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# BUSINESS &ECON CALC MATH 112

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This 55 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 Staff in Fall. Since its upload, it has received 27 views. For similar materials see /class/192104/math-112-university-of-washington in Mathematics (M) at University of Washington.

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Date Created: 09/09/15

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Worksheet 5 dollars quantity dollars quantity MATH 112 Lecture to Accompany Worksheet 6 97 1 14 1 MATH 112 Lecture to Accompany Worksheet 11 time minutes MATH 112 Lecture to Accompany Worksheet 14 The following is the graph of a function fx on the interval from x 1 to z 10 MATH 112 Lecture to Accompany Worksheet 18 Example To earn a bit of extra cash7 you start selling dried fruit and nut mixtures to hungry shoppers at the Fremont Sunday Flea Market You sell two varieties 0 Mostly Nuts7 which contains 25 fruit and 75 nuts 0 Fruitilicious7 which contains 55 fruit and 45 nuts For every pound of Mostly Nuts you sell7 you make 35 pro t and for every pound of P ruitilicious7 you make 60 pro t Your supply of fruit is limited to 90 pounds a day your supply of nuts is limited to 120 pounds a day Let x be the amount of Mostly Nuts that you make in pounds and y be the amount of Fruitiliciousl that you make in pounds Key Question How much of each mixture should you make in order to maximize pro t Another Example Gina inherits a large sum of money and a bunch of pet cages from an animal loving aunt She decides to rescue some unwanted pets from a shelter She has 20 cages that can each house either a bunny or a ferret She does some research and nds that7 on average7 it costs 060 a day to feed one ferret and 080 a day to feed one bunny Gina can buget no more than 1440 a day for pet food But cuddliness is an issue for Gina She gures that bunnies are twice as cuddly as ferrets That is7 ferrets are each worth one cuddle unit7 while bunnies are each worth two Determine how many of each pet Gina should adopt to maximize cuddliness while staying within her budget and without buying more cages MATH 112 Lecture to Accompany Worksheet 19 Example 1 Example 2 4ng 8 r 7 31 6 i 5 speed speed mphW mph 1amp7 r r r w 2 1 o 1 1 1 1 1 2 3 4 0 1 2 3 4 5 6 t hours t hours Example 3 80 4 70 60 3 d 39i 39 39 quot d39t 5 spee 7 1S ance mph 2 r i r r r 39 miles 40 i r r N 30 1 20 10 01 1 1 1 1 1 01 1 1 1 1 0 1 2 3 4 5 1 2 3 4 t hours t hours lnterval1 071 1 172 1 273 1 374 time 1 0 1 1 1 2 1 3 Distance Distance covered covered in that by that interval time 12 MATH 112 Lecture to Accompany Worksheet 20 TRTC From MRMC Interval 075 5710 10715 15720 20725 25730 30735 35740 Area under M0 on that interval 1 H 0 5 10 15 20 25 30 35 40 Area under MO from 0 to q Interval 075 5710 10715 15720 20725 25730 30735 35740 Area under MB on that interval 1 Area under MR from 0 to q MATH 112 Lecture to Accompany Worksheet 21 Interval 071 172 273 374 475 576 677 778 879 9710 Integral off96 on that interval Math 112 Spring 2005 Lecture Materials MATH 112 Lecture to Accompany Worksheet 2 dollars quantity in reams MATH 112 Lecture to Accompany Worksheet 4 distance miles time minutes car As car B73 speed miles minute time minutes MATH 112 Lecture to Accompany Worksheet 5 dollars quantity dollars quantity MATH 112 Lecture to Accompany Worksheet 6 97 H D 00 4 CT CT I 00 K O MATH 112 Lecture Questions to Accompany Worksheet 6 What does f 2 represent What is the value of f 2 Where does the graph of f cross the z axis What happens to the graph of f at those places How are the graphs of g and f related What does g 1 represent What is the value of g 1 Where does the graph of h cross the x axis What7s happening to the graph of h at those values of x Find an interval on which h z is negative H CH H H H K What does that mean about the graph of hz7 Find an interval on which h z is positive What does that mean about the graph of hz7 Find an interval on which k is decreasing and positive Which of these two graphs could be the graph of for z 0 to z 4 Where does k cross the x axis Where does k have horizontal tangents Where does have horizontal tangents Find the intervals on which is increasing MATH 112 Lecture to Accompany Worksheet 10 Recall the derivative rules f 3x674z2x7 fs Note that the variable in the function need not be