CALCULUS III CAT1 3RD OCTOBER 2025, ANSWER ALL THE QUESTIONS
(a) Use L Hospital rule to evaluate lim
(�)→(∞)�2�
�3(3mks)
(b) Show that the limit does not exist lim
(�,�)→(0,0)�6+�2
��3(3mks)
(c) Find the linear approximation to z at point (1,2) if z = 2�3+2�3−9�� (3 marks)
(d) Find all the first order partial derivatives for the following function.
f(x,y,z) = 2�3�2+2�3�4−9�5�� (4 marks)
(e) Determine the point on the plane �−�+�−3=0closest to the point 3, −2 ,4 .( 3 marks)
(f) Find the maximum and minimum values of � �,�,� =��� subject to the constraint �+�+�=1.Assume
�,�,�≥0(4mks)
(g) Verify clairaut’s theorem for �(�,�) = �10 − �7�4+ ���(�
�)(3mks)
(h) Find the tangent plane and normal line to �2+�2+�2=99at the point 2, −2,8 (4 marks)
(i) Use Rationalizing Technique to evaluate lim
�→24− 18−�
�−2 (3 marks)
CALCULUS III CAT1 3RD OCTOBER 2025, ANSWER ALL THE QUESTIONS
(j) Use L Hospital rule to evaluate lim
(�)→(∞)�2�
�3(3mks)
(k) Show that the limit does not exist lim
(�,�)→(0,0)�6+�2
��3(3mks)
(l) Find the linear approximation to z at point (1,2) if z = 2�3+2�3−9�� (3 marks)
(m)Find all the first order partial derivatives for the following function.
f(x,y,z) = 2�3�2+2�3�4−9�5�� (4 marks)
(n) Determine the point on the plane �−�+�−3=0closest to the point 3, −2 ,4 .( 3 marks)
(o) Find the maximum and minimum values of � �,�,� =��� subject to the constraint �+�+�=1.Assume
�,�,�≥0(4mks)
(p) Verify clairaut’s theorem for �(�,�) = �10 − �7�4+ ���(�
�)(3mks)
(q) Find the tangent plane and normal line to �2+�2+�2=99at the point 2, −2,8 (4 marks)
