math2004_ft06_test2a_sol

math2004_ft06_test2a_sol - MATH 2004 TEST 2 October 2006...

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MATH 2004 TEST 2 October 2006 Version A I N S T R U C T I O N S This test has one page and is worth 30 marks . It cannot be taken out of the exam room. Test duration: 50 minutes . Write your full name and student number onto the exam booklet provided by your tutor. Show all your work in this booklet and explain carefully your solutions. No calculators are allowed. 1. A 3-leaved rose is given by the following equation in polar coordinates: r = cos(3 θ ). [7 marks] (a) Find the area bounded by one leaf of this curve. [3 marks] (b) Find the slope of the curve at θ = π 4 2. A particle travels along the trajectory with the following position function: r ( t ) = 1 + sin( t ) , cos( t ) , cos( t ) - e 2 t [2 marks] (a) Find the velocity and acceleration vectors at t = 0. [1 marks] (b) Find the speed of the particle at t = 0. [7 marks] (c) Find the curvature of the trajectory at t = 0. 3. Find the equation of the tangent plane to the surface [10 marks] z = ln ( x 2 y - 1 ) at a point with x = 1, y = 2.
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Solutions 1. (a) Find the area bounded by one leaf of r = cos(3 θ ): A = 1 2 θ 2 θ 1 r 2 ( θ ) dθ.
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