Here is a link to the pdf:

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

Video on Determining derivatives using the power rule:

Here is a link to the pdf:

https://drive.google.com/a/unity.edu/file/d/0B2k1BmkveWWxc2pqNGhtdFVRVUtSRVNBTjhFdzBiUQ/view

A worksheet for derivative calculation:

Lab 3 Part I, Calculus I

Unity College, Carrie Diaz Eaton

Consider an environmental indicator which can be described by the following equation:

P(t)=et

where t is in years.

1. Sketch a graph of P(t) and from that, sketch what you expect P’(t) to look like.

2. Write an equation for P(t+∆t)

3. Try to calculate the derivative algebraically using the limit definition of derivative

4. Create a spreadsheet that calculates the derivative numerically at 21 different points of the graph:  t = 0, .5, 1, 1.5, … 10

(Hint: Create a table for with 21 rows (t = 0, .5, 1, 1.5, … 10) and 8 columns: t,P(t),  then for∆t=1 P(t+∆t), P/∆t=[P(t+∆t)-P(t)]/∆t, then for∆t=.1 P(t+∆t), P/∆t=[P(t+∆t)-P(t)]/∆t, and for∆t=.01 P(t+∆t), P/∆t=[P(t+∆t)-P(t)]/∆t.)

5. Results Graph a scatterplot of P(t) and your estimates of P’(t) based on the numerical work performed in #4.

6. Discussion Do the visual results based on visual, algebraic and/or numerical work agree?

7. Discussion Contrast the initial and long-term value of the instantaneous rate of change.  

Discussion Suppose a different function was hypothesized for P(t), such as t^2 or e^(-t).  How would you modify your spreadsheet to approximate the graph of the derivative?

Educational video on how to get a tangent line equation:

This educational video is on Taking derivatives :

Another video on more advanced derivatives:

video on the basics of derivatives:

This Khan Academy video is to help better understand the equation for a derivative determining the slope at any given time:

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.