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# ALGEBRA WITH APPL MATH 111

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This 9 page Class Notes was uploaded by Addison Beer on Wednesday September 9, 2015. The Class Notes belongs to MATH 111 at University of Washington taught by Andrew Loveless in Fall. Since its upload, it has received 11 views. For similar materials see /class/192075/math-111-university-of-washington in Mathematics (M) at University of Washington.

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Date Created: 09/09/15

Math 111 Week 1 Review This review is not all inclusive You are expected to know how to do all the problems in the homework Worksheet 1 Intro to rates of change Distance and Speed 1 A secant line is a line that goes through two points on the graph A diagonal line is a line that goes through the origin 2 Know the different types of speed a Actual Speed the speed on a speedometer the slope ofa tangent line that touches the distance graph This is studied in Math 112 Calculus change in distance since t 0 change in time since t 0 average speed from the beginning to now the slope of a secant line through the graph at t 0 and the current time change in distance b Average Overall Trip speed c Average Speed over an Interval Incremental Speed change in time average speed between two times the slope of a secant line through the graph at the two times 3 Important Notes a Pay attention to the graph that you are given in a problem Sometimes you are given a distance graph and sometimes you are given a speed graph Carefully write out what you know and use the de nitions above to answer the questions b Ruler Tricks i To compute ATS or AS A Draw the secant line Pick two easy to read7 points and compute the slope ii When asked about ATS x the ruler at the point on the graph at t 0 and rotate to answer the question iii When asked about AS over an interval sequentially try the intervals for example 0 to 10 then 10 to 20 then 20 to 30 etc Once you get close to the correct answer ne tune your answer by trying in between these intervals Worksheet 2 Flow in and out of a reservoir 1 Be careful with the type of graph you are given Sometimes you are given the incremental ow out of or into a reservoir and other times you are given overall ow In either case it is a good idea to make a table to verify that you understand what is given 2 If ow in is at a constant rate for example 3 thousand gallons every hour then you can draw a straight line that represents the ow in 3 Big Topics a When given an incremental change graph make a table of overall change and graph it This makes it easier to answer questions b The largest vertical distance that ow out is above ow in gives the maximum shortage This type of analysis will reappear when we discuss revenue cost and pro t Overall Rate of Change of AX Incremental Rate of Change of AX Amount AX AX F l 140 AXl I ormu a 0 Formula x2 x1 0 Graph Slope of diagonal 0 Graph slope of secant thru on graph of AX X1 and X2 on graph of AX Average Trip Speed mph mpm ft per sec etc Average Speed mph mpm ft per sec etc Dt Distance Dt ATS I I AS 2 Deg Dr1 assuming D00 else M I t2 t1 Total Revenue TRq price X q Average Revenue per item ARM 2 M Marginal Revenue or per item MRq TRq1TRq ATR A q since Aq1 Total Cost TCqVCqFC Average Cost per item Mo Ly Also BEPzLowest value of AC Marginal Cost or per item MCqTCq1TCq ATC A q since Aq1 Variable Cost VCq cost X q Average Variable Cost per item C61 L0 61 Also SDPzLowest value of AVC Marginal Cost or per item MCqVCq1VCq AVC Aq sinceAq1 slope afsecant thru q and q1 0R slope of tangent at q depending on the scale of the q axis WAN M Lequot quot A MCQUT L n QAfm77 ET u lt TCaw TC am 5qwcu 1 1Za i U5 xdf 77 a 5 2 q o u J O 0 9 0 9 RNA urA m 4y m EilhoA 1 171 39 mPJr CunATJA ap 1 TM 7 Tag FZ q51gt E 7394 Sana T v o P 1 1 w ix 97399 q L 00 P003 9 A Us ZDCAQ39O ir Oo ZJUl01ki iq W er Z 39 S 9 V L MINO mm ab TaampltTL ZQ a 1w7gt0 gt 5 33 Abm gg EgzmuLX or mud may JWZUO T 1 7 7 254Lz 754 5 Kr4L i 927 3 4757 7 Flow UD QM H N EM 7amp1 W051 5mm A Agov Z A q 2qma qH hub 3e 0 7 H J2 ATS Ngmns It JWZO OWZ 5 tgt w W th L t ra 0 i W 7 W 93912 O qZS t gvzp A Our I J P4561 3945 6791 ACG3Z7 05 TKO 3 0 5 03 Ehim j TCRL 75339F zigong 39 a a 3 mega 33 A gi 7129 4 NJ mug M13 J 113 mm may Mmagw 5 TCSO 3 13801 20 7 39 330 35 TC5 733 207 0 