math100_L3_midterm_98

# math100_L3_midterm_98 - (4) (a) (16 marks)For the function...

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Math 100 L3 Introduction to Multivariable Calculus Mid-Term Test 1998 (1) (20 marks) Sketch and classify the following surfaces: (a) z = x 2 - y 2 (b) y = - x 2 - z 2 (c) z = p x 2 + y 2 (d) z = x 2 + y 2 (e) x 2 = y 2 + z 2 (2) (24 marks) Find all relative maxima, relative minima, and saddle points for the function f ( x, y ) = 2 x 3 - 3 x 2 y + 3 y - 2 . (3) (a) (16 marks) Find the line that normal to the hyperbola x 2 a 2 - y 2 b 2 = 1 at a point ( x 0 , y 0 ) on the curve. (b) (12 marks)Find all the points in the hyperbola at which the normal line passes through the origin.
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Unformatted text preview: (4) (a) (16 marks)For the function u ( ξ ) = 1-ξ , where ξ = ξ ( x, t ) is deﬁned by x = ξ + (1-ξ ) t. Show that u satisﬁes u t + uu x = 0 . (b) (12 marks)Prove that the function u deﬁned by ± u = f ( ξ ) , x = ξ + f ( ξ ) t, is a solution of the equation u t + uu x = 0 for any diﬀerentiable function f . (Hint: use the implicit diﬀerentia-tion.) 1...
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## This note was uploaded on 09/30/2010 for the course MATH 100 taught by Professor Qt during the Fall '09 term at HKUST.

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