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Course: MAC 2233, Spring 2011School: University of Florida
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11 MAC Project 2233 In this project, we explore the average and instantaneous rates of change of total cost, total revenue, and total prot with respect to the number x of items produced/sold. In particular, note the interpretations of the calculations, and how they might impact important decisions. 1. Suppose the total cost C (x) of producing x books is C (x) = 1200 + x2 20x, represented by $1200 in storage...
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11
MAC Project 2233
In this project, we explore the average and instantaneous rates of change of total cost, total revenue, and total prot with respect to the number x of items produced/sold. In particular, note the
interpretations of the calculations, and how they might impact important decisions.
1. Suppose the total cost C (x) of producing x books is C (x) = 1200 + x2 20x, represented by
$1200 in storage costs, x2 dollars for materials and labor, and a $20 rebate from the government for each book produced.
What is the total cost of producing 0 books? 20 books? What is the average change in cost per
book of producing the rst twenty books?
avg change: 20 dollars per book
What is the average change in total cost as x changes from 25 to 30. What does this imply
about the extra cost incurred by producing each of books number 26 through 30 ?
avg change: 35 dollars per book
What is the total cost of producing 35 books? 36 books? What, then, is the extra cost incurred
from producing the 36th book?
addl coost: 51 dollars for the 36th book
Calculate the instantaneous rate of change in total cost when x = 35. (Indicate the units for
your answer.) How close is this to the cost you calculated above?
inst of rate change: 50 dollars per book ...difference of $1
2. In general, the more items one produces, the less revenue one obtains per item from their sale.
Suppose we can sell our x books for 230 x dollars each. Thus, the total revenue from selling
x books is R(x) = x(230 x). What, then, is the total revenue from 30 books? 60 books?
Write a formula for the function P (x), that represents the total prot (revenue minus cost)
obtained from selling x books.
prot: P (x) = 2x2 + 250x 1200
What is the total prot from 30 books? 60 books?
Find the break-even pointsthat is, the values of x where we obtain zero prot.
break-even: x = 5, 120 books
Graph P (x) and calculate the instantaneous rate of change of P (x) in general at x = a. (Indicate the units for your answer.)
inst. rate: 250 4a
Looking at your graph, what is the relationship between the slope of the tangent line to the
graph of P (x) and the point where the maximum occurs?
inst. rate of change is zero at vertex a = 62.5 books
Based upon your previous answer, what production level x yields the maximum prot? Does
this agree with what you already know from precaclulus about a parabola?
max prot: $6612 dollars at 62 or 63 books
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Project 19MAC 223321. Examine the function h(x) = x 3x 3 . What is the domain of h(x) ? What are its intercepts?Calculate h (x), and use a number line to determine the interval(s) on which the function isincreasing/decreasing.What are the local extr
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Project 20/21MAC 22331. Note that the information learned in this lecture should prepare you for curve sketching!Recall this function from a prior project:2f (x) = 3x( 7 x2 ) 3f (x) =7(3 x2 )(7 x2)13f (x) =28x( 9 x2 )43( 7 x2 ) 3(Again, i
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Project 22/23MAC 22331. Examine the function h(x) =27x2. What is its domain ?x (x + 1)3Determine the horizontal asymptotes and vertical asymptotes, if any, for the graph of h(x) .Does the function have any removable discontinuities?What are the in
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Project 22/23MAC 22331. Examine the function h(x) =27x2. What is its domain ? x = 0, 1x (x + 1)3Determine the horizontal asymptotes and vertical asymptotes, if any, for the graph of h(x) .Does the function have any removable discontinuities?27xan
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Project 24MAC 22331. (From lecture) You are required to construct a closed rectangular box with a surface area of 48square feet so that the length is twice the width. What dimensions will maximize the volumeof the box?2. Sketch the graph of the funct
University of Florida - MAC - 2233
Project 24MAC 22331. (From lecture) You are required to construct a closed rectangular box with a surface area of 48square feet so that the length is twice the width. What dimensions will maximize the volumeof the box?Max volume when width is 2, leng
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University of Florida - MAC - 2233
Project 25/26MAC 22331. Carefully sketch the graph of f (x) = 4 + e(x2) below , including any intercepts and asymptotal lines, AND state its domain and range in interval notation.Domain:Range:Formula: g (x)=Now suppose g (x) is the function obtained
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Cummulative Distribution Function of Bids for All Pay AuctionSpring 2011100.00%90.00%80.00%70.00%60.00%2 bidders5 bidders10 bidders50.00%40.00%30.00%20.00%10.00%0.00%05101520253035404550Bid556065707580859095100
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Project 29MAC 22331. Examine the function h(x) =3 ln(x). What are its domain and intercepts ?xCalculate h (x), and h (x) . Use a number line to determine the interval(s) on which thefunction is increasing/decreasing, concave up/down. Are there any
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UCSB - ECON - 177
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UCSB - ECON - 177
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UCSB - ECON - 177
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MAC2233 Chapter 3 Review1. Find the derivative of h(x) = (x 2)(2x + 3).2. Find the derivative of f (t) = 3t4 2 + t5 .2t24x3. Find the derivative of f (x) = 3x x+2+2 .22t4. Find the equation of the tangent line to y = t4 3+2+1 at t = 0.t25. Th
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University of Florida - MAC - 2233
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UCSB - ECON - 177
2 Bidder, V=0 and V=3060Average Reserve5040V=0V=303020100123456789 10 11 12 13 14 15 16 17 18 19 20Round
University of Florida - MAC - 2233
MAC2233 Chapter 6 Review1. Find the indenite integral7x3 dx.2. Find the indenite integral6 x 9x2 + 5ex dx.3. Find the indenite integral2x2 7dx.x34. Find the indenite integral(x + 2)3 dx.5. Given f (x) = ex 2x and f (0) = 2, nd f (x).6. Find th