named x 99110q9 7 5g2 9q 7 3 5 9M PROFIT ANALYSIS USING DERIVATIVES WS 10 You manufacture hats The TR and TC functions are TR Rq 7375q2 285q TC Cq 2q3 7 4q2 3g 1hSdndreds TR where q is given in hundreds of hatgf dollars and TR and T0 are in hundreds T0 of dollars The graphs give TR and T0 for 0 to about 300 hats quantity in hundreds 1 Write out a formula for pro t7 Pq 2 If the formula for pro t was a quadratic and its graph was a parabola that opens downward7 describe how you would nd the maximum pro t Explain why that method is inappropriate in this case 3 If you had nice graphs of TR and T07 give at least two ways to nd the quantity that maximizes pro t 4 Since pro t is not a quadratic and we don t have nice graphs of TR and T07 we7ll need to get formulas for MR and M0 to nd where pro t is maximized Recall that7 even though the MB is by de nition the slope of a secant line7 we often approximate MR using the slope of a tangent line to TB The slope of the tangent line is given by the derivative Find formulas for MR and MC 11 CT CT 5 00 Give a verbal description of the graphs of MR and MC Sketch rough graphs of MR and MC Find the quantity that maximizes pro t What is the maximum pro t What quantity maximizes total revenue What is maximum total revenue What quantity gives the smallest marginal cost What is the cost of the 121i hat MATH 112 Lecture to Accompany Worksheet 11 time minutes MATH 112 Lecture to Accompany Worksheets 15 and 16 The following is the graph of a function f on the interval from x 1 to z 10 15W 7 7 r r 14 Two things are happening at z 3 1 Q Narne another value of z for which properties 1 and 2 are both true 9 Whats happening at z 6 and z 8 Q What7s happening at z 1 Q atz107 So7 fx has minimum values at z interval from x 1 to z 107 OR We say that7 on this interval 7 7 o f has minimum values at z x o f has a minimum value at z 9 Where does fx have local maximum values on this interval 9 Where does fx have a global maximum value on this interval 9 What is the global maximum value of f on this interval x andz On the is the absolute smallest value of andx Def We use the collective term optimum values to refer to maximum and minimum values FACT The optimum values of a function g on the interval from x a to z b occur TWO APPLICATIONS I You sell things You set the price per thing depending on the size of the order If a customer orders q things7 then the price per thing is p hq q 7 70 1225 For example7 if a customer orders 10 things7 then p h10 102 7 7010 1225 625 and the customer pays 625 per thing for their order Your revenue for that order is R price quantity 625 10 6250 The function hq that gives the price per thing is called a demand function lts graph is a demand curve 25 39 hq39 q2 7 70 1225 Goal Find maximum possible revenue on the interval from q 0 to q 35 15 Step 1 Compute A q and nd where it equals 0 Step 2 Compute Aq at the endpoints of your interval and at the points you found in Step 1 Step 3 Draw a rough sketch of Aq and pick out the optima II We have a function f if 7 gzg 18x2 25 z 7 gzg 18 We de ne a function Sx as the slope of the diagonal line from the origin to the point x That is7 Goal Find the maximum and minimum values of Sx on the interval from x 1 to z 7 MATH 112 Lecture to Accompany Worksheet 18 LINE FITTING Situation The following is a chart containing the rst midterm exam scores and nal course grades for eight students in Math 112 Next to the chart is a graph of these data Student Exam Course Grade 4 70 2 O 3 3 39 i 4 Course 7 r r r 5 10 gradeZ 39 6 26 7 28 1 o 8 33 0 7 I i i l i l i l 0 10 20 30 40 50 m score Goal We want to nd a formula that will predict77 a student7s approximate course grade from hisher rst exam score This formula will be a linear function gt mt b where m and b are chosen to give the line that ts the data best The variable t represents the rst midterm exam score and the value of gt will be our prediction of the course grade Phase 1 Given a speci c line7 nd a number that indicates how wellthe line ts the data Let gt 01t 7 1 Sketch the graph of gt above How well do you think it ts the data not well7 fairly well7 very well To quantify this7 complete the chart below 239 1 2 3 4 5 6 7 8 The number 9a is what our formula would predict for the course grade for student number 239 The number 51 7 9a is the error in using the formula to make the prediction We want to add up all of the errors Since some of the