Math 1230

Fall 2006

**Take home portion of Exam 1 (30 pts)**

Be sure to show all work for full credit.

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 |E_{T}|<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.