03 531 erm 35 Ht 2 H 2 if 133 39 138 l z my N 6 Q As sfo 39kcl quot1 4 Wu cm m quotJ M 4 mwrga 1Ac3Soo Admquot 3 71 12720 5 slate w A Math 111 Week 10 Review This review is not all inclusive You are expected to know how to do all the problems in the homework Worksheet 23 The Natural Logarithm Solving for time 1 The natural logarithm z lny is the inverse of the exponential function y ea So ln 6 z and elm z 2 You need to use lnz when you are solving for a variable that is in the exponent of an equation For example when you are solving for time in any compounding problem and when you are solving for 7 in continuous compounding problemsl Here are two examples to help you recognize when to use logarithms and when to use roots a Solve 103 40 Variable in the exponent You will have to use logarithms o SOL U T ON 31 4 ln3 ln4 z ln3 ln4 M 9 ln3 So I m 126186 b Solve 513 25 Variable in the base You will NOT use logarithms you will have to use the cube root 0 SOLUTION 13 5 I 53 z 513 513 is the same as So 1 117099761 3 Here are two typical examples that occur in bank problems expect problems like these on the nal exam a You deposit 500 into an account that pays 6 annually compounded continuouslyl HOW long does it take the account to double in size 0 FORMULA At 500603905 TRANSLATION Solve 1000 500e03905 SOLUTION 2 005 ln2 lneo3905 1112 010m M t 005 So t m 1115525 years b You deposit 200 into an account that pays 9 annually compounded quarterlyl 0W long does it take the account to triple in size 0 FORMULA At 200 1 0 TRANSLATION Solve 600 200 1 0 SOLUTION 600 200 110225 3 110225 1113 1111 0225 1113 4tln110225 t ln3 41n10225 So t m 1213436 years Final Exam Comments 1 The nal exam will have 6 to 8 questions You will have 3 hours Although you should have more than enough time you still should use your time effectively Look through the entire exam and make sure to start with the problems that are easiest for your 2 The nal exam is comprehensive with emphasis as follows a 23 Final Problems WORKSHEETS 19 EXAMlNlNG GRAPHS 0 Understand when and how to nd change in height slope of secant lines and slopes of diagonal lines from a graph Be able to use reference lines 0 Know everything about distanceASATS ow in ow out to a reservoir and TRTCARACAVCpricePro tMCMR problems 0 Know functional notation and how to translate b 23 Final Problems WORKSHEETS 1016 USING ALGEBRA 0 Understand when and how to nd change in height slopes of secant lines and slopes of diagonal lines from a formula 0 Know everything about the different scenarios and how to use formulas to go between distanceASATS ow in ow out and TRTC etc 0 Know how and when to use the vertex formula and the quadratic formula c 23 Final Problems WORKSHEET 1723 SEQUENCFS AND INTEREST 0 Know the formulas and how to use additive and multiplicative sequences In particular know how to do CAF and bacteria problems 0 Know how to decipher bank information Simple interestDiscrete CompoundingContinuous Compound ing i oBe able to answer questions that ask the following Find the future value M Plug in and get the value that comes out the lefthand side At Find the present value M Plug in and solve for the principal P How long M7 When M7 Find the time Use logarithms to nd the time t Find the rate M For continuous compounding you will need to use logarithms Otherwise you will need to use roots Find the change in value M7 Find the interest earned M 977 change in value interest earned NEW OLD 977 Find the proportionate change in value proportionate change Find the percentage change in value M percentage change W X 100 Find the multiplier M multiplier proportionate change 1 3 For additional review and practice look through the review sheets for the previous weeks Also you should work through all the old nal exams and you should look back at any of the homework that you struggled with Math 111 Week 3 Review This review is not all inclusive You are expected to know how to do all the problems in the homework Worksheet 6 Functional NotationTranslation 1 Know how to translate all of the following in different settings a the y coordinate of the f graph at the value 17 b IBLAH BLAH units after 17 z7BLAH BLAH units before 17 c 7 fa change in height of the f graph from I a to z 127 d ff slope of the diagonal line to the f graph at 17 b 7 a e 7 3 gt 7 slope of the secant line to the f graph from I a to