Week 6
Nonlinear models

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

Nonlinear models: Do we even need them? 📹 Screencast 6:03|📘 PDF: Oh yes we do! Many phenomena are nonlinear. Linear models are only common because they are easy to work with, not because the universe is mostly linear.
Function Fundamentals 📹 Pencast 10:30|📘 PDF: What even is a function? Functions as blocks. Function notation. Not all functions are equations. And vice versa. Is this curve a function? The vertical line test.
Function families: The big three 📹 Pencast 9:56|📘 PDF: Quadratics, Exponentials (and their kin Logarithmic and Logistic frunctions), and Sinusoids (a small subset of the Trignonometric functions).
Quadratic functions 📹 Pencast 6:13|📘 PDF: You'll encounter them in three common forms. Each makes different information about the function apparent.
Example: From English to equation (Charity event) 📹 Pencast 6:40|📘 PDF: What kind of a word problem gives rise to a quadratic?
Data to functions 📹 Pencast 6:25| 📘 PDF: Two points specify a line. What about a parabola? An exponential function? A sinusoid? In general you need as many points as your function has parameters.
Graphing by the method of successive transformations (GBTMOST) 📹 Pencast 16:42| 📘 PDF: This is the rare skill that will impress your future colleagues! 😎 Look at an equation, identify the family, and quickly sketch it to show them what it looks like.
GBTMOST: The reciprocal transformation 📹 Pencast 6:31| 📘 PDF: Most of the transformations we make use of in GBTMOST are simple. Dealing with reciprocals takes a bit more time, but can be done systematically in 3 steps:
1) Circle points where y = 1 or -1;
2) Draw vertical dashed lines where $y=0$ (to show where there will be asymptotes); and
3) Fill in the regions between these by having the reciprocal do the opposite of the function (i.e.function goes up⇒reciprocal goes down, function goes down⇒reciprocal goes up).
Handout: Transformations cheatsheet 📘 PDF: A compact summary of the effect of the common transformations on a function.
GBTMOST: Four examples 📹 Pencast 13:57| 📘 PDF: 1. Understanding $y=mx+b$.
2. $y=3(x+4)^2-2$
3. $y=-5×2^{-x}+3$
4. $ y=-3\sin(x-(\pi/2)+5 $

Exercises

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

Exercise Set 6 Part 1: GBMOST Problems and Solutions.
Exercise Set 6 Part 2: Quadratic models Problems and Solutions.

Supplementary resources

Function transformations covers most of the transformations with nice diagrams (but doesn't do f+g, f-g and 1/f).

5.1 Quadratic Functions Textbook treatment of quadratic functions from OpenStax Algebra and Trigonometry by Jay Abramson and others.

