Week 7
Applications of the derivative

Business

The start of the week email.

⦿ Monday: Optimization

Review (from Week 6): Finding maxima and minima

The derivative is the key to finding the maxima and minima of non-linear functions because it enables us to identify where these extreme values might be found.

The Extreme Value Theorem guarantees us that every continuous function on a closed interval has both a maximum and a minimum value. It guarantees us that they exist, but doesn't tell us where to find them. So where do we need to look? There are three locations we need to check (not unlike the way we had to check the vertices of the feasible set in LP problems).

  1. Stationary points, i.e. places where the value of the derivative is 0.
    → We find these by taking the derivative, setting it to 0, and solving.
  2. "Other critical points", i.e. places where the derivative does not exist, i.e. cusps, vertical tangent lines, vertical jumps, and horizontal gaps.
    → We find these by looking at the formula for the derivative for operations that might produce non-real values, e.g. a division by 0, or taking an even root of a negative number.
  3. The endpoints of the interval.
    → These are sometimes given explicitly in the problem, but if not are inferred from the problem domain, e.g. if we are working with physical percentages then the range limits are 0 and 100.
Example: How large a web to weave? Pencast 11:37 |PDF
A spider needs to weave a web large enough to catch insects, but not so large that it expends more energy constructing it than it can recoup by eating its catch.
Example: Territorial birds Pencast 13:32 |PDF
A territorial bird needs a territory large enough to meet its needs. But defending that territory takes time and energy. What is the sweet spot in territory size?
Example: Cough-cough Pencast 10:22|PDF
We cough to expel matter caught in our throats. To do so we constrict our trachea to increase the momentum of air moving through it, the better to carry away the offending matter. How much should we constrict it? Do our bodies know this?
Example: The perfect moose Pencast 18:46|PDF
What is the best size for a moose to be?

Exercises

If you're comfortable with the topics above you're ready to work on:

Supplementary resources

In Calculus: Early Transcendentals by D. Guichard
§5.2 Extrema of a Function (a more formal way of saying "finding maxima and minima").
§5.7 Optimization Problems.

In Calculus Volume 1 by Gilbert Strang and Edwin Herman
§4.3 Maxima and Minima
§4.7 Applied Optimization problems

⦿ Wednesday: Phase Space and Differential Equations

The mathematical landscape Pencast 5:41 |PDF
Differential equations Pencast 14:11 |PDF
Differential equations (DEs) are by far the most common mathematical tool used to model dynamic systems. Unfortunately there is no single technique for solving them. Instead their solutions rely on a wide variety of disparate techniques (and many cannot be solved symbolically at all!). Paradoxically while they are difficult to solve, it is relatively easy to test a proposed solution to see if it is correct. And that is how we will dip our toes into the DE waters: by learning how to test solutions.
A quick attempt to put the math you have learned in this course (and your life before this course!) into context.
Lynx-hare dynamics Screencast 5:07

The predator-prey relationship between Lynx and Snowshoe Hare is one of the most famous and most studied. And originally developed using data from the northern fur trade!

The Lynx sensitivity and adaptations to hare populations are remarkable.

Phase space Pencast 5:37 |PDF
Phase spaces are the standard tool used to model dynamic systems' behaviour. A phase space is a space (think coordinate system) in which all possible states of a system can be represented. The dynamic behaviour of the system can then be described as a trajectory or path through its phase space.
Lynx-hare model Pencast 12:56 |PDF
A common sense statement of the interactions between Lynx and Snowshoe Hare results in a simplified Lotka-Volterra model that reproduces the dynamics observed in wild Lynx-Hare populations.
Lynx-hare: But what does the data say? Slide Show | PDF
Great model! But does it have anything to do with reality? Let's sanity check our model, by seeing what the trajectories of actual lynx-hare populations look like. Do they match our model's output?
Isoclines Pencast 14:47 |PDF
Isoclines (or nullclines) are lines along which a population does not change. The system equilibirum occurs where the isoclines intersect. Sketching the isoclines gives us a quicker way of determining the system behaviour.
Modelling general two-species competition Pencast 15:12 |PDF
Our Lynx-hare model is a simplified instance of the general category of two-species competition. Here's a more complete model, and a survey of the system dynamics it can capture.
Phase space examples Slide Show | PDF
Dynamic systems have many behaviours beyond that of our Lynx-hare stable system. Here's a look at a few others including 3-D, and bistable systems.

Exercises

If you're comfortable with the topics above you're ready to work on:

Supplementary resources

The first section of A quick guide to sketching phase planes shows how to combine changes in the x and y directions. (Note that they refer to isoclines as nullclines.)

7.2: Phase Space Visualization does not take quite the same approach we did, but might be a useful supplementary approach. (Note that they refer to isoclines as nullclines.)

Section 5-6 : Phase Plane has some nice phase plane sketches, but done in the context of general differential equations.

We are dipping our toes so shallowly into differential equations that I hope no supplementary resources will be required. If you are curious about DEs and would just like to know a bit more about them in general the Wikipedia article Differential equation is a good starting place.

⦿ Friday: Heritage Day Holiday

Assignment

Assignment 7 is posted. Due by 1 pm Monday March 2.

Submit PDF files of work done by hand, or in a word processor. (Let me know if you are unsure how to create these.) Email is preferred, but you can also submit your work through Moodle.
Excel or other binary files should be sent as file attachments.
Google sheets can be submitted by sharing a link to them.
Work done in Desmos can be submitted by sharing a link to your saved graph.

Name the file you submit A7_firstname.pdf, e.g.

A7_Tim.pdf

If you have to submit multiple files, add a suffix number to each, e.g.

A7_Tim_1.pdf
A7_Tim_2.pdf