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# BUSINESS &ECON CALC MATH 112

UW

GPA 3.76

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This 10 page Class Notes was uploaded by Addison Beer on Wednesday September 9, 2015. The Class Notes belongs to MATH 112 at University of Washington taught by Alexandra Nichifor in Fall. Since its upload, it has received 11 views. For similar materials see /class/192056/math-112-university-of-washington in Mathematics (M) at University of Washington.

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Date Created: 09/09/15

Math 112 Part I Review WSl WS 11 Very Important On a test unless otherwise stated always show your work explain your answers If you work from a graph clearly amp carefully draw and mark any points or lines you use on the graph brie y describe the process you used in the work area for that question I General Concepts This part of the course is all about derivatives Notation The derivative of a function y fx is denoted by f39x or On graph f 39m is the slope of the tangent line to fx at the point x In Main Examples 0 The derivative of distance is the instantaneous speed That is we can measure speeds as slopes of tangent lines to the graph of distance 0 The derivative of TR is MR approximately That is you can think of MR as slopes of tangents to the graph of TR 0 The derivative of TC or VC is MC approximately That is you can think of MC as slopes of tangents to the graph of TC II We have two methods for computing derivatives You must be able to do both 1 the long way to compute f 39m a compute the slope of the secant line through the graph of fx atxmampxmh W h slope of secant b Simplify this expression to cancel the h in the denominator c let h go to 0 to get the slope of the tangent line f 39m 2 using the derivative rules This should be your default method do this unless you re told otherwise Review the 3 rules we learned and have them handy on your sheet of notes Ill Relationshi between ra hs of fx and f39x Given the graph of fx you should be able to determine amp sketch the general shape of the graph of the derivative f39m If fx is increasing then f39x is positive the graph of f 39x is above the xaxis If fx is decreasing then f39x is negative the graph of f39x is below the xaxis If fx has a horizontal tangent then f39x 0 the graph of f 39x is intersecting the xaxis Vice versa given the derived graph ie the graph of f 39x you should be able to determine the general shape of fx If f39x is positive then fx is increasing If f39x is negative then fx is decreasing If f39a 0 that is the derived graph crosses the xaxis at xa then fx has a horizontal tangent line at xa This may mean that fx has a local max peak a local min valley or neither at xa depending on the sign of the derivative before and after the point xa Rules for Rewriting Exponents give examples of each I Jab x Add the derivative rules with examples WS 2 Two versions ofMR and MC Main Idea IfWe re given the graph of TR or TC and if the units are large that is a change ofl unit on the xaxis is not legible then the slope of the tangent line approximates MR or M but We have to make sure We compute it in the correct unitsll Example You own a print shop The graph in the handout from the package of lecture handouts on the class website shows the graph of your total revenue TR From the graph An order ofl ream results in a total revenue of TR1475 An order of2 reams results in atotal revenue of TR 2 8 An order of3 reams results in a total revenue of TR310 21 What is the marginal revenue at 2 reams 1 Recall the de nition of MR The marginal revenue at quantity q is the change in TR as we move from q to ql sell one more unit MRqTRq1TRq Here we re working with q in reams so 1 unit1 Ream Al MR2TR3 TR2 lO82 2 Recall the other wav to think of MR MRq slope of secant line thru the TR graph at q and q1 Al TR3 TR2 10 8 3 2 lream MRZ 2 ream NOTE Both versions of MR as increment and as slope give the same numerical value but the units are different 0 As increment MR is in 0 As slope MR is in per unit 211d Scenario Suppose you start selling paper in units of 100 pages That is 1 unit1 hundred pages HP and l ream 500 pages 5HP lHP l5 reams02 reams l Q2 Compute MR at 10 HP 2 reams both ways l 1 As Increment recall 1 unitlHP MRlO HP TRll HP TRlO HP TR22 reams TR2 reams 05 2 As slope 0 Draw a secant through TR at 2 reams and 22 reams 0 Find two good points on this line 0 3 and 28 0 MRslope5225 Huh How come it does not match the value 05 above What are the units 250 is in per Ream Aha Should be per uniti ie l per HPI So the problem is that we re not done we need another step 0 convert 250 per ream t0 per HP 250 Ream250 5 HP05 per HP 3rd Scenario Suppose you now start selling by the page That is 1 unit1 page and l ream 500 pages 1 page l5000002 reams etc l Q3 Compute MR at 500 pages 1 ream 1 As Increment with l unitlpage MR500 TR501 TR500 TR1002 reamsTRl reams Impossible to read Points differing by 1 page are too close 2 We have to compute it as a slope 0 Do your best to draw a secant through TR at l Reams and at 1002 Reams It should look like the tangent at l ream Find two good points on this line 0 09 and 2611 0 MRslopel l09260388 per Ream 0 Now convert to the correct units 388 ream 388 500 pages 000776 page A3 When 1 unit 1 page MR500000776 per page Finally suppose you have no xed costs FC0 but it costs you 2 to produce every 3 Reams l Q What quantity maximizes your pro t Recall Pro t is maximal When we have 1 largest difference vertical gap between TR and TC with TR above TC 2 MRMC matching slopes transition from MRgtMC to MRltMC Since our costs are 2 for each 3Reams every extra Ream costs 23 per Ream so the Marginal Cost is constant MC a 23 per Ream Further TC 0 23 023 a with q measured in Reams So TC is a diagonal line of slope 23 Now it s time to notice that the question is not explicit enough Are we assuming that the quantity q must be a whole number of reams or can we sell in HP s or pages 1 If q must be a whole number of reams it s enough to compare the vertical distance at ql 2 3 4 5 etc Reams We see that the max gap is at q3 or q4 Reams So if we sell by the ream our max pro t is about 8 when we sell q3 or q4 Reams 2 If we sell by the sheet of paper then we d have to compare all intermediate points too which is tough to do well by eye alone Here it s best to use the second method nd MRMC ie nd a quantity where the slopes match Since TC is already a line we want a point on the graph of Total Revenue TR where the tangent is parallel to TC Use Rolling Ruler at about q35 Reams orl750pages Max pro t 82 at ql750 pages

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