Joseph M. Mahaffy
 Math 124: Calculus for the Life Sciences Spring 2016
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Computer Laboratory for Math 124

A Lab Manual is being developed with much of the first part completed and online. All labs need to begin with the Lab Cover Page. For grading of the Labs and Lab policy see the Lab Policy Page. Also, there is a Laboratory Guidelines page for more details on what is expected. Documentation is provided to create good graphs in Excel 2010. Also, there is additional documentation on what is expected through the link to the Good Graph Document. Here are a couple special Excel spreadsheets: the Graphing Template and the Discrete Models. For help with Maple, there are a couple of Maple help sheets: Maple Help document and Introduction to Maple.

Below is a list of the labs and a brief summary of the problems.

Lab 1 (Help page)

1. Intersection of Line and Quadratic (A2). Graphing a line and a quadratic and finding significant points on the graph.
2. Cricket Thermometer (A3). Listening to crickets on the web, then using a linear model for relating to temperature.
3. Weak Acids (C2). Solving for [H+] with the quadratic formula, then graphing [H+] and pH.

Lab 2 (Help page)

1. Lines, Cubic and Rational Functions (CD1). Introduction to Maple for solving equations. Graphing and finding points of intersection, asymptotes, and intercepts.
2. Concentration and Absorbance (B2). Linear model for urea concentration measured in a spectrophotometer. Relate to animal physiology.
3. Growth of Yeast (C3). Linear model for the early growth of a yeast culture. Quadratic to study the least squares best fit.

Lab 3 (Help page)

1. Exponential, Logarithm, and Power Functions (E1). Study the relative size of these functions. Finding points of intersection.
2. Dog Study (D3). Use an allometric model to study the relationship between length, weight, and surface area of several dogs.
3. Allegheny Forest (E3). Model volume of trees as a function of diameter or height. Compare linear and allometric models.

Lab 4 (Help page)

1. Length of Day and Temperature (B5). A sine or cosine function is used to approximate the length of the day and average temperature over a year for a particular city.
2. Malthusian Growth and Nonautonomous Growth Models (F4). Census data for a particular country is analyzed for trends in their growth rates. Models are compared and contrasted to data, then used to project future populations.
3. Malthusian Growth (F2). Data for two countries presented with a discrete Malthusian growth model used for analysis.

Lab 5 (Help page)

1. Model for Breathing (G2). Examine a linear discrete model for determining vital lung functions for normal and diseased subjects following breathing an enriched source of argon gas.
2. U. S. Census models (H3). The population of the U. S. in the twentieth century is fit with a discrete Malthusian growth model, a Malthusian growth model with immigration, and a logistic growth model. These models are compared for accuracy and used to project future behavior of the population.
3. Weight and Height of Girls (I2). Data on the growth of girls is presented. Allometric modeling compares the relationship between height and weight, then a growth curve is created.

Lab 6 (Help page)

1. Tangent Lines and Derivative (J1). Secant lines are used, then the limit gives the tangent line. Rules of differentiation are explored.
2. Growth of Fish (I4). Use von Bertalanffy's equation for estimating the length of fish with some fish data to find growth in length of a fish.
3. Oxygen consumption of Triatoma phyllosoma (J2). Cubic polynomial is fit to data for oxygen consumption of this bug. The minimum and maximum are found.

Lab 7 (Help page)

1. Female Body Temperature and the Menstrual Cycle (J6). A cubic polynomial model and a sine or cosine are fit to data on the female body temperature over one month. Timing of ovulation is related to points of inflection, and the maximum and minimum temperatures are found.
2. Radioactive Isotopes (K6). Certain radioactive isotopes are used for medical imaging. Exponential functions are used to study the decay of these isotopes. The derivative is used to find a maximum and point of inflection.
3. Fourier Fit to Population (D3). Data on lynx or hares gathered by the Hudson Bay company are fit with a series of trigonometric functions, providing increasing accuracy with additional functions.

Lab 8 (Help page)

1. Growth of Pacific Fish (L1). The von Bertalanffy equation is used to find the length of Pacific fish, then an allometric model relates the length to the weight. The chain rule of differentiation is used to find the maximum weight gain as a function of age.
2. Tumor Growth (K5). The growth of a tumor is studied by creating the logistic and Gompertz growth functions from tumor data, then these models are simulated and compared to the literature.
3. Optimal Volume (A1). A box is formed from a rectangular piece of paper, and optimal dimensions are determined.

Lab 9 (Help page)

1. Optimal Trough (D1). A trough with a cross-section in the shape of an isosceles trapezoid is optimized for volume.
2. Optimal Tent Size (A4). A pyramidal shaped tent is cut from a square piece of canvas with maximal volume in two ways.
3. Optimal Foraging (A3). A study of seagulls dropping clams is examined for optimal foraging strategies.