z 127 2 In the DistanceSpeed setting we have the following a Dt the distance traveled at time t7 b tBLAH BLAH minutes after t t7BLAH BLAH minutes before t c Db 7 Da distance traveled from from t a to t 127 d w the average trip speed at t overall rate e w the average speed from t a to t 127 incremental rate 3 In the Total Revenue setting we have the following a TRq the total revenue for quantity 47 b qBLAH BLAH quantities after 4 q7BLAH BLAH quantities before 47 c MRq TR 1 7 TRq change in TR from L to 4 17 the cost to produce the q 1 iteml d w the average revenue at q7 the average price per item for 47 e MRq W the marginal revenue at 47 Worksheet 7 Reference Lines 1 Overall Rate of Change Questions ATS AR AC a If you are asked Find the overall rate of change at t 107 6 nd W then you draw a line through the graph at t 0 and t 10 and you nd the slope b If you are asked Find the time when the overall rate of change is 5 units per minute7 Le solve w 5 then you need to draw a reference line that has a slope of 5 and nd the corresponding time c NOTE To draw a line with slope 5 start with the point 00 To get another point pick an zvalue and multiply by 5 to get the y value That is at z 30 the y value is 150 So you would draw a line through 00 and 30150 2 lncremental Rate of Change Questions AS MR MC a If you are asked Find the incremental rate of change from t 10 to t 207 6 nd W then you draw a secant line through the graph at t 10 and t 20 and you nd the slope b If you are asked Find a 10minute interval where incremental rate of change is 8 units per minute7 ie Find t and t10 so that W 8 then you need to draw a reference line that has a slope of 8 Then you need to roll or slidel your ruler until to nd a 10minute interval that has a secant line with a parallel slope 3 Change in the Value Questions a If you are asked Find the change in height from t 4 to t 97 6 nd D9 7 D4 then you approximate how much the graph has gone up or down from t 4 to t 8 b If you are asked Find a 5minute interval where the change in height is 507 ie nd t and t 5 so that Dt 5 7 Dt 50 then you have a couple options i Option 1 At various 5minute intervals approximate the change in height If it is close to 50 then try moving your interval slightly until you get the right answer This is tedious and imprecise but it can work i Option 2 Turn it into a slope problem by dividing both sides by 5 Dividing by 5 gives w 10 Now you can use the reference line technique discussed above for incremental rates of change Worksheet 8 Breakeven price and shutdown price H E0 5 F The breakeven price is the lowest price at which a pro t is possible It is the same as the lowest possible average cost AC This is true because if the average price per item is more than the average cost per item then a pro t is possible for some quantity Since average cost is the slope of a diagonal line to the total cost graph we can use the following method a Fix your ruler at the origin and rotate to visualize slopes of diagonal lines to the TC graph b The lowest slope of a diagonal line to the TC graph the lowest value of average cost the lowest average price at which a pro t is possible the breakeven price c Warning You are not being asked for the quantity that gives the lowest slope The breakeven price is the slope itself FC Fixed Cost the cost of producing zero items such as costs for space rental of equipment etc Thus FC is the total cost at zero Or simply TC0 FC VC Variable Cost the cost of producing 4 items not including the xed costs Thus TCq FC VCq V001 l1 AVCq the slope of the diagonal line to the VC graph7 The shutdown price is the lowest price at which it is still possible to pay off some of the xed cost It is the same as the lowest possible average variable cost AVC Since average variable cost is the slope of the diagonal line to the variable cost graph we use the following method a Fix your ruler at the origin and rotate to visualize slopes of diagonal lines to the VC graph b The lowest slope of a diagonal line to the VC graph the lowest value of average variable cost the average price at which it is possible to make more than the variable cost ie make some xed cost back the shutdown price c Warning You are not being asked for the quantity that gives the lowest slope The shut down price is the slope itself

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