calculus
PRO J E CT II
MTH 151, FAL L 2021
Gini index
Due: Nov 30, 2021
Work in groups of two or three on this assignment. The report must be typed and well organized. It must be written in complete sentences and all the argu- ments must be justified. The report can not exceed three pages. This assignment will be graded both for mathematical quality and expository quality. The textbook provides good models of mathematical expository style. The handout Writing in Mathematics (available on canvas) provide additional advice. In particular,
• Write the paper for an audience of advance algebra students.
• Begin your paper with a brief introduction. You should restate the prob- lem in your own words and as you understand it.
• Shows mastery of the meaning of the functions and their properties.
• Is mathematically correct as well as grammatically correct.
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• Is well organized, coherent, and well written.
• Explain why you included the mathematical calculations or graphing method you did.
• Mix necessary mathematical equations and expressions with sentences of logic and explanation.
• Use present tense in explanations. Use ‘we’ to indicate the reader and author working together to solve the problem.
• Graphs and tables can often be very helpful. Any graph and/or table needs to be clearly labeled (both axes and key points) and the text needs to refer to it and explain what it shows.
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Economists use a cumulative distribution called a Lorenz curve to describe the distribution of income between house- holds in a given country. Typically, a Lorenz curve is defined on [0, 1] with endpoints [0, 0] and [1, 1] and is continu- ous, increasing, and concave upward. The points on this curve are determined by ranking all households by income and then computing the percentage of households whose income is less than or equal to a given percentage of the total income of the country. For example, the point (a/100, b/100) is on the Lorenz curve if the bottom a% of the households receive less than or equal to b% of the total income. Absolute equality of income distribution would occur if the bottom a% of the households receive a% of the income, in which case the Lorenz curve would be the line y = x . The area between the Lorenz curve and the line y = x measures how much the income distribution differs from absolute equality. The coefficient of inequality is the ratio of the area between the Lorenz curve and the line y = x to the area under y = x . Another name for the coefficient of inequality is Gini index (G).
Figure 0.1: Landing a plane.
We sometimes read that the inventors of calculus were Sir Isaac Newton (1642–1727) and Gottfried Wilhelm Leibniz (1646–1716). But we know that the basic ideas behind integration were investigated 2500 years ago by ancient Greeks such as Eudoxus and Archimedes, and methods for finding tangents were pioneered by Pierre Fermat (1601–1665), Isaac Barrow (1630–1677), and others. Barrow––who taught at Cambridge and was a major influence on Newton––was the first to understand the inverse relationship between differentiation and integra- tion. What Newton and Leibniz did was to use this relationship, in the form of the Fundamental Theorem of Calculus, in order to develop calculus into a systematic mathematical discipline. It is in this sense that Newton and Leibniz are credited with the invention of calculus.
Read about the contributions of these men in one or more of the given references and write a report on one of the following three topics. You can include biographical details, but the main thrust of your report should be a description, in some detail, of their methods and notations. In particular, you should consult one of the sourcebooks, which give excerpts from the original publications of Newton and Leibniz, translated from Latin to English.
N The Role of Newton in the Development of Calculus
N The Role of Leibniz in the Development of Calculus
N The Controversy between the Followers of Newton and Leibniz over Priority in the Invention of Calculus
References
1. Carl Boyer and Uta Merzbach, A History of Mathematics (New York: Wiley, 1987), Chapter 19.
NEWTON, LEIBNIZ, AND THE INVENTION OF C ALCULUSW R I T I N G P R O J E C T
What is the percentage of total income received by the bottom of the households? Find the coefficient of inequality.
; 68. On May 7, 1992, the space shuttle Endeavour was launched on mission STS-49, the purpose of which was to install a new perigee kick motor in an Intelsat communications satellite. The table gives the velocity data for the shuttle between liftoff and the jettisoning of the solid rocket boosters.
(a) Use a graphing calculator or computer to model these data by a third-degree polynomial.
(b) Use the model in part (a) to estimate the height reached by the Endeavour, 125 seconds after liftoff.
50% households receive of the income, in which case the Lorenz curve would be the line . The area between the Lorenz curve and the line measures how much the income distribution differs from absolute equality. The coefficient of inequality is the ratio of the area between the Lorenz curve and the line to the area under .
(a) Show that the coefficient of inequality is twice the area between the Lorenz curve and the line , that is, show that
(b) The income distribution for a certain country is repre- sented by the Lorenz curve defined by the equation
L!x" ! 512 x 2 ! 7
12 x
coefficient of inequality ! 2 y1 0 #x " L!x"$ dx
y ! x
x1
y
0
1 y=x
y=L (x)
(1,!1)
y ! xy ! x
y ! x y ! x
a%
WRITING PROJECT NEWTON, LEIBNIZ, AND THE INVENTION OF CALCULUS | | | | 399
Event Time (s) Velocity ( ft%s)
Launch 0 0 Begin roll maneuver 10 185 End roll maneuver 15 319 Throttle to 89% 20 447 Throttle to 67% 32 742 Throttle to 104% 59 1325 Maximum dynamic pressure 62 1445
Solid rocket booster separation 125 4151 1. Show that Gini index G =
∫ 1 0 (x − L(x ))d x∫ 1
0 xd x
2. Show that the Gini index is twice the area between the Lorenz curve and the line y = x , that is, show that
G = 2 ∫ 1
0 (x − L(x ))d x
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3. The income distribution for a certain country is represented by the Lorenz curve defined by the equation
L(x ) = 5 12
x 2 + 7 12
x
What is the percentage of total income received by the bottom 50% of the households? Find the coefficient of inequality.
GR AD I N G RU BR I C
Mathematical content (30%) Is the mathematics consistent and correct? Is it at a level of sophistication appropriate for this class? Are topics and ideas introduced with sufficient explanation? If there are pictures, figures or examples, are they accurate, appropriately used and do they support the text?
Clarity of mathematical exposition (30%) Are topics presented in a log- ical order? Does the paper achieve an appropriate balance of concise- ness and explanation? Are complicated parts/proofs (if any) broken into steps?
Style (40%) Is the paper clearly written, in paragraph form? Is the gram- mar, spelling, and sentence construction correct? Does the introduction serve its purpose? Is the paper readable, does it flow?
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