Week 9
Applications of the derivative

Business

The start of the week email. No need to reread it if you received it. It's just linked here in case anyone accidentally deleted it (it's happened before 🥺). Definitely let me know if you didn't receive it though so we can figure out why not.

Monday: Optimization

Review (from Week 7): Finding maxima and minima

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?

Wednesday: Phase space

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.

Friday: Differential Equations

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.

The mathematical landscape 📹 Pencast 5:41 |📘 PDF

A quick attempt to put the math you have learned in this course (and your life before this course!) into context.

All the exercises in one place

All due by 4 pm Monday March 8.

Exercise Set 8 Part 1: Optimization Problems and Solutions.

Exercise Set 8 Part 2: Phase space Problems and Solutions.

Exercise Set 8 Part 3: Differential equations Problems and Solutions.

Assignment

Assignment 7 is posted. Due by 4 pm Friday March 12. Note that the problems are worth different numbers of marks. Use your time wisely.

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