Another sequence worksheet to go with the first one:

Calculus I Exploration, Sequences and Limits, part II

The following exercise is designed to arrive at rules for determining long-term behavior directly from a sequence generator function without building a table.

1. Make a hypothesis about the long-term behavior of sequences generated by the difference equation xn+1=λxn, x0.  Try to come up with some generalizations: sequences that look like _____ will do _____ long-term.  You may have several hypotheses, depending on the values of xn, x0, and λ.

2. Test at least one other example for each hypothesis formed.  Does it agree, disagree?  If it disagrees, go back to step 1.  When it agrees, move to step 3 for each hypothesis.

3. Create a written argument that explains why the general hypothesis is true.  You may use an example to illustrate, but it must be a general explanation for any example that would fall in that case.

4. Create a comprehensive, yet minimal list of “rules” and explanations derived in 1-3.  Cut and paste into the text box provided under the assignment link on Canvas.

Your blog assignment:

Record your observations from this exploration.  Also write down your resulting minimal list of hypotheses with verbal justifications about why they are true.

Here is a worksheet for educational purposes on Sequences:

Calculus I Group Exploration 2, Sequences and Limits

Team member names: ________________________________________________________

The following exercise is designed to arrive at rules for determining long-term behavior directly from a sequence generator function without building a table.

1. Count off a,b,c,d,e,f.  All a’s group up and take sequence a, etc.

   1. an+1=1.05an, a0=543

   2. an+1=0.95an, a0=543

   3. an+1=1.05an, a0=-543

   4. an+1=0.95an, a0=-543

   5. an+1=-1.05an, a0=543

   6. an+1=-0.95an, a0=543

2. Have one person create a Google Spreadsheet.  Make your spreadsheet documents viewable to anyone with a link.  Invite your group members.

3. Make a table and a scatterplot of the data.  Use good axes labeling.  Infer the long term behavior/limit.

4. What is the general solution to the difference equation?  Check your answer using the spreadsheet.  

Add the link and your results (Your difference equation, the general solution, the limit, and the graph) to the class results page here. Please note you must be signed into Unity College Google Apps to edit the class results page.

This video goes over 2 basic calculus concepts: