Dr. Latina’s Challenge Problems

1)

For k = 0, 1, 2, 3…., define where

1) Show that diverges

2) Since diverges, the functions have no common domain. Can

they converge in some sense other than pointwise convergence?

Hint: Start by proving as many basic properties as you can for the natural stump function

2)

Definition:

Show that

3)

(a) Let be a piecewise continuous function for

(b)

(c)

(d)

Show

4)

The Pump Problem

In a flat, barren desert owned by an underdeveloped nation, the government of that nation wishes to establish three settlements. As in the diagram above,

they are located at points A, B, and C. The distance between each pair of settlements is known.

It is also known that there is an abundant source of water deep beneath the desert floor. Drilling and pumping is an expensive proposition. They decide to drill in one location (W) and pump the water through pipelines to each settlement. The problem is to determine the location of W that will minimize the total amount of pipe used.

Note: This problem could also be phrased in terms of a source of oil pumped to three refineries.