MAT2100 Assignment 2, University Calculus: Comprehensive Solutions

This document presents a comprehensive solution set for Assignment 2 in a Calculus course, likely MAT2100, at a university. The solutions cover a range of calculus topics, beginning with basic problem-solving and progressing to more complex concepts. Specific problems addressed include Fourier series representation, total differential equations, and the evaluation of double integrals, line integrals, and mass calculations. The document provides step-by-step solutions, demonstrating the application of various calculus techniques and theorems. The assignment addresses topics such as even functions, curl of a vector field, and integration over specified regions. The solutions are presented with clear mathematical notation and explanations, making it a valuable resource for students studying calculus.

Assignment 2, MAT2100
Name of the Student
Name of the University
Author Note

Table of Contents
Answer to question 1:.................................................................................................................................3
Answer to question 2:.................................................................................................................................6
Answer to question 3:.................................................................................................................................7
Answer to question 4:...............................................................................................................................10
Answer to question 5:...............................................................................................................................16
Answer to question 6:...............................................................................................................................16
Answer to question 7:...............................................................................................................................18
Answer to question 8:...............................................................................................................................18
Answer to question 9:...............................................................................................................................24
Answer to question 10:.............................................................................................................................27

Answer to question 1:

Answer to question 2:
Simplifying

[b]
Answer to question 3:
[a]

[b]

[c]

Answer to question 4:
[a]
[b]
The function is defined as
[i] not defined in the period of (-10, -2];
[ii] –t^2 in the period of (-2,0]
[iii] t^2 in the period of [0,2)
[iv] not defined in the period of [2, -10)
[c]
The function is an even function.

[d]
Average value of the function on its period
= 1
0+2 ∫
−2
0
−t2 dt + 1
2−0 ∫
0
2
t2 dt
= −1
2 [ t3
3 ]−2
0
+ 1
2 [ t3
3 ]0
2
=-4/3 + 4/3
=0
[e]
The Fourier Series representation is
xT (t )=a 0+∑
n=1
∞
(an cos ( nπ t
2 )+bn sin( nπ t
2 ))
Since, the function is even,
xT (t )=a 0+∑
n=1
∞
¿ ¿
= ∑
n=0
∞
¿ ¿
The average value is 0
And the value of an is 1
2 ¿
= 1
2 ¿
Now,