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AMERICAN UNIVERSITY OF THE MIDDLE EAST
 
MA261 –  Multivariate Calculus,  Fall 2020, Section: O8
Assignment 3,  Due: Dec 8th
 
Instructions:  There are 4 questions. Please answer clearly, show all your work step by step.
 
 
Student Name/ID:                                                                                                                     
 
Student Name/ID:                                                                                                                     
 
 
 
Q1 [25 pts]: Justify your answers clearly.
 

  1. a) [9 pts] Find the domain of the function

�(�, �) = √1 − �2 − �2 + ln(�2 − �).
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

  1. b) [9 pts] Sketch the domain that you found in part (a).

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

  1. c) [7 pts] Write down a point that lies in the domain of the function in part (a). Evaluate the function at this point.

 
Q2 [25 pts]: Justify your answers clearly.
 

  1. a) [9 pts] Show that limit does not exist

 
lim
 
 
 
3��
 
(�,�)→(0,0) �2 + �2 + 3��
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

�2+�2+3��
  1. b) [7 pts] Is the function �(�, �) =          3 ��      

continuous at the point (0, 0)? If so, justify your
 
 
answer. If not, find a point that �(�, �) is continuous.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

c)

[9 pts] Determine the set of points at which the function �(�, �) =   2 + � + �  
2+cos �
is continuous.
 
Q3 [25 pts]: Justify your answers clearly.
 

  1. a) [7 pts] Find the first partial derivatives of �(�, �) = (2� + 3�)√3� + 2�.

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

  1. b) [11 pts] Find the ����  where �(�, �, �) = 2��2 cos(1 − ��2).

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

c)                                                                                                      𝜕   𝑓                                                                                                                                

2
[7 pts] Find a point ��(�, �)  such that         (�, �) = 0   where �(�, �) = 4�3�(2� + 7�).
������
 
Q4 [25 pts]: Justify your answers clearly.
 
 

a)

[13 pts] Use Chain Rule to find  𝑑�
𝑑𝑡
where
 
� = �sin⁡(2�+3�),⁡⁡⁡⁡⁡⁡⁡⁡� = 𝑡 ln 𝑡 ,⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡� = ��√𝑡
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

b)

[5 pts] Use the answer in part (a) to evaluate   𝑑�  when⁡𝑡 = 1.
𝑑𝑡
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

  1. c) [7 pts] What is the difference between Chain Rule Case 1 and Case 2? Explain your answer.