Math 3066 -- Assignment 24: Limits & Derivatives

Due: November 30th

The idea of a limit is one of the "big ideas" in higher level mathematics.  The concept of a limit is accessible to middle school children who have access to technology like a spreadsheet.  Using Excel, solve this problem:

Imagine a cube shaped box that measures 1 foot on each side.  Stacked on this box is a box with all the dimensions exactly half of the box below.  So the second box measures 1/2 foot on each side and the total stack height is 1.5 feet.  This process of stacking a box half as big on all dimensions as the top box continues.  The question is how many boxes will be required to make a stack 10 feet tall?  How about 5 feet tall? 

Extensions of this problem are to change the size of that initial box, change the proportions of the "new" box (i.e. what if the new box is 90% of the original), another pattern is 1/2, 1/3, 1/4, 1/5, ... and to look for patterns in the boxes.

In class we will look at three problems.  The first half of your assignment is to print out a short paragraph explaining why the limit exists (when it exists).  Your print out should include the Excel tables showing the limits (or lack thereof).

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The derivative has several interpretations.  In this lesson we will explore the slope of curved line, at some point on the line, as that line curves.  We will do this by constructing tangent lines at various points.  The second half of your assignment will be made in class following a brief lecture.  Please complete the assignment and turn in a written solution to Dr. Frauenholtz.