear Functions Food for thoughtUsing only the terms from the previous slide—slope, y-intercept,x-intercept, m, x0, y0, etc—consider the following:Food For Thought:When are two lines parallel?Food For Thought:When are two lines perpendicular?RemarkThe answers are in the book: Section 1.1, Examples 8 and 9.Lines, Supply and Demand, Revenue vs. Cost (University of Illinois, Urbana-Champaign)Math 234, Calculus for Business January 18th, 2017 4 / 15Section 1.2, Linear Functions and Applications Supply and DemandDefinitionThe demand function,p = D(q)relates quantity, q, of a product that buyers are willing to buy at price, p.Note: higher prices lea d s to lower demand (Law of Demand).DefinitionThe supply functionp = S(q)relates the price, p,atwhichproducersarewillingtosupplyq-many units.Note: higher prices mea ns produ ce rs make more supplies (Law of Supply).Lines, Supply and Demand, Revenue vs. Cost (University of Illinois, Urbana-Champaign)Math 234, Calculus for Business January 18th, 2017 5 / 15Section 1.2, Linear Functions and Applications Supply and DemandDefinitionEconomic equilibrium occurs when supply equals demand:D(q)=p = S(q)The equilibrium quantity and price are the values of p and q,respectively, that solve the above equation.RemarkThe Law of Supply AND Demand posits that, left to itself, themarket trend towards equilibrium quantity and price.The equilibrium quantity q and price p are the coordinates of thepoint where the graphs of the s up p l y and demand functions interse ct.Lines, Supply and Demand, Revenue vs. Cost (University of Illinois, Urbana-Champaign)Math 234, Calculus for Business January 18th, 2017 6 / 15Section 1.2, Linear Functions and Applications Food for thoughtRecall that D(q)=p = S(q) gives equilibrium quantity and price.Food For Thought:What is the economic interpretation of D(q) < S(q)?Food For Thought:What is the economic interpretation of D(q) > S(q)?RemarkThe answers are in the book: Section 1.2, the discussion after Example 2.Lines, Supply and Demand, Revenue vs. Cost (University of Illinois, Urbana-Champaign)Math 234, Calculus for Business January 18th, 2017 7 / 15Section 1.2, Linear Functions and Applications Supply and Demand ExampleExampleSuppose supply and demand functions for math textbooks are given byp = S(q)=15q and p = D(q) = 120 25qFind the equilibrium quantity and price.Solution (Step 1: Compute Equilibrium Quantity)Equilibrium occurs when supply equals demand,D(q)=S(q)120 25q =15q120 =35q200 = q.The equilibrium quantity is q = 200 un i ts. Let’s now find equilibrium price.Lines, Supply and Demand, Revenue vs. Cost (University of Illinois, Urbana-Champaign)Math 234, Calculus for Business January 18th, 2017 8 / 15Section 1.2, Linear Functions and Applications Supply and Demand ExampleExampleSuppose supply and demand functions for math textbooks are given byp = S(q)=15q and p = D(q) = 120 25qFind the equilibrium price and quantity.Solution (Step 2: Compute Equilibrium Price)To find equilibrium price, plug in q = 200 to either p = S(q) or p = D(q).If there are no mistakes, then they should output the same value.p = S(200) =15⇥ 200 p = 120 25⇥ 200p = 40 p = 120  80p = 40The equilibrium price is $40.Lines, Supply and Demand, Revenue vs. Cost (University of Illinois, Urbana-Champaign)Math 234, Calculus for Business January 18th, 2017 9 / 15Section 1.2, Linear Functions and Applications Revenue vs. CostDefinitionThe cost function, C(x), of producing x units of a product isC (x)=mx + bwherem is the cost to produce each item (marginal cost), andb is the initial cost to set up (fixed cost).RemarkThe book uses a change of independent variables—from usi n g q to using xto represent units of a product— emphasize a change of topics.Lines, Supply and Demand, Revenue vs. Cost (University of Illinois, Urbana-Champaign)Math 234, Calculus for Business January 18th, 2017 10 / 15Section 1.2, Linear Functions and Applications Revenue vs. CostDefinitionThe revenue function, R(x), of selling x units of a product isR(x)=pxwhere p is the price per unit on the market.DefinitionThe profit function, P(x), for selling x units is revenue R(x) minus costC (x),P(x)=R(x)  C (x).Lines, Supply and Demand, Revenue vs. Cost (University of Illinois, Urbana-Champaign)Math 234, Calculus for Business January 18th, 2017 11 / 15Section 1.2, Linear Functions and Applications Revenue vs. CostDefinitionThe break-even point occurs when revenue equals cost,R(x)=C (x).The value of x that sol ves eithe r equation is the break-even quantity.RemarkEquivalently, the break-even point also occurs when the profit is zero,P(x)=0,because of the definition P(x)=R(x)  C (x).Lines, Supply and Demand, Revenue vs. Cost (University of Illinois, Urbana-Champaign)Math 234, Calculus for Business January 18th,