Lab 10 (Help page)

1. Updating functions for Beetle Populations (L4). The updating functions for the logistic, Beverton-Holt, Ricker's, and Hassell's models are compared to beetle data and studied using the tools from the course. Discrete simulations are run to compare to data.
2. Discrete Models for Birds (L2). Discrete models for the growth of a population of birds is studied. The models that are compared are the logistic growth model, logistic growth model with emigration, and a cubic model with the Allee effect.
3. Continuous Yeast Growth (L2). Data are fit for a growing culture of yeast. Derivatives are used to find the maximum growth in the population.

Lab 11 (Help page)

1. Cell Study (F4). Compute the volume and surface area of different cells, then study their growth with a Malthusian growth law. Learn more about exponential growth testing a statement by Michael Crichton.
2. Malthusian and Logistic Growth Models (G1). The solutions of these models are explored with their slope fields using Maple.
3. Newton's Law of Cooling (G2). Newton's law of cooling is applied to a situation where a cat is killed by a car, and the time of death needs to be found.

Lab 12 (Help Page)

1. Growth of E. coli (H1). Two theories for the growth of the cytoplasm or mass of bacteria are compared.
2. Lead Exposure in Children (H2). Differential equations are used to find the level of lead in children during their early years.
3. European Population Model (J1). A time-varying Malthusian growth model is used to help study the declining growth rates in several European countries.

Lab 13 (Help Page)

1. Cadium and Smoking (K1). The cumulative exposure to cadium is explored over many years. The effect of this carcinogen is analyzed for a nonsmoker exposed through diet and a smoker, where Cd is absorbed through the lungs.
2. Blood Flow in an Artery (J4). Poiseuille's law for flow of fluids is applied to small arteries. Integrals are used to derive relationships for the velocity of blood in arteries.
3. Insect Population (I2). Polynomials and Fourier series are used to approximate a population survey. Definite integrals are used to find average populations.
1. Plankton in the Salton Sea (J3). The logarithm of the populations are found, then fit with a quartic polynomials. Extrema are found to find peak populations.
2. Tides (C2). Four cosine functions are fit to the October 2000 tide tables for San Diego and analyzed. Minima and maxima are explored.
3. SIR Model for Influenza (L3). A discrete dynamical system with susceptible and infected individuals is compared to CDC data for the spread of influenza. The model is used to examine different strategies to lessen the effect of the disease.
4. Carbon Monoxide in a Room (I1). Machinery produces CO, which builds up in a room. Exposure levels are found by solving a differential equation exactly and numerically.
5. Predator-Prey (J3). The Lotka-Volterra model is studied with data on a specific predator and prey system. Parameters are fit to the model, and the model is analyzed.

Other

1. Island Biodiversity (E2) Fit an allometric model through data on herpetofauna on Caribbean islands.
2. Malthusian Growth Model for the U. S (F1). Java applet used to find the least squares best fit of growth rate over different intervals of history. Model compared to census data.
3. Bacterial Growth (G1). Discrete Malthusian and Logistic growth models are simulated and analyzed.
4. Logistic Growth for a Yeast Culture (H1). Data from a growing yeast culture is fit to a discrete logistic growth model, which is then simulated and analyzed.
5. Logistic Growth Model (H2). Simulations are performed to observe the behavior of the logistic growth model as it goes from stable behavior to chaos.
6. Flight of a Ball. Data for a vertically thrown ball is fit, then analyzed (I1). Average velocities are computed for insight into the understanding of the derivative.
7. Circadian Body Temperature (J4). A cubic polynomial is fit to data for human body temperature as it varies over a 24 hour period. A maximum and minimum are found.
8. Graphing a polynomial times an exponential (K1). Graphing the function and its derivative. Maple is used to help find extrema and points of inflection for this function.
9. Drug Therapy (K3). Models comparing the differences between drug therapies. One case considers injection of the drug, while the other considers slow time release from a polymer.
10. Radiocarbon Dating (E3). Radioactive decay of 14C can be used to date ancient objects, using a simple linear differential equation.
11. Euler's and Improved Euler's Methods (F2). Numerical solutions of two differential equations are studied.
12. Atmospheric Pressure (F1). A simple model for atmospheric pressure is examined.
13. Pollution in the Great Lakes (F3). A simple model for build up and removal of toxic substances from the Great Lakes is studied.
14. Drug Absorption (G3). Two models for drug absorption are examined to show the difference between injected drugs and ones delivered using a polymer delivery system.
15. Nonlinear Cell Growth (G4). A culture of cells is growing in a nonlinear and time-dependent manner. Solutions are found exactly and numerically.
16. Flight of a Ball (H3). The flight of a ball in two dimensions is studied for optimal distance and angle of trajectory.
17. Malthusian and Logistic Growth (I3). The Malthusian and Logistic growth models are applied to data for cultures of Paramecium.
18. Model for Gonorrhea (I5). Euler's method is used to examine a model for the spread of gonorrhea.