Mathematics Assignment - Desklib.

This Mathematics assignment includes solved questions on topics like integration, Lorenz curve, Gini coefficient, etc. The assignment also provides real-life examples of the Lorenz curve and its interpretation. The references used in the assignment are Rogawsky & Adams, Stewart, and Stroud & Dexter.

Running Head: MATH 1
Mathematics Assignment
Student’s Name
Institutional Affiliation

Math 2
Assignment 14
Question 1
y= ( 5 x2−2 ) 2
x
x 0 0.2 0.4 0.6 0.8 1
y 0 0.648 0.576 0.024 1.152 9
0 0.2 0.4 0.6 0.8 1 1.2
0
1
2
3
4
5
6
7
8
9
10
X Axis
y Axis
y= ( 5 x2−2 ) 2
x= ( 25 x4 −20 x2 + 4 ) x
y= ( 25 x5−20 x3 +4 x )
Integrating,
∫
0
1
( 25 x5−20 x3 +4 x ) dx= 25 x6
6 −5 x 4+ 2 x2 , at x=0∧x=1
Area
( 25
6 −5+2 )−0= 7
6 units2

Math 3
Question 2
y=5 x−1
x 1 1.5 2 e=2.718
y 5 3.33 2.5 1.839
0.5 1 1.5 2 2.5 3
0
1
2
3
4
5
6
Area under the curve
X Axis
Y Axis
Integrating
∫
1
e
(5 x¿¿−1) dx=¿ ln 5 x at x=1∧e ¿ ¿
Therefore, area under the curve
ln 5 ( e )−ln 5=1units2
Question 3
∫ 3 x
x2+ 8 dx=∫3 x ∙ ( x2 +8 )−1
dx
let u= ( x2 +8 )
du
dx =2 x
Therefore,
du
2 =xdx

Math 4
∫3 x ∙ ( x2 +8 )
−1
dx=∫3 ∙ ( u ) −1 xdx
But , xdx = du
2
∫3 ∙ ( u )−1 du
2 =1.5 ln u+C=1.5 ln ( x2 +8 ) +C
Question 4
∫12 x √ 2 x2 +7 dx=∫12 x (2 x2 +7)
1
2 dx
Let , 2 x2 +7=u
du
dx =4 x
Therefore,
du
4 =xdx
∫12 x (2 x2 +7)
1
2 dx =∫ 12 x (u)
1
2 dx
But , xdx = du
4
∫3 (u)
1
2 du=2(u)
3
2 +C=2(2 x2 +7)
3
2 + C
Assignment 15
x 0 0.2 0.4 0.6 0.8 1
L 0 0.077 0.216 0.416 0.677 1

Math 5
0 0.2 0.4 0.6 0.8 1 1.2
0
0.2
0.4
0.6
0.8
1
1.2
Lorenz curve
% population
% in co m e
b.
At x=0.1,
L= 23
30 ( 0.1 )2+ 7
30 ( 0.1 )=0.031
The bottom 10% of the population earn 3.1% of the total income.
c.
Area under triangle,
¿ 1
2 b h=0.5 ×1× 1=0.5 sq . units
Area under curve
∫
0
1
( 23
30 ( x ) 2+ 7
30 ( x ) ) dx=( 23
9 0 ( x ) 3+ 7
6 0 ( x2 ) ) , 1,0
A=23
90 ( 1 )3 + 7
60 ( 12 )−0=0.372 sq .units
Area between curve and hypotenuse
0.5−0.372=0.128 sq . units

Math 6
Gini coefficient = Areabetween curve∧hypotenuse
Areaof triangle = 0.128
0.5 =0.256
d.
Real Lorenzo curve
https://www.ons.gov.uk/peoplepopulationandcommunity/personalandhouseholdfinances/
incomeandwealth/compendium/wealthingreatbritainwave4/2012to2014/
chapter2totalwealthwealthingreatbritain2012to2014
The curve referred to above illustrate inequality in Great Britain for 4 wealth components. The
data used was collected between the year 2012 and 2014. All the components were graphed and
the curves compared to the 45 degree line. The closer the curve is to the 45 degrees diagonal, the
lesser the inequality and vice versa. From the research, the highest inequality was recorded in net
financial wealth. On the contrary, physical wealth showed more equality.
References
Rogawsky, J., & Adams, C. (2015). Calculus: Early Transcendentals (3rd ed.). W. H. Freeman.
Stewart, J. (2016). Calculus (8th ed.). Boston: Cengage Learning.
Stroud, K., & Dexter, B. (2013). Engineering Mathematics 7th Edition. Palgrave.