errors might be negative and some might be positive7 we square the errors to avoid the cancelling out that can occur We take the average of the squared errors 81 9t12 82 9amp2 58 i 92 1 8 g 25139 902 This number is the meari squared error for the line gt 01t 7 1 The smaller the MSE7 the better the t Phase 2 Find the line with the smallest mean squared error Let gt mt b be a candidate for the line with the smallest mean squared error The MSE for this line depends on the values of m and b We want to nd the values of m and b that minimize MSE MATH 112 Lecture to Accompany Worksheet 14 t Pt 1 3310 2 4030 3 5680 40007 r i 7 4 7940 SSOOfF 39 39 300027 9 5 8740 5 250027 7 3 i r r 39 pop 6 12760 mowi 7 7 i 7 7 7 7 20000 150007 8 21890 300 i 50027 r 7 9 28680 i 0 4 4 4 4 4 4 4 4 4 4 1 2 3 4 5 6 7 8 9 10 10 37570 time 11 77 10 g o 397 9 7 0 O O 8 O 7 7 6 mltPlttgtgt5 4 3 2 1 0 4 4 4 4 4 4 4 4 4 4 1 2 3 4 5 7 8 9 10 time 239 t1 21 It tlzl 1 1 810 1 6561 810 2 2 830 4 6889 166 3 3 864 9 746496 2592 4 4 898 16 806404 3592 5 5 908 25 824464 454 6 6 945 36 893025 567 7 7 990 49 9801 693 8 8 999 64 998001 7992 9 9 1026 81 1052676 9234 10 10 1053 100 1108809 1053 Ely55 221 9323 St 385 22 8754975 2122i 5355 1 Eb m E 10b2 3857712 255bm 4 29323b 4 25355m 8754975 52 3857712 11bm 7186461 4107177 8754975 67E2b11m718646 67E77m11b71071 3b 771 The smallest possible Mean Squared Error occurs when 6E 6E 7 0 d 7 0 6b an am This occurs when b 7807 and m 02756 So7 the best tting line is 2 027561 7807 20 MATH 112 Lecture to Accompany Worksheet 19 Example To earn a bit of extra cash7 you start selling dried fruit and nut mixtures to hungry shoppers at the Fremont Sunday Flea Market You sell two varieties 0 Mostly Nuts7 which contains 25 fruit and 75 nuts 0 P ruitilicious7 which contains 55 fruit and 45 nuts For every pound of Mostly Nuts you sell7 you make 35 pro t and for every pound of P ruitilicious7 you make 60 pro t Your supply of fruit is limited to 90 pounds a day your supply of nuts is limited to 120 pounds a day Let x be the amount of Mostly Nuts that you make in pounds and y be the amount of Fruitiliciousl that you make in pounds Key Question How much of each mixture should you make in order to maximize pro t Another Example Gina inherits a large sum of money and a bunch of pet cages from an animal loving aunt She decides to rescue some unwanted pets from a shelter She has 20 cages that can each house either a bunny or a ferret She does some research and nds that7 on average7 it costs 060 a day to feed one ferret and 080 a day to feed one bunny Gina can buget no more than 1440 a day for pet food But cuddliness is an issue for Gina She gures that bunnies are twice as cuddly as ferrets That is7 ferrets are each worth one cuddle unit7 while bunnies are each worth two Determine how many of each pet Gina should adopt to maximize cuddliness while staying within her budget and without buying more cages 21 22 MATH 112 Lecture to Accompany Worksheet 20 Example 1 Example 2 40L 8 7 my 6 5 speed speed mph my 39 39 39 39 mph 1amp7 r r r r 2 1 o l l l l 1 2 3 4 0 1 2 3 4 5 6 1 hours 1 hours Example 3 8 4 7 6 3 5 speed 39 39 39 39 39 distance4 mph 2 r r 39 miles r r 3 1 2 r 1 r 0 l l l l l 0 l l l l 0 1 2 3 4 5 0 1 2 3 1 hours 1 hours lntervall 071 l 172 l 273 l 374 time l 0 l 1 l 2 l 3 Distance Distance covered covered in that by that interval time 23 MATH 112 Lecture to Accompany Worksheet 21 TRTC From MRMC 24 Interval 075 5710 10715 15720 20725 25730 30735 35740 Area under M0 on that interval 1 Area under MO from 0 to q Interval 075 5710 10715 15720 20725 25730 30735 35740 Area under MB on that interval q H 0 5 10 15 20 25 30 35 40 Area under MR from 0 to q 25 MATH 112 Lecture to Accompany Worksheet 22 Interval 071 172 273 374 475 576 677 778 879 9710 Integral off96 on that interval 26 27 Math 112 Winter 2005 Additions to the Text 28 29 Math 112 Worksheet 18A Multivanablc Functions In the previous sections of the course7 you worked with functions that consisted of only one variable For example7 Rq 25q705q2 is a formula for Total Revenue Its only input is quantity Q In this section7 you will explore functions that have multiple inputs Since each input is represented by a different variable7 these functions are called multivariable functions Introduction to Multivariable Functions Example During 20017 Puget Sound Energy utilized two methods for calculating the electrical bill for residential users