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 .