29 Now differentiate this equation getting D x 2 x 7 2 x 2 7 x 16 D x 0 x 3 5

29 now differentiate this equation getting d x 2 x 7

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Now differentiate this equation, gettingD0(x) =2x-72x2-7x+ 16.D0(x) = 0x= 3.5.Note thatD(0) = 4, D(3.5) =7.5.Whenx= 3.5,y=3.5, we have the minimum distance7.5..3.5 Business and Economics ApplicationsAverage cost: letCbe the total cost, andxbe the number of units produced, then¯C=Cx.Price elasticity of demandη: = (rate of change in demand) / (rate of change in price).Letxbe quantity demanded,p(x) be the price. Thenη=p(x)xp0(x).For a given price, the demand is elastic if|η|>1; the demand is inelastic if|η|<1;the demand has unit elasticity if|η|= 1.Example 73A store has been selling H¨aagen-Dazs bar at the price of$3.00 per bar and, atthis price, the store can sell 60 bars per day. If the store raises its price, it will sell 2 fewerbars per day for each$0.4 increase in price. Assume thatxbars cost the store0.1x2+ 0.4.a. Find the demand equation (pas a function of quantityx)b. Find the minimum average cost.c. Find the maximum revenue.d. Find the maximum profit.e. Find the intervals on which the demand is elastic, inelastic, and of unit elasticity.f. Use e to describe the behavior of revenueSolution: a.x= 60-2(p-3.000.4),x= 60-5p+ 15,p(x) = 15-0.2x.0x75.b.C(x) = 0.1x2+ 0.4.¯C=Cx= 0.1x+0.4x.¯C0(x) = 0.1-0.4x2.¯C0(x) = 0x=±2.Sox= 2.¯C00(2) = 100>0we have minimum average cost 0.8 whenx= 2.30
c.R(x) =xp(x) = 15x-0.2x2. R0(x) = 0x=150.4= 37.5.Maximum revenue isR(37.5).d.P(x) =R(x)-C(x) = 15x-0.3x2-0.4. P0(x) = 0x= 25. Maximum profit isP(25).e.η=p(x)xp0(x)=15x-0.2-0.2=-75x+ 1.|η|= 1x= 37.5,|η|>1x <37.5,|η|<1x >37.5.f. The revenue is increasing in (0,37.5) and decreasing in (37.5,75).3.6 AsymptotesDefinition 16The linex=ais called a vertical asymptote of the curvey=f(x)if atleast one of the following statements is true:limxa-f(x) =±∞,limxa+f(x) =±∞,limxaf(x) =±∞.Example 74limx01x2=,limx1-1(x-1)2=-∞.Example 75Verify VA from graph, sketch graph.Definition 17Iflimx→±∞f(x) =L, theny=Lis a HA (Horizontal Asymptote).Example 76Find HA:f(x) =2x2-1x2,P(x)Q(x),x2+ 1x.Example 77Verify HA from graph, sketch graph.31
3.7 Curve SketchingThe following checklist is intended as a guide to sketching a curve y = f(x) byhand.Not every item is relevant to every function. For instance, a given curve might not havean asymptote or possess symmetry. However, the guidelines provide all the information youneed to make a sketch that displays the most important aspects of the function.A. DOMAINB. INTERCEPTSC. SYMMETRY1.EVEN FUNCTION: f(-x) = f(x) for all x in D. the curve is symmetric about they-axis. This means that our work is cut in half.2.ODD FUNCTION: f(-x) = -f(x) for all x in D. the curve is symmetric about theorigin. This means that our work is cut in half.D. ASYMPTOTESHORIZONTAL: limx→±∞f(x) =L, theny=Lis a HA.

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