Summer 08 Midterm 1

Summer 08 Midterm 1 - MATH 32A Practice Midterm 1 Summer...

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MATH 32A: Practice Midterm 1 Summer 2008 – Dr. Frederick Park 1
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1. (20 points) Find a vector function that represents the curve C of intersection of the surface z 2 = x 2 + y 2 and the plane 2 z = 1+ y for z 0. Graph the curve and indicate the direction in which the curve C is traced as the parameter t increases from your parametrization of C. 2
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2. (20 points) Let P1: 3 x + 2 y - z = 4 and P2: 2 x - 3 y + 4 z = 16 be 2 planes in R 3 . (a) Show that the planes are neither parallel nor perpendicular. (b) Find the equation of the line where the planes intersect. (c) Find the angle between the two planes. (You may use the back of this sheet if additional space is needed.) 3
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3. (20 points) Let C be the curve defined by the parametric equations x = 2 - t 3 , y = 2 t - 1 , z = ln t . (a) Find the point where C intersects the xz plane. (b) Find parametric equations for the tangent line at (1 , 1 , 0). (c) Find the equation of the plane containing the point (1,1,0) in the direction of the tangent line to C at (1 , 1 , 0). e.g. the tangent vector to C at (1,1,0) determines the direction of the plane. Such a plane is called the Normal Plane to the curve C . 4
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4. (20 points) Do as indicated (a) Find a vector perpendicular to the plane containing the points A (1 , 0 , 0), B (2 , 0 , - 1), and C (1 , 4 , 3). (b) Find the area of triangle ABC . 5
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5. (20 points) Determine the point where the tangent lines to the curve r ( t ) = h sin πt, 2 sin πt, cos πt i at the points t = 0 and t = 1 / 2 intersect. Lastly, find the angle between the two lines. (Note: the parameters for each line may be different.) skip this page 6
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Weds July. 9th, 2008 Page 1/7 MATH 32A: FIRST MIDTERM PRACTICE EXAM SOLUTIONS SUMMER 2008 – Copyright Dr. Frederick Park 1
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Weds July. 9th, 2008
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