Lab Index

A Computer Resources page is being developed to provide additional Lab material. There is a hyperlink to Maple on rohan. For grading of the Labs and Lab policy see the Lab Policy Page. Documentation is provided to create good graphs in Excel 2010. A copy of the cover sheet is available. Here are a couple special Excel spreadsheets: the Graphing Template and the Discrete Models.

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

Lab 1 (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. 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.

Lab 2 (Help Page)

1. Optimal Volume (A1). A box is formed from a rectangular piece of paper, and optimal dimensions are determined.
2. Optimal Tent Size (A4). A pyramidal shaped tent is cut from a square piece of canvas with maximal volume in two ways.
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.

Lab 3 (Help Page)

1. Optimal Foraging (A3). A study of seagulls dropping clams is examined for optimal foraging strategies.
2. Length of Day (B3). A cosine function is used to approximate the length of the day over a year.
3. Body Temperature and Menstrual Cycle (B4). A sine or cosine function is used to fit data on the body temperature variation of a woman over a month.

Lab 4 (Help Page)

1. Optimal Trough (D1). A trough with a cross-section in the shape of an isosceles trapezoid is optimized for volume.
2. Fourier Fit to Population (D3). Data on hares gathered by the Hudson Bay company are fit with Fourier series.
3. Tides (C2). Four cosine functions are fit to the October 2000 tide tables for San Diego and analyzed. Minima and maxima are explored.

Lab 5 (Help Page)

1. Atmospheric Pressure (F1). A simple model for atmospheric pressure is examined.
2. 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.
3. Radiocarbon Dating (E3). Radioactive decay of 14C can be used to date ancient objects, using a simple linear differential equation.

Lab 6 (Help Page)

1. Malthusian and Logistic Growth Models (G1). The solutions of these models are explored with their slope fields using Maple.
2. Nonlinear Cell Growth (G4). A culture of cells is growing in a nonlinear and time-dependent manner. Solutions are found exactly and numerically.
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 7 (Help Page)

1. Euler's and Improved Euler's Methods (F2). Numerical solutions of two differential equations are studied.
2. 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.
3. 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.

Lab 8 (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 9 (Help Page)

1. 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.
2. Cadium and Smoking (K2). 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.
3. Insect Population (I2). Polynomials and Fourier series are used to approximate a population survey. Definite integrals are used to find average populations.

Lab10 (Help Page)

1. Flight of a Ball (H3). The flight of a ball in two dimensions is studied for optimal distance and angle of trajectory.
2. Model for Gonorrhea (I5). Euler's method is used to examine a model for the spread of gonorrhea.
3. 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.

Lab11 (Help Page)

1. Malthusian and Logistic Growth (I3). The Malthusian and Logistic growth models are applied to data for cultures of Paramecium.
2. Pollution in the Great Lakes (F3). A simple model for build up and removal of toxic substances from the Great Lakes is studied.

Lab12 (Help Page)

1. Continuous Yeast Growth (L2). Data are fit for a growing culture of yeast. Derivatives are used to find the maximum growth in the population.
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. Nutrient Transport (I4). The effects of surface to volume ratio on limiting the growth of a cell is studied.
4. European Population Model (J1). A time-varying Malthusian growth model is used to help study the declining growth rates in several European countries.
5. Harvesting Fish Populations (J2). The logistic growth model with harvesting is studied for a population of game fish.
6. Spruce Budworm Outbreak (K1). A qualitative analysis of a differential equation that models the outbreak of the spruce budworm.