Practice Midterm 2 - MATH 32A SECOND PRACTICE MIDTERM...

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MATH 32A: SECOND PRACTICE MIDTERM EXAMINATION Summer 2008 1
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1. (20 points) The C be a curve defined by the position function r ( t ) = < sin 2 t, t, cos 2 t > . (a) Calculate the equation of the Normal plane at the point (0 , π, 1). It’s easier than you think. (b) Calculate the equation of the Osculating plane at the point (0 , π, 1) (c) Is the Osculating plane perpendicular to the Normal plane? Explain. 2
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2. (20 points) For fun one day, Ronald fires a projectile from a Bazooka with an initial speed of 500 m/s and an angle of elevation 30 degrees while his colleague Wenye looks on. Here, the gravitational constant g = 9 . 8 m/s 2 (You may use the back of this sheet if additional space is needed.) (a) Find the range of the projectile. (b) Find the maximum height. (c) Find the speed of impact. 3
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3. (20 points) Find the curvature of the elliptical helix defined by the function r ( t ) = h 3 cos t, 2 sin t, t i at the points P 1 = (3 , 0 , 0) and P 2 = (0 , 2 , π/ 2). Here, you may us the fact that the curvature κ is given by: κ = | T 0 ( t ) | | r 0 ( t ) | = | r 0 ( t ) × r 00 ( t ) | | r 0 ( t ) | 3 . 4
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4. (20 points) Let f ( x, y ) = 1 / (1 + x 2 + y 2 ) (a) State the domain and range of f ( x, y ) (b) Draw the level curves of f ( x, y ). Specifically, for
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