Math 166

Summer 2002

Final Exam

 

  1. (21 pt) Evaluate the following integrals:

a)         b)              c)

  1. (10 pt) Determine if the following sequences converge or diverge:

a)       b)  where  for all

  1. (21 pt) Determine if the following series converge or diverge:

a)         b)    c)

  1. (15 pt) Consider the ellipse

a)      Find the area enclosed by this ellipse (does your answer make sense?).

b)      Find for the upper half of this ellipse (the coordinate of the centroid of the upper half of this ellipse).

c)      Find the volume of ellipsoid obtained by revolving the upper half of this ellipse about the axis (does your answer make sense?).

  1. (10 pt) Consider the polar equation

a)      Sketch the graph of the equation.

b)      Find the area outside  but inside the circle

  1. (8 pt) Sketch the graph of the curve given by the parametric equations  
  2. (8 pt) Find the center, radius, and interval of convergence of the power series
  3. (6 pt) Recall that  Use this to find  with error less than
  4. (5 pt) Suppose that you are trying to find  and you find  How large must you choose  so that the error bound for Simpson’s rule is less than or equal to  the error bound for the trapezoid rule?
  5. (6 pt) Find a Maclaurin series for the function . Explain why this function has infinitely many derivatives at all points.

 

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