Test 2_MAT 2030_11SS

Test 2_MAT 2030_11SS - r (0) = h 1 , , i with initial...

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Name : MAT 2030, Calculus III, Test 2 No calculators needed or allowed! Questions : (50 + 5 points) (1) Find an equation of the plane that passes through the point (6 , 0 , - 2) and contains the line x = 4 - 2 t , y = 3 + 5 t , z = 7 + 4 t . [6 pts] (2) Find the domain of the vector function given by r ( t ) = h 4 - t 2 , e - 3 t , ln( t + 1) i . [6 pts] (3) Find parametric equations for the tangent line to the helix with parametric equations r ( t ) = 2 cos t i + sin t j + t k at the point (0 , 1 , π/ 2). [8 pts] (4) Find r ( t ) if r 0 ( t ) = 2 t i + 3 t 2 j + t k and r (1) = i + j . [8 pts] (5) Show that C = 2 πr , in which C is the circumference of a circle with radius r . [6 pts] (6) A moving particle stars at an initial position
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Unformatted text preview: r (0) = h 1 , , i with initial velocity v (0) = i-j + k . Its acceleration is a ( t ) = 4 t i + 6 t j + k . Find its velocity and position at time t . [8 pts] (7) Find and sketch the domain of the function given by f ( x, y ) = p 9-x 2-y 2 , on which also sketch the level curves for k = 0 , 1 , 2 , 3. [8 pts] Additional : Find and sketch the domain of the function given by g ( x, y ) = 1-x 2 + p 1-y 2 . [5 pts] (No partial credit) Show all your work in blue/green book! 1...
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