Wednesday

About cancelling in fractions 📹 Pencast 6:58| 📘 PDF: Why is it that we can cancel things on the tops and bottoms of fractions? How do we do it? Watch out for plus signs!
Laws of exponents 📹 Pencast 4:37| 📘 PDF: All of them.
Exponential model forms 📹 Pencast 9:41| 📘 PDF: You will encounter exponential models expressed in three equivalent, but different-looking, forms. Here they are.
Example: Bacterial culture 📹 Pencast 9:25| 📘 PDF: Bacteria multiply very quickly. See last year's class' estimates, then see the actual answer.
Example: Daily improvement 📹 Pencast 4:09| 📘 PDF: If you could improve at something by just 1% a day how much better would you get at it in a year?
Example: Radium-226 📹 Pencast 6:21| 📘 PDF: While bacterial populations grow, radioactive elements decay, but changing the sign on x changes the formula for exponential growth into one for exponential decay.
The forgotten truth about logarithms 📹 Pencast 2:51| 📘 PDF: Logarithms are just exponents! Honest.
What base should I use? Can I change my mind? 📹 Pencast 6:13| 📘 PDF: Computer scientists say 2, because then information is measured in bits. Natural scientists say 10 because they're familiar with scientific notation. Mathematicians say e, because it simplifies working with derivatives and integrals as we will be doing for the next three weeks. It's easy to change from one base to another, but many people forget how.
Laws of Logarithms 📹 Pencast 5:30| 📘 PDF: All the rules for logarithms are just restatements of the rules for exponents. Total plagiarism. 🙃
Inverse functions 📹 Pencast 5:34| 📘 PDF: A function's inverse undoes whatever it does. And vice versa.
$f^{-1}(x)$ is not $1/f(x)$: In a really unfortunate choice of notation, $x^{-1}$ means $1/x$, but $f^{-1}(x)$ means $f$'s inverse function NOT $1/f(x)$. I apologize on behalf of mathematicians. 🥸
Logs and exponentials are inverses 📹 Pencast 9:18| 📘 PDF: The fact that logs and exponentials are inverses gives us our main tools for manipulating them algebraically.
Solving for x in log and exponential equations 📹 Pencast 8:23| 📘 PDF: How to solve for x when it is:
1) in the base of an exponential expression → use a fractional power (root)
2) in the exponent of an exponential expression, → take logs
3) inside a logarithm → exponentiate the equation, or use the definition of the logarithm.
Example: Candy bar inflation 📹 Pencast 8:22| 📘 PDF: When I was a boy candy bars cost $0.10. How much will they cost in 2050?
Example: Yukon population growth 📹 Pencast 8:01| 📘 PDF: The news last week was that the Yukon's population has increased 21% in the last ten years to 42,052. When will it reach 100,000?
Doubling time and half-life 📹 Pencast 5:55| 📘 PDF: It turns out exponential quantities double or halve in constant time amounts, i.e. they double or halve every n time units.
Example: Plutonium decay 📹 Pencast 5:31| 📘 PDF: How to work with a half-life value.
Example: Logistic function plot 📹 Pencast 8:15| 📘 PDF: Our pièce de résistance of GBTMOST: Sketching the logistic function.
Example: Bird population recovery 📹 Pencast 9:12| 📘 PDF: Applying the logistic function to an island bird population.
Example: Modelling a lobster harvest 📹 Screencast 5:33: Using Desmos to fit a function by eye. Desmos is a an online graphing tool we will be using a lot in this segment of the course.

Back to modelling: A summary

Choose the form of model to use (e.g. linear, quadratic, exponential, log, logistic).

Q. How do we choose a model form?
A: By recognizing its fundamental shape if you have a bunch of data. If you only have a few data points then theory and experience are your guides.
Fit the model, i.e. find the model parameters.
If you have just a few points, substitute as many as necessary into your model form, until you have as many equations as unknowns, and then solve for the model's parameters.
If you have a bunch of points, fit the model interactively by eye, using a tool like Desmos.
Use the model.
Now that you have a specific model you can use it to calculate the values you need to answer the question you have.

Exercises

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

Exercise Set 6 Part 4: Sinusoidal models Problems and Solutions.

Supplementary resources

Amplitude, Period, Phase Shift and Frequency provides a good overview of the general sine function and the role of its parameters.

All the exercises in one place

Exercise Set 6 Part 1: GBMOST Problems and Solutions. Due by 4 pm Monday February 15.

Exercise Set 6 Part 2: Quadratic models Problems and Solutions. Due by 4 pm Monday February 15.

Exercise Set 6 Part 3: Exponential models Problems, Solutions, and Narrated solutions to problems 5-8. Due by 4 pm Monday February 15.

Exercise Set 6 Part 4: Sinusoidal models Problems and Solutions. Due by 4 pm Monday February 15.

Assignment

Assignment 5 is posted. Due by 4 pm Friday February 26. 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 A5_firstname.pdf, e.g.

A5_Tim.pdf

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

A5_Tim_1.pdf
A5_Tim_2.pdf

Week 6Nonlinear models