Problems (Solusand to problemis mariged * appear at the bock of this book. Problems adapted to one calculus in availabit saint at 1. Which assumption about consumer preferences does by three different consumers (Augn, Barbara, and each of the following individuals violate? Camina), holding everything else constant: . Adam likes basketball more than football; Auon Football more than baseball; and baseball more Barbara Camira than basketball. b. Christina prefers prune juice to orange juice but 02 cannot decide how she feels about grapefruit juice. C. Blake likes superhero comic books but prefers 5 comic books to 10 comic books. 54 2. By assumption, individual preferences must be tran- sitive so that if A is preferred to B, and B is preferred to C. then A is preferred to C. Suppose that Marsha, 17 5 Jan, and Cindy individually have transitive prefer- ences over three goods: oranges, apples, and pears. Compute the marginal utility of X for each of the If Marsha, Jan, and Cindy were to vote on whether three consumers at each level of X. to name oranges, apples, or pears the "fruit of the b. Based on the data in the table, can you tell month," show that it is possible the preferences for whether any of these consumers are violating any the group might not be transitive. of the standard assumptions about preferences? 3. Draw two indifference curves for each of the follow- c. Is it possible that any of these three consumers ing pairs of goods. Put the quantity of the first good have the same preferences, and that columns for the on the horizontal axis and the quantity of the second three consumers differ only because of the arbitrary good on the vertical axis. units that are used to measure utility? Explain Paul likes pencils and pens, but does not care $6. A consumer's utility function is given by U- XY. which he writes with where MU, - F and MU, = X. Rhonda likes carrots and dislikes broccoli. a. What is the utility derived from 1 unit of X and C. Emily likes hip-hop iTunes downloads and 2 units of Y? What is the utility derived from doesn't care about heavy metal downloads. 2 units of X and I unit of F? What is the utility d. Michael only likes dress shirts and cufflinks in 1 derived from 5 units of X and 2 units of Y? to 2 proportions. b. How does the consumer rank the following e. Carlene likes pizza and shoes. bundles? *4. Suppose that John is indifferent between consuming Bundle Quantity of I Quantity of F bundle A. which consists of 4 apples and 1 peach, and bundle B, which consists of 4 peaches and A 2 1 apple. If John were given the choice between bun- 10 dle A and bundle C, which contained 3 peaches and 2 apples, which should be pick? (Hint: Draw an 5 indifference curve or two.) 3 2 5. The following table displays the total utility ((X ) that 2 3 comesponds to the number of units of X consumedMAT 166 - Pre-Calculus Writing Assignment I Students will be expected to answer all questions with the whole story. Perform each task on your paper. If you are asked to explain or define, be sure to use complete sentences, Sentence fragments are meaningless in technical writing. The intent should be to inform the reader, and to convince the reader of the knowledge and competence of the author. Answer the following questions in the order they are given. Answers should be organized on a separate sheet of paper. Answers may be hoped, hand written or a combination of both. If answers to questions are not numbered for lettered), it will be assumed that question I was answered and no other. This assignment is intended as a condensed review of functions. There is a recommended video that may aid in the completion of this assignment, but you may use any resource to aid in the completion of this assignment. Be sure to use a trusted source, and cite any source that you used. If you quote a source, paraphrase the quote in your own words. 1. Read all the directions for this assignment and copy the most important instruction as your answer to this question. 2. Watch the video https /www.youtube.com/watch/v .HOggysawWg and write your favorite color as the answer to this question. 3. Create or find a definition of a function. Include a definition of "domain" and "range" in the answer to this question. (It is recommended that the words "relation" or "pairing" be used in the definition.) (see page 137,138) Use the previous definition of function to explain the "vertical line test" and why it works. Note: A fest and a definition are not the same. 5. Tell why the trigonometric functions adhere to the algebraic definition of a function, and describe the "inputs" and "outputs" of any trigonometric function. 