Solution Assignment: Multivariate Calculus

Instructions:

Complete the following problems related to Fourier Series. Ensure that all steps are clearly shown, and provide proper justifications for each step. Include graphical representations where required. Submit your assignment in a professional format.

Problems

1. Classification of Functions

Check whether the following function are even, odd or neither even nor odd:

(i) 𝑓(𝑥) = { 𝑥 + 4 if − 4 < 𝑥 < 0 ,

𝑥 − 4 if 0 < 𝑥 < 4

solution:

f(x) = x + 4

f(- x) = - x + 4

f(- x) = - (x - 4)

f(- x) = - f(x)

f(x) = x - 4

f(- x) = - x - 4

f(- x)= - (x + 4)

f(-x)=-f(x).

Hence f(-x)=-f(x) So it is odd function.

(ii) 𝑔(𝑥) = 𝑥²|𝑥 + 2| for (−4 < 𝑥 < 4),

solution:

g(x) = x² |x+2|

g(-x)= (-x)² |-x+2|

= x²|2-x|

As it is neither equal given function is neither to g(x) nor -g(x) So, it is a neither even nor odd.

(iii) ℎ(𝑥) = 𝑥|𝑥| for (−1 < 𝑥 < 1).

solution:

h(x) = xlxl

h(-x) = -xl-xl

h(-x) = -xlxl

h(-x)= -h(x)

Hence h(-x)= -h(x) So it is odd function.

2. Graphical Verification

a) Plot the graphs of the functions provided in 𝟏(𝒊 − 𝒊𝒊𝒊) using a graphing tool such as GeoGebra or Desmos.

b) Using these graphs, identify whether each function is even, odd, or neither even nor odd. Provide justifications based on the graphical representations.