at rate and time of use With the rst method7 a customer pays a at rate for each kilowatt hour of electricity used7 no matter what time of day the usage occurs Puget Sound Energy charged 005 per kilowatt hour for at rate usage 1 Customer 1 chooses the at rate method a Determine the charge for 2000 kilowatt hours to Customer 1 b Develop a formula 01z that gives the charge to Customer 1 for z kilowatt hours On the other hand7 the time of use method charges different amounts per kilowatt hour depending on when the electricity is used During peak times7 when the demand is highest7 the charge per kilowatt hour is higher than that at off peak times The time of use system has three rates peak hours 0062 per kilowatt hour for usage between 6 am and 10 am and between 5 pm and 9 pm daytime hours 00536 per kilowatt hour for usage between 10 am and 5 pm offpeak hours 0047 per kilowatt hour for usage between 9 pm and 6 am 2 Determine the charge for Customer 27 who chose the time of use method and who had the following usage during a one month period Time of Usage Kilowatt hours Rate Charge Peak Hours 1000 0062 Daytime Hours 700 00536 Off Peak Hours 300 0047 Total Charge 30 In order to make a formula for the charge to Customer 2 one variable will not be enough We7ll need three variables one for each time category 3 Let x be the electricity usage during peak hours y be the usage during daytime hours and 2 be the usage during off peak hours all for a one month period Develop a formula for 02zy z the charge to Customer 2 Note Just like formulas with only one input the variables are listed in parentheses The stuff in the parentheses is like the marquee at a movie theater Now showing the formula Charge for Customer 2 starring the variables 2 y and z 4 Suppose Customer 2 purchases smart appliances for a total of 3000 These smart appliances dishwasher washerdryer water heater etc can be programmed to operate during off peak hours Customer 2 also adjusts patterns of electrical usage so that the meter reading per month is now given by the following Time of Usage Kilowatt hours Rate Charge Peak Hours 500 0062 Daytime Hours 300 00536 Off Peak Hours 1200 0047 Total Charge How many months will it take for Customer 2 to recoup the investment in smart appliances That is how long will it take before Customer 2 has saved a total of 3000 on electric bills Example Another example of a function that has multiple inputs is the formula used to compute the balance of a certi cate of deposit compounded continuously A P6 This function has three inputs the principal P the interest rate r and the time of maturation t However many lending institutions set P equal to 1 The balance can then be found easily by multiplying by the principal Thus the formula becomes Art e and the two inputs are interest rate 7 expressed as a decimal of course and time t in years The following chart showing the balance of an account under different interest rates and at different times is similar to charts found in bank pamphlets Interest Rate 76 31 5 Use the chart to determine the values of A0064 and A00557 3 Remember that r is expressed as a decimal 6 Use the chart to estimate the values of A0056257 3 and A0045325 Rates of Change of Multivariable Functions Rates of change not to be confused with interest rates can be found in multivariable functions in a manner similar to that for functions with only one variable The technique is to look at the change in only one variable at a time That means you have to x the other variables and think about what happens to the function as the remaining variable changes As an example7 think about what the fourth row of the chart represents Each entry in that row is for an account paying 6 interest If you cover up the remaining rows and look only at the fourth row7 you have different values of a function A006t You can think of this as a function of a single variable t The interest rate is xed at 006 In this way7 a multivariable function can be reduced to a function of only one variable 7 Describe what Ar2 represents When you consider rates of change for a multivariable function7 think of it as moving A0053 7 A005 1 1 across a row or down a column For example7 f is the rate of change in the balance at 5 interest as time changes from 1 to 3 years The balance changes as time changes7 while the interest rate stays xed at 5 Evaluating this expression gives A0053 i A005 1 7 11618 710513 2 7 