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tutorial1 - 4. Sketch and describe the surfaces in R 3...

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MATH1211/10-11(2)/Tu1/TNK THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH1211 Multivariable Calculus 2010-11 2nd Semester: Tutorial 1 Date of tutorial classes: January 27–28. (The section/problem numbers in the following refer to those in the textbook.) 1. ( § 1.5 no.18) Give a set of parametric equations for the plane determined by the equation 2 x - 3 y + 5 z = 30. 2. ( § 1.7 no.20) (a) Graph the curve in R 2 having polar equation r = 2 a sin θ , where a is a positive constant. (b) Graph the surface in R 3 having spherical equation ρ = 2 a sin ϕ . 3. ( § 1.9 no.10) Show that the two lines 1 : x = t - 3 , y = 1 - 2 t, z = 2 t + 5 2 : x = 4 - 2 t, y = 4 t + 3 , z = 6 - 4 t are parallel, and find an equation for the plane that contains them.
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Unformatted text preview: 4. Sketch and describe the surfaces in R 3 determined by the following equations. (a) 2 x 2 + 2 y 2 + z 2 = 1 (b) x 2-y 2 + z 2 = 1 (c) x 2-y 2-z 2 = 1 (d) x 2-y 2 + z 2 = 0 5. ( § 2.2 no.12, 16) For each of the following limits, either evaluate it or explain why it fails to exist. (a) lim ( x,y ) → (-1 , 2) 2 x 2 + y 2 x 2 + y 2 (b) lim ( x,y ) → (0 , 0) x 2 x 2 + y 2 6. ( § 2.3 no.18) Find the gradient ∇ f ( a ) where f ( x,y ) = e xy + ln( x-y ) , a = (2 , 1) . (Recall that if f : X ⊆ R n → R then its gradient is the vector ∇ f ( x ) = ± ∂f ∂x 1 , ∂f ∂x 2 , ··· , ∂f ∂x n ² . ) 1...
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