Download Mathematics for Biology - Farmingdale State College

Mathematics for the
Biosciences at
Farmingdale State
Sheldon Gordon
[email protected]
Major Premise
“Biology will do for mathematics in
the twenty-first century what physics
did for mathematics in the twentieth
century”
Major Premise
Almost all math and bio projects start at
the calculus level or above.
But the overwhelming majority of
beginning biology students, both majors
and non-majors, typically are at the college
algebra or precalculus level.
Most of these students have avoided math
as much as possible.
Our Project
Our original plan was to begin developing
the first stages of a new mathematics
curriculum to serve the needs of biology
students, both the bioscience majors and
the non-majors.
This would also impact the level of
quantitative work in the biology courses.
Our Project
The project is a collaborative effort
between Farmingdale State University and
Suffolk Community College.
Farmingdale State brings an exceptional
history of successful efforts in reforming
the mathematics curriculum.
Suffolk brings an outstanding record of
utilizing technology and quantitative
methods in its introduction biology courses
and labs.
The Farmingdale State Project
Our first step was to develop an alternative
to our modeling-based precalculus course
that would focus almost exclusively on
biological applications.
The course would feature a lab component
taught by the biology faculty, so that each
week’s primary math topic would be
accompanied by an experiment requiring
the use of that mathematical method.
Course Topics
Week 1 Behavior of Functions
Week 2 Families of Functions,
Linear Functions
Week 3 Linear Functions
and Linear Regression
Week 4 Exponential Growth
and Decay Functions
Week 5 Exponential Regression
and Power Functions
Week 6 Power Functions and
Polynomials
Intro to Measurements
and Measuring
Linear Growth – part 1
Linear Growth – part 2
Exponential Growth
Exponential Decay
part 1
Exponential Decay –
part 2
Course Topics
Week 7
Week 8
Week 9
Week 10
Week 11
Week 12
Week 13
New functions from Old
Logistic and Surge
Functions
Matrix Models and
Linear Systems
Sinusoidal Functions
and Periodic Behavior
Periodic Functions –
part 2
Probability Models
Probability Models and
Difference Equations
Power Function growth
Logarithmic Functions
Logistic Growth
Surge Functions
Polynomial growth
Periodic Behavior
Probability Model
(genetics)
What Happened Next
To accommodate the lab component, we
had to change the precalculus course from
four to five credits.
Because of that and conflicts with other
courses (intro chemistry), the biology
students did not register for the course and
it did not run.
What We’ve Done Instead
All the labs in the introductory biology
course are in the process of being changed
to dramatically increase the level of
quantitative experience – the new labs will
incorporate most of the labs intended to be
part of the precalculus course.
The math department has officially created
a new four credit precalculus course to
serve the needs of the biology students – the
same math course, but no lab.
What We’ve Done Instead
The math department has created a new
two-semester calculus sequence for biology
students – it will emphasize concepts over
manipulation and will stress biological
applications.
The math department has created a onesemester post-precalculus mathematical
modeling in the biological sciences course
for bioscience majors and applied math
majors.
Some Sample Problems
Identify each of the following functions (a) - (n) as linear,
exponential, logarithmic, or power. In each case, explain
your reasoning.
(m)
(g) y = 1.05x
(h) y = x1.05
x
y
(n)
x
y
0
0
3
5
1
(i) y =
(0.7)t
(j) y =
v0.7
1
(l) 3U – 5V = 14
7
2
2
(k) z = L(-½)
5.1
7.2
9.8
3
3
9.3
13.7
Some Sample Problems
Biologists have long observed that the larger the area of a
region, the more species live there. The relationship is best
modeled by a power function. Puerto Rico has 40 species
of amphibians and reptiles on 3459 square miles and
Hispaniola (Haiti and the Dominican Republic) has 84
species on 29,418 square miles.
(a) Determine a power function that relates the number of
species of reptiles and amphibians on a Caribbean island
to its area.
(b) Use the relationship to predict the number of species of
reptiles and amphibians on Cuba, which measures 44,218
square miles.
Island
Area
N
Redonda
1
3
Saba
4
5
Montserrat
40
9
Puerto Rico
3459
40
Jamaica
4411
39
Hispaniola
29418
84
Cuba
44218
76
Number of Species
The accompanying table and associated
scatterplot give some data on the area (in
square miles) of various Caribbean islands
and estimates on the number species of
amphibians and reptiles living on each.
100
80
60
40
20
0
0
15000
30000
Area (square miles)
45000
(a) Which variable is the independent variable and
which is the dependent variable?
(b) The overall pattern in the data suggests either a
power function with a positive power p < 1 or a
logarithmic function, both of which are increasing and
concave down. Explain why a power function is the
better model to use for this data.
(c) Find the power function that models the
relationship between the number of species, N, living
on one of these islands and the area, A, of the island
and find the correlation coefficient.
(d) What are some reasonable values that you can use
for the domain and range of this function?
(e) The area of Barbados is 166 square miles.
Estimate the number of species of amphibians and
reptiles living there.
The ocean temperature near New York as a function of
the day of the year varies between 36F and 74F.
Assume the water is coldest on the 40th day and
warmest on the 224th.
(a) Sketch the graph of the water temperature as a
function of time over a three year time span.
(b) Write a formula for a sinusoidal function that
models the temperature over the course of a year.
(c) What are the domain and range for this function?
(d) What are the amplitude, vertical shift, period,
frequency, and phase shift of this function?
(e) Estimate the water temperature on March 15.
(f) What are all the dates on which the water
temperature is most likely 60?