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The final examination for math 1220-005 during the spring semester of 2003. The examination covers various topics in mathematics including geometric series, orthogonal trajectories, vector calculations, spherical and cylindrical coordinates, area and volume calculations, parametric equations, series, and convergence tests.

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Pre 2010

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Download Math 1220-005 Final Examination Spring Semester 2003 and more Exams Calculus in PDF only on Docsity! Math 1220-005 FINAL EXAMINATION Spring Semester 2003 Lance L. Littlejohn Name Instructions. Show all work. Partial credit can only be given if sufficient work accom- panies each answer. This examination is out of 75 points. GOOD LUCK! Problem No. Points 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. TOTAL /75 1. (5 Points) Using geometric series, evaluate the infinite series โX n=0 (โ1)n2 n+1 7n . 2. (6 Points) Find, explicitly, the one-parameter family of orthogonal trajectories for the family of curves x2 โ 3y3 = k (k an arbitrary constant). In particular, find this trajectory that passes through the point (x, y) = (1, 2). 2 6. Let C be the curve parametrized by โโr (t) = cos ti+ (t2 โ 2)j+ sin 3tk. (a) (3 Points) Find parametric equations of the tangent line to C at the point (1,โ2, 0). (b) (3 Points) Find an equation of the normal plane to C at the point (1,โ2, 0). (c) (3 Points) Find the curvature ฮบ of this curve at the point (1,โ2, 0). 5 7. Let f(x) = Pโ n=0 (โ1)nx2n 2n+1 . (a) (4 Points) Find the radius of convergence and the interval of convergence of this series. (b) (4 Points) Using geometric series, determine explicitly f(x). (c) (2 Points) Using (b), evaluate the series Pโ n=0 (โ1)n 2n+1 . 6 8. Consider the three points P = (1, 1,โ1), Q = (1, 0, 1), and R = (โ1, 1,โ1). (a) (4 Points) Find an equation of the plane passing through these three points. (b) (3 Points) Find the area of the triangle with vertices at P,Q, and R. 7