Math 1230

Fall 2006

Take home portion of Exam 1 (30 pts)

Be sure to show all work for full credit. If you have trouble reading this, click here to download the word document.

1)             Consider the following: .

a.     Find the general antiderivative.

b.     Determine if   converges or diverges.

2)             Consider the following improper integral: .  Answer the following to find an estimate for this integral within an error of 0.01.

 

a.     Prove this improper integral converges.

b.     Determine a lower bound, a, so that the area under the curve from a to infinity is less than 0.005 (ie the tail end of the integral is small).  In other words find a lower bound so that .

c.     Next consider approximating the area under the curve from 1 to a using trapezoid approximation (ie the improper integral without the small tail end).  Specifically, use the error bound for trapezoid approximation on  to find an n so that |ET|<0.005.  Be sure to explain how you chose K.  If you use a graphing calculator to help determine K, be sure to tell me what function(s) you graphed, what your view window was, and give a rough sketch the graph.

d.     Use the above to estimate   with the trapezoidal method and with an error less than 0.01.  Note you may use Maple to do this.  If you use Maple, be sure to attach a print out of the work sheet.

3)                  

a.     Show that  converges.

b.     Use integration by parts to show that  converges.

c.     Use the above to show that  converges.