Growth and Accumulation
Authors:  Bill Davis, Horacio Porta and Jerry Uhl  © 1999
Producer:  Bruce Carpenter
Publisher:         Distributor:  

Mathematics.

Line functions and polynomials. Interpolation of data. Compromise lines through data. Dominant terms in the global scale.

Science and math experience.

Reading plots. Linear models. Drinking and driving.  Japanese economy cars versus American big cars. Data analysis and interpolation. Data analysis of U.S. national debt and U.S. population in historical context, including plots of yearly growth and the effect of immigration on the growth of the U.S. population. Cigarette smoking and lung cancer correlation. Global scale of quotients of functions studied by looking at dominant terms in the numerators and denominators.

Mathematics.

How to write exponential and logarithm functions in terms of the natural base . While line functions post a constant growth rate, exponential functions post a constant percentage growth rate. How to construct a function with a prescribed percentage growth rate.

Science and math experience.

Recognition of exponential data, exponential data fit, carbon dating, credit cards, compound interest, effective interest rates, financial planning, decay of cocaine in the blood, underwater illumination, inflation.

Mathematics.

The instantaneous growth rate  as the limiting case of the average growth rates . What it means when  is positive or negative. Calculation of  for functions  like , , ,  and .  Why  is the natural logarithm andwhy  is the natural base for exponentials. . Max-min.

Science and math experience.

Relating the plots of  and . Using a plot of  to predict the plot of . Visualizing the limiting process by plotting  and  on the same axes and seeing the plots coalesce as  closes in on . Spread of disease model. Instantaneous growth rates in context.

Mathematics.

The  derivative as the instantaneous growth rate. Chain rule. Product rule as a consequence of the chain rule. Instantaneous percentage growth rate of a function .

Science and math experience.

Another look at why exponential growth dominates power growth and why power grow th dominates logarithmic growth. Logistic model of animal growth. The idea of li near dimension and using it to convert a model of animal height as a function of age to a model of animal weight as a function of age. Learning why the adololes cent growth spurt is probably a mathematical fact instead of a biological accide nt. Compound interest. Making functions with prescribed instantaneous percentage growth rate.

Mathematics.

What it means when  for . 
Why  is not as big (or small) as it can be at  unless .

Science and math experience.

Why a good representative plot of a given function  usually includes all 's at which . Max-Min in one or two variables.. Using the derivative to get best least s quares fit of data by smooth curves. Fitting of Space shuttle O-ring failure dat a as a function of temperature and using the result to explain why the Challenge r disaster should have been predicted in advance.  Data fit by lines a nd by Sine and Cosine waves. Optimal speed for salmon swimming up a river. Desig ning the least cost box to hold a given volume. Analysis of an oil slick at sea. How tall is the dog when it is growing the fastest? Analysis of what happens to  as advances from  to .

Mathematics.

The three differential equations 
   &nb sp;   ,
       
&nb sp;      
and t heir solutions.
The meaning of the parameters  and  in the three differen tial equations. Why it's often a good idea to view logistic growth as toned down exponential growth.

Science and math experience.

Models based on these differential equations. Why radio active decay is modeled by the differential equation .  Logistic versus exponential growth. Biological principles behind carbon dating. Growth of U.S. and world populations: Malthusian versus logistic models. Calculation of interest payments resulting from buying a car on time. Ma naging an inheritance. Wal-mart sales. Pollution elimination, data analysis, spe culating on why dogs and humans grow faster after their birth than they are at t he instant of of their birth, but horses grow fastest at the instant of their bi rth.  Newton's law of cooling. Pressure altimeters.

Mathematics.

The race track principles:
If  and  for , then  for .
If  and  is approximately equal to  for , then  is approximately equal to  for .
If  and  for , then  for .
Eul er's method of faking the plot of a function with a given derivative explained i n terms of the race track principles.
Euler's me thod of faking the plot of a the solution of a differential equation explained i n terms of the race track principles.

Science and math experience.

Using the race track principle to explain why, as  advances from , the plots of solutions of 
      & nbsp; and  
will run close together in the case th at  is small relative to . Why  for  and related inequalities. Estimating how many accurate decimals of  are needed to get  accurate decimals of . The error function. Calculating accurate v alues of  and .

Mathematics.

Plots of numerical approximations to solutions of first order differential equat ions.  Qualitative analysis of first order differential equations and systems of first order differentitial equations.

Science and math experience.

Analysis of the predator-prey model. Cycles in the predator-prey model. Drinking and driving model. Variable interest rates. Michaelis-Menten Drug equation. War games based on Lanchester war model including a simulation of the Battle of Iwo Jima. Harvesting in the logistic model. SIR epidemic model. The idea of chaos.

Mathematics.

Parametric plotting of curves in two dimensions. Parametric plotting of curves a nd surfaces in three dimensions. Derivatives for curves given parametrically.

Science and math experience.

Circular parameterization (polar coordinates) and other parameterizations. Proje ctile motion. Cams designed by sine and cosine wave fit. Predator-prey plotting. Parametric plottting of circles and ellipses. Elliptical orbits of planets and asteroids.  Plotting of circles, tubes and horns centered on curves in three dimensions.  Equilibrium populations in the predator-prey model . Modifications of the predator-prey model. The effect of poisoning predators wi th application to spraying insecticides.

Mathematics.

Integrals defined as area measurement as done in E. Artin's MAA notes written in the 1950's. Approximations by trapezoids.

Science and math experience.

Integrals of functions given by data lists. Using known area formulas for triang les, trapezoids and circles to calculate integrals. Odd functions. Trying to bre ak the code of the integral by taking selected functions , putting
      & nbsp; 
and plotting        and  .
on t he same axes for small 's. Plotting and guessing a fo rmula for . Plottingt and guessing a formula for .  Estimating the acr eage of farm field bordered by a river.

Mathematics.

If  is given by  then .
The fundamental formula .

Science and math experience.

Relating distance, velocity and acceleration through the fundamental formula. Ge tting the feel of the fundamental formula by using it to calculate integrals by hand. Relating
        
to the solution of the differential equa tion 
        with .
Very brief look at the "indefinite integral,"  
Measuring area between curves. T he error function, , and other functions defined by integrals.  Measurements of acculumulated g rowth. Coloring ceramic tiles.  

Mathematics.

Measurements based on slicing and accumulating: Area and volume; density and mas s. Measurements based on approximating and measuring: Arc length. Measurements b ased on the fundamental formula: Accumulated growth.

Science and math experience.

Volumes of solids with no special emphasis on solids of rotation. Volume measure ments of curved tubes and horns. Eyeball and precise estimates of curve lengths.   Filling water tanks. Harvesting corn. Voltage drop. Another look at linear dimension. Work. Present value of a profit-making scheme.  Catf ish harvesting. Designing an 8 fluid ounce logarithmic champagne glass.