Math53MT1(Frenkel)-2

Math53MT1(Frenkel)-2 - 3. Consider the space curves dened...

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Mock Midterm Exam – Multivariable Calculus Math 53, 2009. Instructor: E. Frenkel 1. (a) Sketch the curves given by the equations (in polar coordinates) r = 2 cos θ and r = 1. For each curve, list all points of intersection between the curve and the coordinate axes (using rectangular coordinates). (b) Find the area of the region that lies inside the ±rst curve and outside the second curve. 2. (a) Find an equation of the plane that contains the points (1 , 2 , 3) , (1 , 1 , 1) , ( - 2 , 2 , 4). (b) Write down the parametric equations of the line passing through the points (4 , 2 , 3) and (0 , 0 , 1). (c) Determine whether the above line and plane intersect. If they do intersect, ±nd the point of intersection.
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Unformatted text preview: 3. Consider the space curves dened by the parametric equations x = t 2 + 1 , y = t-1 , z = t 3 + t + 2 and x = 2 s-2 , y = s-2 , z = 3 s-2 . Find the point of intersection of these curves. 4. Sketch the surface x 2-y 2 4-z 2 4 = 1 . Write down and simplify the integral that computes the surface area of the part of this surface bounded by the planes x = 1 and x = 5 2 . 5. Show that the function x 2 y 2 x 4 + y 4 does not have a limit at ( x, y ) = (0 , 0). 6. Write down the equation of the tangent plane to the graph of the function f ( x, y ) = x 3-3 x + y 2 + 2 y at the point (3 , 2). 1...
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