Exponential function Lab.pdf

The exponential function and some of its
uses in oceanography
•What is the exponential function?
•Three examples of its use in Oceanography
•Growth of phytoplankton.
•Attenuation of light.
•Phytoplankton growth and light.

The exponential function:
f(x)=Ae
(Bx) , e=2.71828183
2.71828183…
Note: any power function can be
translated to exponentials:
a x =(e ln(a) ) x = e xln(a)
= 1+
x
+
x
2
2
+
3
x
3!
+ ....
e x ( )
•Derivative proportional to the function
which means in solve the differential
equation:
df x
dx
=
Bf
( x)

Probabilities and the exponential function:
You have 100 pennies.
1. Toss them and take out all that are ‘head’.
2. Keep tossing and write down the number of
heads you got per toss.
3. When done, use Excel plot the number of
heads you got as a function of tosses.
4. Fit an exponential function to the data.
What is the exponent of the best fit? Is it
sensible?
5. ln{1/2}=-0.69.
6. What if you had 100 dice. How would the
curve change?

Coin toss results:
Probabilities and the Exponential Function
120
100
80
y = 92.634e -0.6586x
R 2 = 0.9947
Group 1
60
40
20
0
0 1 2 3 4 5 6 7 8
Toss #
Probabilities and the Exponential Function
120
100
80
y = 98.185e -0.6612x
R 2 = 0.9547
Group 2
60
40
20
0
0 1 2 3 4 5 6 7 8
Toss #

Phytoplankton growth:
You have a series of data from a
phytoplankton growth experiment (thank you
Kate) where chlorophyll fluoresence (chl
F )
was used as an indicator of phytoplankton
biomass.
day
1 2 6 8 10
Chl F
0.34
1.17
16.53
30.1
37.4
Plot the data of chl F as function of time
and ‘best’ fit:
P(t)=P(t=0)e μt
How would you go about calculating the
doubling (halving?) time of the
phytoplankton?
Doubling time:
P(t 2 )=P(t=0)exp(μt 2 )=2P(t=0)
exp( exp(μt 2 )=2 t 2 =ln(2)/μ.
ln(2)~0.7

Phytoplankton growth results:
Phytoplankton Growth
80
70
60
y = 0.3386e 0.5305x
R 2 = 0.9315
50
40
30
20
10
0
0 2 4 6 8 10 12
Days
Doubling time = 1.08 days

Phytoplankton growth:
How fast to phytoplankton concentration
increase during the spring bloom?
1. GoMOOS data from E01 for 2007.
2. Based on the chlorophyll data estimate the
growth rate? µ = 0.2962
Phytoplankton Growth rate (GOMOOS data)
14
12
10
y = 0.8377e 0.2962x
R 2 = 0.9564
8
6
4
2
0
0 1 2 3 4 5 6 7 8 9 10
Day

How does it compare to lab cultures? What
may be the reason for
agreement/disagreement?
The growth rate is much slower than lab
cultures. Some reasons for the discrepancy
include: limited light/nutrients and/or
predation by grazers.

Transmission of light:
Beer’s s law and the exponential function:
Measure the intensity of light at the receiver when
only water is in the tank.
Add drops of dye (such that total drops in the tank= 1,
2, 4, 8, 16, 32). Measure light intensity in each case.
Light meter
Light source

Plot the ratio of light intensity with dye divided by
light intensity in tap water as function of drops of
dye.
Transmission of Light
1600
1400
1200
y = 952.05e -0.1678x
R 2 = 0.9123
1000
800
600
400
200
0
0 2 4 6 8 10 12 14 16 18
# of drops
Could you use your results to predict how
many drops of dye are in another tank?
Yes!
This is the basis of spectroscopy!

Scientists use underwater
spectrophotometer to investigate the
ocean:

Light attenuation in the environment:
E(λ,z)=E(λ,0)e (-k(λ)z)
Blue ocean
Coastal ocean
Inland pond

Basic
instrument to
measure water
quality. The
secchi disk:
Secci depth: Depth at which disk disappears.

Saturated growth:
How much carbon is fixed at a given light level?
P=P max (1-exp(-I/I k ))
P max =P max (T,N)

Summary:
•The exponential function can be used to describe many
processes in the ocean. Knowing it and how to
manipulate it allows one to better predict and
understand the environment.
•It turns out that e x with x imaginary is simply related
to the sine and cosine functions simplifying much the
proof of trigonometric identities facilitate the
formulation and study of waves.
•If learning by humans followed the model:
Dk/dt=μk
imagine what a little change in knowledge acquisition
rate (μ) can do to society!
•But remember, things can’t keep growing exponentially
forever; after a finite time we reach the carrying
capacity of the system/environment…

Teaching the exponential function
Resource: http://faculty.gvsu.edu/goldenj/exponential.html
Some highlights
The exponential function and probability:
http://faculty.gvsu.edu.goldenj.badpenny.html
Exploring growth patterns:
http://score.kings.k12.ca.us/lessons/growth.html
In marine sciences we see the exponential function showing up
in many applications:
1. Light decrease with depth in the ocean.
2. Phytoplankton growth when supplied with a given light
level of light and nutrients.
3. Nuclear decay.