Calculus Lab: 3/08/2022

An important recap:
  • One way to interpret the derivative is as the instantaneous rate of change of a function (with respect to the independent variable).

  • The sign of the derivative is key:
          When \(f^\prime > 0\), \(~ f \mbox{ is } \uparrow\).     When \(f^\prime < 0\), \(~ f \mbox{ is } \downarrow\).

 

Overall objective of this lab: To design "rate of change" models for different types of growth processes.

(I) Modeling some simple rates of growth

Consider 2 different applications:
  1. Total (cumulative) sales at the Earlham coffeeshop (Cafe 1847) as a function of time:
    \(t\) = time in days, \(S(t)\) = total sales in $.
  2. Total amount in a certificate of deposit (CD) account as a function of time: \(t\) = time in days, \(A(t)\) = total amount in $.
Explorations:

Note: Some of these explorations will ask you to conjecture things about the underlying application that you have no knowledge or information about. This is not a problem for our purposes here. Even without detailed knowledge, we can gain an amazing amount of insight by thinking in broad terms, for instance, about the type of functions involved -- linear, quadratic, exponential, trigonometric, etc. In addition, we're not asking you to use any knowledge about compound interest formulas, or exponential growth formulas that you may have seen before. The goal here is to simply relate changes in the underlying function to its derivative. E.g., \(S'(t)=4\) means that \(S\) is increasing at the rate of $4 per day; \(S'(t)=0.3S\) means \(S\) is increasing by 30% each day.
[Pause and test your understanding: What does \(S'(t)=-4\) mean? What about \(S'(t)=-4 + 0.3S\) mean?   Or, \(S'(t)=-4 - 0.3S\)?]

  1. Sketch graphs showing the qualitative behavior of the 2 functions \(S(t)\), \(A(t)\). Use your intuition and general knowledge to do this. Give a brief explanation for why the choices you made seem reasonable to you.

  2. The rate of change of functions is often easier to estimate than their actual values. For each case above, think of (and discuss!) the factors that cause the function to change from one day to the next.
    Coffeeshop: Suppose today the sales were $4200. How much might they be tomorrow? Why? The point here is to consider average or "typical" scenarios -- not minor fluctuations or weird exceptions!
    CD account: Today there is $100 in it. By how much would it grow tomorrow? Why?

  3. The average change in one day can be a good approximation to the instantaneous rate of change. Use your answers from (2) to write a mathematical statement of the rate of change for each of the functions \(S(t)\) and \(A(t)\). In other words, we want to find expressions of the form: \(S'(t)=\) stuff, and \(A'(t)=\) stuff. Here "stuff" can be a constant, or a function of \(t\) and/or a function of the y-value itself (\(S\) or \(A\)).
    E.g., \(S'(t)=4\)   or   \(S'(t)=4t-3\)   or   \(S'(t)=0.3S\)
Here is an example showing how your lab report could discuss these explorations:
We will begin with a series of explorations for modeling the growth rates in 2 relatively simple applications: ....
  1. ...
  2. ...
Let \(s(t)\) denote the total cumulative sales in dollars at Cafe 1847. The graph below shows a plausible relationship between \(s\) and \(t\). This form of the relationship makes sense if we assume ...
...

To be more specific suppose, for example, today's sales at Cafe 1847 were $4200. If we assume today was a typical day at the cafe, then $4200 would be reasonable to take as the average daily sales ...
...

The average change over a short period may be a good approximation for the instantaneous rate of change. With that in mind, the derivative of \(s(t)\) might be reasonably approximated by \(s^\prime(t) =\) .....

(II) Modeling the population growth of Richmond

Objective: To use current population data for modeling future trends in the population of Richmond, Indiana.

Strategy and data that we will assume:

We want to model the population \(P\) as a function of time \(t\) in years. But, it is nearly impossible to directly model population growth with any accuracy. Instead we will focus on modeling the derivative, since changes in population are much easier to model.

To summarize, we want to find: \(P'(t)=\) a function of \(P,t\) and other stuff.

The minimum data we will require is current population, together with an estimate of births and deaths. We will assume:

* At present (i.e, \(t=0\)), population = 36,000.
* Number of births in the past year = 720.
* Number of deaths in the past year = 570.

Warmup exercise: Suppose \(P'(4)=186\). Interpret what that would mean in this application context and give its units.

Population Model A:

Suppose the only source of population change is births and deaths, and there are no constraints of any kind on how large the population can get.

  1. Estimate the annual growth rate at \(t=0\) by making use of the data we assumed above. Give reasons. Notice that this value, in calculus lingo, is \(P'(0)\).
  2. In the absence of any constraints, the population will grow in the exact same way as a CD in a bank account! That means the rate of change \(P'\) will be proportional to the current population \(P\). Use this fact to build a model for \(P'\). In other words, we know that \(P'=kP\) for some constant \(k\). Find \(k\) and explain how you found it.
  3. Use the Apple Grapher or Sage (see below) to solve your model and graph the result. Predict the population 100 years from now. [NOTE: You may need to ask me how to solve your model on the grapher -- it is easy to do, but the process is not obvious. Here is a screenshot showing how to implement and solve a similar model in the Apple Grapher.]

    An alternative to using the Apple Grapher is a free online software option called the Sage cell server. It looks like the box below. It offers a very convenient way to do short computations and graphics without ever leaving this page. The code shown in the box below runs Population Model A. Fill in the missing parameter values (k =,   p0=), then click "Evaluate" and see!

Population Model B:

Now we will assume there are some constraints on how large the population can grow. In particular, let us assume the town has some maximum "carrying capacity" due to a variety of factors (housing, transportation, schools, etc.). Suppose the carrying capacity is 200,000 people.

The annual growth rate at \(t=0\) is known from Model A, and will not be affected by the carrying capacity. We also know the value of the constant \(k\) in the previous model (i.e., \(P'=kP\)), which will be useful for building our new model.

A simple way to account for the carrying capacity is to extend the previous model as follows:
                                \(P'=kP(1-P/C)\)
where \(C\) is the carrying capacity and \(k\) is the same constant as before.

  1. Explain how/why this model accounts for a carrying capacity of \(C\).
    Hint: Think of the sign of \(P'\), and how it is affected when \(P\) changes relative to \(C\). What happens when \(P<\)\(C\), \(P>\)\(C\), \(P=C\)?
  2. Use the Apple Grapher (or the Sage code below) to solve your model, and predict the population 100 years from now.
Just FYI: Here is a screenshot showing how to implement and solve a similar model in the Apple Grapher.

The Sage code shown in the box below will run Population Model B after you fill in the 3 missing parameter values (k =,   p0 =,   cc =).   Enter those values and click "Evaluate" to see results!

(III) A review of trigonometry via guided exploration

(No need to turn in for grading. But please be sure to complete by March 11 class, since lots of items in future classes will depend on your understanding of these basic ideas.)
We want to prepare ourselves to do calculus with trigonometric functions. From previous experience, it is known that many students need to refresh their skills in trig. Thus it is important that everyone work through this review, which is no more than a bare-bones minimum to try to get you more comfortable.

Work through this list of guided explorations.