1. [5] Let . Use the definition of the limit and the Binomial Theorem to calculate

   

2. [5]a)

   Let Find the derivative of f, i.e., , using any method whatsoever.

   [5]b)

   Evaluate .

3. [5]a)

   A function y = f(x) is defined implicitly by the relation

   

   Find the equation of the tangent line to the graph of y = f(x) at the point .

   [5]b)

   Calculate at the point .

4. Find the derivatives of the following functions.

   [5] a)

   ,

   [5] b)

   

5. Evaluate the following limits (Show all work).

   [5] a)

   ,

   [5] b)

   .

6. [10] The problem of modeling planetary motion in the case of two bodies has been known since the time of Kepler and Newton. Using classical approximations it is known that planets will travel in ellipses with the sun at one focus. Asteroids or comets tend to travel in highly eccentric orbits (resembling parabolae) in the plane of the solar system with the sun at their focus.

   Let's assume that a planetary body is orbiting the sun (which is assumed to be very close to the origin) in a fixed almost circular orbit given by

   

   Let's say that an asteroid is approaching the sun in an orbit whose equation is given by

   

   see the figure below.

   

   Use Newton's method to find the two expected points of crossing of these two orbits, i.e., the two possible collision points, to three significant digits.

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