2 005525 Thus7 the incremental rate of change of the balance at 5 between year 1 and year 3 is 005525 dollars per year 8 a Determine the incremental rate of change in the balance at 45 between year 2 and year 5 b Determine the incremental rate of change in the balance for rates between 45 and 6 for year 3 c Determine the overall rate of change in the balance at 5 over 4 years 9 a Use the result of 8a and the value of A00453 to estimate A0045325 How close is your answer to that in 6 b Use the result in 8b and the value of A00553 to estimate A0056253 How close is your answer to that in 6 32 Note The rates of change in 8 a and b are not quite the same thing Both are rates of change in the balance But in part a time is changing and in part b the interest rate is changing The rate of change across a row is different than down a column This is evident in the units of the rates of change 10 Use the units to explain the difference between the two rates of change Derivatives of Multivariable Functions In the examples above we were able to treat multivariable functions just like regular functions when evaluating them and determining rates of change We just had to take the precaution of dealing with only one variable at a time A similar approach is used when taking the derivative of a multivariable function The derivative rules for single variable functions work here as well but the rules are applied to one of the variables at a time while the others are considered xed For example let7s consider the balance function Ar t equot We7ll start by thinking of the time as xed and nd the rate of change of A where r is the only variable 0 First we need to indicate which of our multiple variables were allowing to change Here we are thinking of time as xed and we are allowing the interest rate to change We take the derivative of A with respect to r The derivative of A with respect to the interest rate r is denoted The delta 3 is used to indicate that the function is a multivariable function The notation fTA would indicate that the function A involves only one variable Next think of the other variable t as being xed Although we will not be assigning the variable t a numerical value we treat the variable t as if it were a number Some people like to indicate that a variable is xed by marking the variable with an asterisk We could write the formula as Ar t em The asterisk is a reminder that time is xed Finally we take the partial derivative using rules from previous worksheets Remem ber we treat r as the variable and t as a xed number Applying the derivative rule for exponential functions we get 6A7 ifrttt 6T 6 The last part is the derivative of rt Recall for example that the derivative of z 3 3x is simply 3 Since we are treating t just like a number the derivative of rt t r is simply t 3A 11 Compute W 33 As another example7 lets look at the multivariable function we computed for the time of use electricity charges 02z7y72 0062z 00536y 00472 To compute think of everything without an x in it as a number Even though the terms 00536y and 00472 have a variable in them7 they dont have an x in them So7 we treat those two terms like constants Thus7 902 7 i 0062 M i 12 Compute 3370212 and For practice7 consider the function Ebm b2 347712 2833bm i 561 7154771 1023 Then7 the partial derivative of E with respect to m is computed by treating all of the Us as numbers 6E 7 68m 2833b 7154 3771 3E 13 Compute 3 Notation You may also see partial derivatives denoted with subscripts For example7 the partial derivative of E with respect to m might be written Em b7 m 68m 2833197154 14 Let ay x3 xy y2 7 z 7 y 5 Compute fmzy and fyy 34 35 MATH 112 Homework for Worksheet 14 0 Do 1 27 and 3 as written 0 Most of the table for 4 is done below Fill in the missing values in the table and complete the plot of the data in 4 You are plotting time vs lnvalue 0 Let ti be the time and 51 be the lnValue in the 2 row of the table Then7 12 55501 25114157 2t 11254 5 56422 Ems 23221 7 Use these values and the process of Worksheet 18 to nd the best tting line for these data 7 On the grid from 4 sketch the line you found i If we let ft be the value of the stock at time t then you have just found a formula for lnft In subsequent questions7 use this equation when it tells you to use the equation for lnft7 which you found in Problem 677 0 Do 8 97 107 117 and 12 as written 36

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