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(MATH100)midterm-1999.pdf
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MATH 100 Introduction to Multivariable Calculus Mid-term Test
October 20, 1999.
Student Name:
Student ID:
1.
(20 marks) Sketch the surface z = x2 +4y2 + 1.
2.
(25 marks) Let 1 be the closed curve obtained by intersecting two surfaces x2 + y2 = 1 and x + y + z = 1; 2 be the closed curve obtained by x2 + y2 = 1 and x + y . z = 5. Show that both have the same arc length.
3.
(25 marks) Find the tangent plane of the surface
sin(x + y) + sin(y + z) + sin(z + x)=0
at (0, 0, 0).
4. (30 marks) Find all relative maxima, relative minima, and saddle points
for the function
f(x, y)=3x 2 . 3xy + y 3 .
sketch paper sketch paper sketch paper sketch paper sketch paper sketch paper sketch paper sketch paper
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