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Sample Test 1

# Sample Test 1 - Math 200 Sample Test 1 No books calculators...

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Math 200, Sample Test 1 No books, calculators or notes allowed. Answer all questions. 1. [10] Let u ¯ = 3 i ¯ + j ¯ + 4 k ¯ and v ¯ = i ¯ 3 j ¯ + 2 k ¯ . Find (a) u ¯ · v ¯ (b) | u ¯ | (c) u ¯ × v ¯ (d) The unit vector in the direction of u ¯ v ¯ (e) Sketch u ¯ v ¯ Solutions: straightforward de fi nitions. 2. [10] Find the equations of the following planes: (a) Through the point (2 , 4 , 1) with normal vector n ¯ = 2 i ¯ +3 j ¯ +4 k ¯ (b) Through the points (1 , 3 , 2) , (3 , 1 , 6) , and (5 , 2 , 0) . Solutions: Examples 4 and 5, section 13.5 3. [10] (a) Find the equation of the sphere, centre (1 , 4 , 3) and radius 5 . What is the intersection of the sphere and the xz plane? (c) Find the equation of the sphere that passes through the point (4 , 3 , 1) and has centre (3 , 8 , 1) . Solutions: 13.1, exercises 11 and 13. 4. [10] A woman walks due west on the deck of a ship at 3mph. The ship is moving north at a speed of 22mph. Find the speed and direction of the woman relative to the surface of the water. Solution: 13.2, exercise 31. 5. [10] Evaluate the following integrals. I = Z x cos(4 x 2 1) dx, J = Z e 1 x ln( x ) dx Answers Z x cos(4 x 2 1) dx = 1 8 sin ¡ 4 x 2 1 ¢ + k Z e 1 x ln( x ) dx = 1 4 e 2 + 1 4

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