21D-SQ02

# 21D-SQ02 - MATH 21 D (Section A) Quarter: Spring Date: June...

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MATH 21 D (Section A) NAME: Quarter : Spring SECTION: Date: June 08, 2002 1 2 3 4 5 total 6 Final Problem 1. (30 pts; estimated time: 20 mn ) Find the dimensions of the box of largest volume whose surface area is to be 6 square inches (see ﬁgure 1). Problem 2. (40 pts; estimated time: 25 mn ) Denote by S the surface of equation: z = - 1 2 x 2 + y 2 + 2 x + 1 . (a) Verify that the point (0 , 1 , 2) belongs to the surface S . (b) Verify that the vector i + j + 4 k is tangent to the surface S at the point (0 , 1 , 2). (c) Find the parametric equations of the vertical plane passing through the point (0 , 1 , 2) and containing the vector i + j + 4 k . (d) Find the parametric equations of a curve drawn on the surface S whose tangent at the point (0 , 1 , 2) is the vector i + j + 4 k . Problem 3. (40 pts; estimated time: 25 mn ) (a) Show that the curve parameterized by G ( t ) = e t + e - t 2 i + e t - e - t 2 j = cosh t i + sinh t j , for 0 t a , with a > 0, lies on the parabola x 2 - y 2

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## 21D-SQ02 - MATH 21 D (Section A) Quarter: Spring Date: June...

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