Calculus Assignments: MAT 142 - Montgomery County Community College

This document contains solutions to two Calculus assignments for MAT 142 at Montgomery County Community College. The first assignment focuses on applying the Fundamental Theorem of Calculus, the Power Rule, and integration techniques to solve definite and indefinite integrals. The second assignment involves analyzing a Lorentz curve, calculating income distribution percentages, determining the coefficient of inequality, and providing a real-world example of a Lorentz curve with income inequality data for the USA in 1968 and 2010. Desklib offers a platform for students to access similar past papers and solved assignments.

Running head: CALCULUS 1
Calculus
Name
Institution

CALCULUS 2
Assignment 14
Question 1
∫
0
1
(3 x2 −2)4 x dx
Let 3 x2−2=u so that dx= du
6 x
∫(3 x2 −2)4 xdx=∫ (u )4 x du
6 x = 1
6 ∫(u)4 du= 1
30 (u)5
¿ 1
30 ( 3 x2−2)5
∫
0
1
(3 x2 −2)4 x dx= 1
30 [ (3 x2−2)5 ]0
1
= 1
30 { (3(1)2−2 )5
−(3 (0)2−2)5 }
¿ 1
30 { ( 1 )5 − (−2 )5 }= 1
30 (1+32 ) =33
30 =1.1

CALCULUS 3
Question 2
∫
1
e
3 x−1 dx
∫
1
e
3 x−1 dx=3∫
1
e
1
x dx= [ 3 ln (x ) ] 1
e
=3 ln ( e ) −3 ln ( 1 ) =3
Question 3
∫ 6 x
x2+5 dx
Let x2+5=u so that dx= du
2 x
∫ 6 x
x2+5 dx=∫ 6 x
u
du
2 x =∫ 3
u du=3∫ 1
u du=3 ln ( u )+C

CALCULUS 4
∫ 6 x
x2+5 dx=3 ln ( x2 +5 ) +C
Question 4
∫7 x √ 3 x2 +8 dx
Let 3 x2+ 8=u so that dx= du
6 x
∫7 x √3 x2 +8 dx=∫7 x √u du
6 x = 7
6 ∫ √u du= 7
6 ∫u
1
2 du
7
6 ∫u
1
2 du=7
6 ( 1
1
2 +1 ) u
1
2 +1
+C= 7
6 ( 2
3 ) u
3
2 +C=7
9 u
3
2 +C
7
9 u
3
2 +C= 7
9 (3 x2+ 8)
3
2 +C
∫7 x √3 x2 +8 dx= 7
9 (3 x2+8)
3
2 +C

CALCULUS 5
Assignment 15
Lorentz curve
Part a
The sketch of the Lorentz curve f ( x )= 19
20 x2 + 1
20 x using Desmos graphing calculator is shown
below.
Part b
f ( x )= 19
20 x2 + 1
20 x
f ( 0.60 )= 19
20 (0.60)2+ 1
20 ( 0.60 )=0.342+0.03=0.372
That is, when the population is 60 % the national income is 37.2 %
Part c

CALCULUS 6
The coefficient of inequality is the area bounded by the Lorentz curve and the line f ( x )=x from
x=0 ¿ x=1 as shown in shaded region of the figure below.
The equation of the shaded region is ( fx ) =x− ( 19
20 x2+ 1
20 x )=x −19
20 x2 − 1
20 x
Coefficient of inequality=Area of shaded region=∫
0
1
( x− 19
20 x2− 1
20 x ) dx
∫
0
1
( x− 19
20 x2− 1
20 x ) dx= [ 1
2 x2 −19
20 ( 3 ) x3− 1
20 ( 1
2 ) x
2
] 0
1
¿ [ 1
2 x2− 19
60 x3− 1
40 x2
]0
1
¿ 1
2 (1)2− 19
60 ( 1 ) 3− 1
40 ( 1 ) 2= 19
120
Part d

CALCULUS 7
Figure 1 below shows a real life example of Lorentz curve for USA in the year 1968 and 2010.
The graph shows that there has been an increase in income inequality in the US since 1968.
Figure 1: Lorentz curve (http://mathscinotes.com/2015/01/income-inequality-and-the-gini-
coefficient/)