An Educational video on Sketching derivatives:

an educational handout on discontinuities:

https://drive.google.com/a/unity.edu/file/d/0B_Vlmt-dGhKdUGE0ZnNNQ2VNemc/edit

another video on Limits to infinity:

An educational video With a limit of infinity: 

Here is a handout on limit rules: 

Calculus I                            Carrie Diaz Eaton, Unity College

Special limits of sequences rules

1. When the base changes:

Discrete For an=n-k=1nk, k>0, thennan=0

Continuous or f(x) = x-k, k>0, then xf(x)=0

Example:

2. When the exponent changes:

    For an=a0n, or f(t)=x0t

1. if -1<<1, then nan=0 or tf(t)=0

        Example:

2. if >1, then nan DNE or tf(t) DNE

        Example:

3. Rational expressions:

        For an=bpnp+bp-1np-1+...+b1n+b0cqnq+cq-1nq-1+...+c1n+c0,

1. if p<q, then nan=0

    Example:

2. if p=q, then nan=bpcq

    Example:

3. if p>q and both p and q are positive or negative, then nan= (DNE)

    Example:

4. if p>q and only one of p and q are positive, then nan=- (DNE)

    Example:

4. Sandwich Principle

   1. General case:

    If  nan=ncn=L     and anbncn, then nbn=L

    Example:

2. Special case, Alternating sequences with (-1)n:

        If nan=0,     then nan=0,     otherwise the limit DNE

        Example:

Here is a educational video on limits:

Here is a handout on Limit properties:

Calculus I                                     Unity College

Properties of Limits

For real numbers, a (a can also be ), c, L, and M, given that taf(t)=L, tag(t)=M, then

WARNING!  CAREFUL OF INDETERMINATE FORMS, like - and , 00!

Note: -0     and     1!!!!!!

Special sequences (KNOW these!):  

n(1/n) = 0   ,      n(e-n) = 0

This project is to go along with differential equations:

Learning Objectives addressed:

1. Understand sequences, limits, and continuity algebraically, numerically, visually, and verbally.

3.   Be able to model simple scenarios of change through either difference equations or differential equations.

5.   Recognize limits and derivatives in the practical and professional world, particularly in environmental and life science.

6.   Be able to use a computer algebra system and spreadsheet system to investigate or evaluate given problems.

7.   Work in groups to investigate problems and communicate solutions on an introductory level.

Project methods:

A small lake trust in Maine is undergoing an investigation that has both environmental and economic impacts.  The exact numbers presented here have been simplified for ease of analysis.  

To build up a population of trout in a small lake, 200 young trout are added each year. In addition, the population increases its own numbers by 20% each year. Let xn denote the size of the population after n years.

a) If x0 = 1200, determine the largest n such that when xn <2800.

b) Once xn =2800 the lake is no longer stocked and fishermen will catch 600 fish per year. What is the fate of the population?

Now extend this problem.

c) Suppose you have only have enough money to stock the lake for the 3 years it takes to reach 2800.  What level of fishing could be sustained when the population reaches 2800?

d)  Suppose you want to harvest 600, and that is your main goal.  How long should you stock the lake before switching to allowing permitted fishing?

e) Take the recommendation from part d.  What if you accidentally had overestimated your reproduction rate, and it is really only 15%?  What happens to your population long-term following the guideline recommended in part d?

Communicating the solution:

Consider the above, and write a 2-3 page, group report, outlining the management problem and your solution.  In so doing, take into consideration all of the mathematical work and discussion from above. Write the report as if you are writing a recommendation to the lake trust board, using summarizing graphs as necessary.  Submit through Canvas as a url (GoogleDocs).

Components of a “lite” management report for this lake (see rubric on Canvas):

1. Title describing the charge of the report

2. Background on the known information prior to the study.  Statement of intent for the report.

3. Investigation under financial constraints: (a-c)

4. Investigation under ecotourism constraints: (d)

5. Possible pitfalls, violation of assumptions (e)

6. Overall recommendation

7. Appendix which includes a summary of work (may be a separate file)  This part will not be graded, however, if there is a mistake in any of the calculations, this gives me the opportunity to go back and see how the mistake was made and determine appropriate partial credit.

Team work:  If you do not contribute in pre-defined, agreed upon ways, you will not receive credit.  Regardless of any individual’s participation, the entire team is responsible for handing in a full report. Please acknowledge all team members on the final report, and do the participation quiz after submitting the report.

This video walks through how to compute equilibrium for differential equations. 

Here is another educational video on different equations:

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.

Difference Equations are the basis for Discrete-Time dynamical systems Here is an Educational video on the topic

This video goes over 2 basic calculus concepts:

The blog posting tips are for how to write a reflective blog post in a mathematical setting: 

Tips for Creating Great Blog Posts in a Math Class

Judy Williams, Unity College

Note: Most of these ideas are for professional blogs, but can be used for reflectionary blog writing/journaling as well

• Use a short but effective title that hints at your controlling idea.

  • http://www.twelveskip.com/guide/blogging/1247/blog-post-title-templates-that-work

• Be creative about a hook to draw readers into your blog post.

  • http://beafreelanceblogger.com/blog-content-writing-hooks/

• Be sure you have a controlling idea and state it early in your blog post, maybe just after your hook.

• If there are multiple components to an assignment (e.g. an article to read, learning from your math class, and your own experience), do some prewriting on each separately and then look for connections.

• Clearly and concisely explain the similarities and differences for your readers.   Don’t presume they “get it.”

• Start with the mathematical concept you’re working on in class and show how the article illustrates that concept. Back this up with an example from your own life that also illustrates the concept.

• Use images, video, etc. to accompany text and break it up into accessible chunks. Make sure all images, etc., are visually appealing and have a clear connection to your topic.

  • https://blog.bufferapp.com/free-image-sources-list

• Use bullets, sub-headings, formatted text to make reading easy. Be concise! Avoid repetition!

  • http://writtent.com/blog/format-perfect-blog-post-10-tips/

• Return to your hook and central idea in closing.

  • http://www.copyblogger.com/closing-call-to-action/

• Invite comments. Consider including a call to action, if appropriate.

http://en.blog.wordpress.com/2012/04/12/how-to-get-more-comments/

A variety of graphs and calculations can come from the Christmas bird count It makes a good time series graph. 

Here is a link:

http://netapp.audubon.org/cbcobservation/

Nothing works better to check an answer then a software that can derive and intergrate almost anything 

Here is a link:

http://www.wolframalpha.com/

Khan Academy is great for learning mathematical based concepts.

Here is a link to it: 

https://www.khanacademy.org/

This link is to the hand out:

https://drive.google.com/a/unity.edu/file/d/0B2k1BmkveWWxZTQyZTQ0NzctYTBlNC00Y2RjLWIxYmYtMjg1NDkwMWMwMWU0/view

Colvin G. 2010. Talent is Overrated: What Really Separates World-Class Performers from Everybody Else. Penguin Publishing Group. 1-240.

This Book is about how practice makes perfect and can be directly applied to mathematics.

Here is a link to the book:

http://www.barnesandnoble.com/w/talent-is-overrated-geoff-colvin/1100833086

A great book for quick formative assessment techniques.

Tools I may want to use

Chapter from "Doing Bayesian Data Analysis: A Tutorial with R, JAGS, and Stan (Second Edition)" by John K. Kruschke

Proxy Access