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Calculus & Mathematica: 151 Course Description
Derivatives: Measurments of Growth Rate
1.01 Growth
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.
1.02 Exponential Growth
Mathematics
How to write exponential and logarithm functions in terms of the natural base e. 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, and inflation.
1.03 Instantaneous Growth
Mathematics
The instantaneous growth rate f'[x] as the limiting case of the average growth rates (f[x+h] - f[x])/h. Calculation of f'[x] for functions f[x] like xk, Sin[x], Cos[x], e, and Log[x]. Why Log[x] is the natural logarithm and why e is the natural base for exponentials. What it means when f'[x] is positive or negative. Max-min.
Science and Math Experience
Relating the plots of f[x] and f'[x]. Using a plot of f'[x] to predict the plot of f[x]. Visualizing the limiting process by plotting f'[x] and (f[x+h} - f[x])/h on the same axes and seeing the plots coalesce as h closes in on 0. Spread of disease model. Instantaneous growth rates in context.
1.04 Rules of the Derivative
Mathematics
The derivative as the instantaneous growth rate. Chain rule. Product rule as a consequence of the chain rule. Instantaneous percentage growth rate 100 f'[x]/f[x] of a function f[x].
Science and Math Experience
Another look at why exponential growth dominates power growth and why power growth dominates logarithmic growth. Logistic model of animal growth. The idea of linear 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 adolescent growth spurt is probably a mathematical fact instead of a biological accident. Compound interest. Making functions with prescribed instantaneous percentage growth rate.
1.05 Using the Tools
Mathematics
What it means when f'[x]
0 for x = a. Why f[x] is not as big (or small) as possible at x = a unless f'[a] = 0.
Science and Math Experience
Why a good representative plot of a given function f[x] usually includes all x's at which f'[x] = 0. Max-Min in one or two variables.. Using the derivative to get best least squares fit of data by smooth curves. Fitting of Space shuttle O-ring failure data as a function of temperature and using the result to explain why the Challenger disaster should have been predicted in advance. Data fit by lines and by Sine and Cosine waves. Optimal speed for salmon swimming up a river. Designing 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 x/e^x as x advances from 0 to infinity.
1.06 Differential Equations of Calculus
Mathematics
The three differential equations y'[x] = r y[x], y'[x] = r y[x] (1 - y[x]/b) y'[x] = r y[x] + b and their solutions. The meaning of the parameters r and b in the three differential 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 y'[x] = r y[x]. 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. Managing an inheritance. Wal-Mart sales. Pollution elimination, data analysis, speculating on why dogs and humans grow faster after their birth than they are at the instant of their birth, but horses grow fastest at the instant of their birth. Newton's law of cooling. Pressure altimeters.
1.07 Race Track Principle
Mathematics
The race track principles:
If f[a] = g[a] and f'[x]
g'[x] for x
Euler's method of faking the plot of a function with a given derivative explained in terms of the race track principles. Euler's method of faking the plot of a solution of a differential equation explained in terms of the race track principles.
Science and Math Experience
Using the race track principle to explain why, as x advances from 0, the plots of solutions of y'[x] = r y[x] and y'[x] = r y[x] (1 - y[x]/b) will run close together in the case that y[0] is small relative to b. Why Sin[x]
x for x
1.08 More Differential Equations
Mathematics
Plots of numerical approximations to solutions of first order differential equations. Qualitative analysis of first order differential equations and systems of first order differential 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.
1.09 Parametric Plotting
Mathematics
Parametric plotting of curves in two dimensions. Parametric plotting of curves and surfaces in three dimensions. Derivatives for curves given parametrically.
Science and Math Experience
Circular parameterization (polar coordinates) and other parameterizations. Projectile motion. Cams designed by sine and cosine wave fit. Predator-prey plotting. Parametric plotting 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 with application to spraying insecticides.
Significant Changes From the Traditional Course
- Student writing, plotting and experimentation is the stock in trade of the course. Calculus is experienced as a course in measurements heavily intertwined with other parts of science and the world.
- Emphasis on linear and exponential growth from the beginning before calculus begins. Linear functions are those with constant growth rates; exponentials are those with constant percentage growth rates.
- Students learn at the very beginning that exponential growth dominates power growth without appeal to the mysticism of L'Hospital's rule or any other calculus ideas.
- Functions are not studied for their own sake, but rather for the measurements they make. Students analyze real data.
- The derivative is introduced as a measurement of the instantaneous growth rate. As a result, the idea that functions with positive derivatives are increasing functions available to the student immediately without waiting for the Mean Value Theorem. The interpretation of the derivative as the slope of the tangent line is delayed.
- Students work with real world data on applications they find to be important. Financial calculations recur on a regular basis.
- Students learn the meaning of the derivative as a measurement at the same time they are learning to calculate derivatives. This idea is reinforced by many student-produced plots and analyses of the graphs of f[x] and f'[x] on the same axes.
- Although there is no formal "epsilon-delta" presentation of limits, students experience the limiting process visually by plotting the average growth rates f[x + h] - f[x])/h and the instantaneous growth rate f'[x] as functions of x and watch what happens to the plots as they make h close in on 0.
- The active form of the mean value theorem called the Race Track Principle introduced. Euler's method is explained in terms of the Race Track Principle.
- Following Poincare, differentiation of functions two variables with respect to each variable is done with no particular fanfare.
- Students do serious work with mathematical models involving derivatives. The benefits are twofold: Working with the models reinforces the idea of what the derivative is and the students can experience the tentacles of calculus in outside the traditional calculus classroom.
- Biological models are favored over physical models at the beginning because the derivative measures growth and growth is a natural biological process.
- Linear dimension makes a decisive entrance into calculus.
- Logistic growth is studied in some detail.
- Qualitative analysis of the solutions simple differential equations and the solutions of simple systems of differential equations enters a calculus course for the first time. Reasons: Studying them reinforces the meaning of the derivatives and they beautifully show the scope of calculus in science. Students experiment with predator-prey, spread of infection and Lanchester war models and try to explain the results in terms of derivatives.
- Parametric plots in two and three dimensions are studied in the first course because they provide the student with needed plotting freedom for the what's to come.
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