6. Create or find a definition of a "one-to-one function". (p 357) 7. Use the previous definition to explain the "horizontal line test" and why it works. Note: A rest and a definition are not the same 8. Create or find a definition of "inverse functions". (p 359) 9. Given that the function /(* has an inverse function . (F); a. If /(2)=120 , Find / (120) b. If the domain of /(x) is xE[0, 1000) . what is the range of (E)? c. If the domain of ()represents time in years after 2010, and the domain of / (*) represents the population of Ourtown, write a sentence to interpret i. f (2)=120 ii. f (500) =7 / (x1=x+5 10. Let 7 and g(x)=7x-5 a. Find (9 (x) and (9 f)(x ) b. Come to a conclusion about the relationship between the functions f(x ) and g(x) explain how you arrived at the conclusion. and 1 1. What is the most important concept you have learned remembered by doing this assignment?Question 4 (Further Processing) Company D manufactures four products from an input of a raw material to Process 1. Following this process, product U is processed in Process 2, product V in Process 3, product Win Process 4 and product X in Process 5. The normal loss in Process 1 is 10% of input, and there are no expected losses in the other processes. Scrap value in Process 1 is $0.5 per litre. The costs incurred in Process 1 are apportioned to each product according to the volume of output of each product. Production overhead is absorbed as a percentage of direct wages. Data in respect of a particular month Process Total 1 2 3 4 5 $'000 $'000 S'000 $'000 S'000 $'000 Direct Material at $1.25 100 100 per litre Direct Wages 48 12 8 4 16 88 Production Overhead 66 Product U V W X litres litres litres litres Output 22000 20000 10000 18000 $ Selling Price 4 3 2 wuIt Estimated Sales Value at end of 2.5 2.8 1.2 Process 1 Required: Suggest and evaluate an alternative production strategy which would optimize profit for the month. It should not be assumed that the output of Process 1 can be changed.El . 3 ID . : |Company C is considering a proposal to use the services of a press cuttings agency. t the moment, press cuttings are collected by a junior member of the marketing department, who is also responsible for ofce administration (including lling), travel bookings, a small amount of proof reading and making the tea. The total annual cost of employing this person is $15000 per annum. There is concern that the ability of this person to produce a comprehensive file of cuttings is limited by the time available. She has calculated that she needs to spend about two hours of her seven and a half hour day simply reading the national and trade press, but usually only has about five hours a week for this job. Press subscriptions currently costs $350 per annum and are paid annually in advance. The assistant makes use of a small micro-che device for storing cuttings. The cuttings are sent to a specialist rm once a month to be put into che. Company C pays $45 each month for this service. The micro-fiche reader is leased at a cost of sins per calendar month. This lease has another 2? months to run. The cutting service bureau has proposed an annual contract at a cost of $1253. Several existing users has conrmed their satisfaction with the service they receive. W Should Company C outsource its press cuttings work? Please explain. 3 (Variant of Exercise 2.7.1, 7th ed., p. 133) Let X1, X2, Xa " fx(x) = e I(0, too) (#); that is, X1, X2, X3 - Exp(A = 1). Consider the following random variables defined as functions of X1, X2, and X3: X1 X1+ X2 Y1 = X1+ X2 Y2 = X1 + X2 + X3' Ya = X1 + X2 + X3. a Use the Jacobian method to obtain the joint PDF fy,v2,Ya(V1, y2, y3) of Y1, Y2, and Ya. b Use the following generalization of Theorem 2.5.1 (see p. 111, 7th ed.) to any num- ber m > 3 of continuous random variables, to show that Y1, 12, and Ya are mutually independent (see p. 120, 7th ed.): Let Y1, . .., Ym be continuous random variables with joint PDF fr,,Ym (y1, . .. , ym), with joint support S = S X ... X Sm, where S1, . .. , Sm are the respective marginal supports of Y1, . .., Ym. Then Yi, -.., Ym are mutually independent if and only if their joint PDF fri,-,Ym(VI, . .. ,Um) can be written as a product of nonnegative functions 91 (V1), - -- ,9m(Um), with each g;(y; ) being a function of y; alone, for j = 1, . .. , m; that is, fri,., Ym(V1, " .. .ym) = 19,(3;)Is, (v;). i=1 where gi(y;) > 0, for all y; ( S,, and g;(y;) = 0, elsewhere, for j = 1, . .., m. The discrete case is analogous. That is, factorize fy,12,13(31, 12, 93) as fri,Ya,Ya (#1, 12,93) - II;-19;(3;)Is, (u;) by spec- ifying the functions g1(31), 92(32), and 93(13), as well as the supports $1, $2, and $3. C Keeping in mind the kernels of PDFs of the special distributions in Lecture 1, simply fill-in the appropriate normalizing constants, when needed, to make the functions g;())Is,(#;) in (b) proper PDFs, and thus - using the definition of mutual indepen dence (p. 120, 7th ed.) - obtain the respective marginal PDFs fr, (y1), fra(y2), and fry(ys) of Y1, Y2, and Y's in the modified factorization fri, Ya,Ya (31, 92, 93) = II;_ fy, (y;). Note that some normalizing constants may be "missing", as they likely cancel each other out in the product; thus, filling in the "missing" normalizing constants should have the overall effect of merely multiplying IT ,9;(y,)Is, (#;) by 1 (and thus, nothing is changed!)