Math 122 - Calculus for
Fall Semester, 2012
1999, All Rights Reserved, SDSU & Joseph M. Mahaffy
San Diego State University --
This page last updated 31-Oct-12
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.
This hyperlink goes to the Main
Lab Page (access to old labs of previous years).
Below is a list of the labs and
a brief summary of the problems.
1 (Help Page)
- 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.
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 (Help Page)
- Optimal Volume
(A1). A box is formed from a rectangular piece of paper, and
optimal dimensions are determined.
- Optimal Tent Size (A4). A pyramidal shaped tent is cut from a
square piece of canvas with maximal volume in two ways.
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.
3 (Help Page)
Foraging (A3). A study of
seagulls dropping clams is examined for optimal foraging strategies.
- Length of Day
(B3). A cosine function is used to approximate the length of the
day over a year.
- 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.
4 (Help Page)
- Optimal Trough (D1). A trough with a cross-section in the shape of an isosceles trapezoid is optimized for volume.
Fit to Population (D3). Data on
hares gathered by the Hudson Bay company are fit with Fourier series.
- Tides (C2). Four cosine functions are fit to the October 2000 tide tables for San Diego and analyzed. Minima and maxima are explored.
5 (Help Page)
Pressure (F1). A simple model for atmospheric pressure is examined.
- 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.
Dating (E3). Radioactive decay
can be used to date ancient objects, using a simple linear differential
6 (Help Page)
and Logistic Growth Models (G1).
The solutions of these models are explored with their slope fields
- Nonlinear Cell Growth (G4). A culture of cells is growing in a
nonlinear and time-dependent manner. Solutions are found exactly and
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.
7 (Help Page)
and Improved Euler's Methods (F2).
Numerical solutions of two differential equations are studied.
- 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.
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.
8 (Help Page)
of E. coli (H1).
Two theories for the growth of the cytoplasm or mass of bacteria are
Exposure in Children (H2).
Differential equations are used to find the level of lead in children
during their early years.
- European Population Model (J1). A time-varying Malthusian growth model is used to help study the declining growth rates in several European countries.
9 (Help Page)
- 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.
- 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.
- 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)
of a Ball (H3). The flight of a
ball in two dimensions is studied for optimal distance and angle of
- Model for
Gonorrhea (I5). Euler's method is used to examine a model for the
spread of gonorrhea.
(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)
and Logistic Growth (I3). The
Malthusian and Logistic growth models are applied to data for cultures
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)
Yeast Growth (L2). Data are fit for a growing culture of yeast.
Derivatives are used to find the maximum growth in the population.
- 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.
Transport (I4). The effects of surface to volume ratio on limiting
the growth of a cell is studied.
Population Model (J1). A
time-varying Malthusian growth model is used to help study the
declining growth rates in several European countries.
Fish Populations (J2). The
logistic growth model with harvesting is studied for a population of
Budworm Outbreak (K1). A
qualitative analysis of a differential equation that models the
outbreak of the spruce budworm.