Project Help Notebook by Keith Stroyan

Help NoteBook Copyright 1998 by Academic Press, Inc. - All rights reserved.
Modify this Notebook to prepare your project report.

Project 6: The Expanding Economy

by A. Student and B. Pupil

with technical help from Da LabMonitor and Help Notebook

My Project Introduction

Write this last.  Begin with the basic question and state your main finding clearly, but with few technical details.  You might even modify the title of this section to make it more interesting.

Beginning Explorations or Computer Experiments

Hint 6.1

Read the hint and the preceding text in the printed project.  Compare that with the calculations below.  Write an explanation of the gerneral computations below and illustrate them with specific numerical examples.

Variables

t = time in years

p = population in number of people

e = average per capita expenditure on energy in $/year

T = total yearly expenditure on energy in $

General total expenditure as a product

Whatever the function of time that gives us  e = e[t]  and  p = p[t], the total expenditure is the product.  We define this in Mathematica by the command that follows.

\(Clear[p, e, T, t]; 
T[t_] := \ e[t]*p[t]\)

Initial example

Explain what this does

p[0]  =  15000000e[0]  =  1000. T[0]

First sample change

Explain what this does

p[1]  = p[0]  +  10000e[1]  =  e[0] +  8\(	T[1]\)

First sample change by arithmetic

Explain why this is the sams as...

(p[0]  +  10000) (e[0] +  8)

General changes

Following are symbolic expressions for new values of   e  and  p  in terms of the old value and a change.  Notice that we do not use the usual Mathematica sytax  f[x_] = (expression in x), but leave the underscore out.  This makes a geneeral experssion when we don't have a formula. You might prefer to simply type the algebra for this project and not use Mathematica to perform the symbolic computations for you.

p[t + Δt]  =  p[t]  +  Δpe[t + Δt]  =  e[t]  +  ΔeT[t + Δt]Expand[T[t + Δt]]

Change in total expenditure

T[t + Δt] - T[t]ΔT = Expand[T[t + Δt] - T[t]]

Change in energy expenditure for existing people

p[t] (e[t] + Δe)  -  p[t] e[t]Expand[p[t] (e[t] + Δe)  -  p[t] e[t]]

New people's old energy

Δp e[t]

Explain the terms in the formula for change in energy in terms of

old peoples new energy

new peoples old energy

new peoples new energy

Hint 6.2

Read the hint the printed project.

Hint 6.3

Read the hint the printed project.  Explain how we get the formula for the instantaneous time rate of change of total energy expenditure,

(dT/dt) =  p[t] (de/dt) +  e[t] (dp/dt)

In particular, what happens to the term representing the rate of change of new people's new energy as the change in time tends to zero?

Δ(T/Δ)tExpand[Δ(T/Δ)t]