Week 7
Modelling change: Derivatives
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
- Two fundamental concepts 📹 Pencast 10:59|📘 PDF
- Two tools from caculus, the derivative and the integral, let us model change and accumulation respectively.
- Sketching the derivative 📹 Pencast 13:45|📘 PDF
- We begin by learning how to sketch the derivative of a function.
- Handout: Sketching the derivative 📘 PDF
- Summary of the steps to sketch a derivative.
- Example: Sketching the derivative of a semicircle 📹 Pencast 4:53|📘 PDF
- An example of sketching the derivative of a semicircle: \(y=\sqrt{c^2-x^2}\)
- Notation and terminology 📹 Pencast 5:47|📘 PDF
- New concepts require new terminology and notation to talk about them and write them down.
- Differentiation rules: The basics 📹 Pencast 19:45|📘 PDF
- Sketching derivatives is easy enough, but the result is a picture not an equation. How do we get the equation of a function's derivative? By applying rules that specify algebraic transformations of the function into its derivative. This first installment covers the basics which are enough to handle polynomial functions.
Exercises
If you're comfortable with the topics above you're ready to work on:
Supplementary resources
I am not aware of a good textbook source for sketching derivatives. Typical calculus texts tell you the derivative is the slope of the tangent line and then assign a wack of sketching problems as exercises and assume students will sort it out. Perhaps because the textbooks don't show how, if you google "sketching derivative" you'll get a slew of youtube videos of people showing you how.
There are two reasonable introductory calculus texts that cover the rest of this week's material.
In Calculus: Early Transcendentals by D.
Guichard
§4.3 covers the basic rules up to polynomials, and has further examples (and provides derivations/proofs
of the rules).
In Calculus Volume 1 by Gilbert Strang and
Edwin Herman
§3.3 covers the basic rules up to polynomials, and has further examples (and provides derivations/proofs
of the rules).
Wednesday
- The derivatives of \(\sin x\) and \(\cos x\) 📹 Pencast 7:28|📘 PDF
- Spoiler alert: The derivative of \(\sin x\) is \(\cos x\) and the derivative of \(\cos x\) is \(-\sin x\).
- The derivative of \(e^x\) 📹 Pencast 5:45|📘 PDF
- Spoiler alert: \(e^x\) is its own derivative! The derivative of \(b^x\) is \(b^x \ln b\). Mathematicians prefer to use \(\ln\) to avoid the accumulation of those arbitrary \(\ln b\) terms.
- The derivative of \(\ln x\) 📹 Pencast 4:36|📘 PDF
- Unexpected! The derivative of \(\ln x\) is \(\frac{1}{x}\). Using a base otehr than \(e\) again generates annoying constants because the derivative of \(\log_b x\) is \(\frac{1}{x \ln b}\).
- Differentiating function combinations: Multiplication and division 📹 Pencast 11:50|📘 PDF
- The differentiation of sums and differences aligned with our intuitions: differentiate them seprately and add or subtract the results. Not so with products and quotients. These rules are unintuitive!
- Differentiating function combinations: Compositions 📹 Pencast 8:48|📘 PDF
- The Chain Rule tells us how to differentiate compositions of functions, i.e. when one function is applied to another, like \(f(g(x))\).
- Handout: Differentiation rules 📘 PDF
- All the rules summarized. Includes brief examples.
- Examples 📹 Pencast 11:04|📘 PDF
- A bunch of examples showing how to apply the rules to functions you might meet.
Exercises
If you're comfortable with the topics above you're ready to work on:
Supplementary resources
In Calculus: Early Transcendentals by D.
Guichard
§4.3 Covers the basic rules up to polynomials, and has further examples (and provides derivations/proofs
of the
rules).
§4.4 does trig functions.
§4.5 Chain rule
§4.6 Log and Exponential functions
In Calculus Volume 1 by Gilbert Strang and
Edwin
Herman
§3.3 Covers the product and quotient rules
§3.5 Covers derivatives of trigonometric functions (including sin and cos).
§3.6 Covers the Chain Rule for handling compositions.
Friday
- More examples 📹 Pencast 12:50|📘 PDF
- Because you can't have too many examples when you're learning something so new.
- Example: Manufacturing antibiotics 📹 Pencast 10:31|📘 PDF
- Producing more doesn't always mean making more profit. Sometimes high levels of production wewar out equipment ro require overtime pay, etc. This problem looks at the nonlinear optimization problem of maximizing the profit from producing an antibiotic.
- The Extreme Value Theorem 📹 Pencast 6:41|📘 PDF
- The Extreme Value Theorem guarantees us there is both a minimum and a maximum under certain circumstances. Unfortunately it doesn't tell us where to find them. A little common sense suggests the three places we should look:
1) stationary points (where \(f'(x)=0\)),
2) other critical points (anywhere \(f'(x)\) doesn't exist), and
3)the endpoints of the interval under consideration. - Existence of derivatives 📹 Pencast 7:00|📘 PDF
- We can go looking for derivatives, but must they exist? Not always. (Just like we can look for the roots of a quadratic, but they may not exist.)
- Example: The Allee effect 📹 Pencast 13:47|📘 PDF
- Exponential and logistic growth models assume even tiny populations can grow and proser. The ecologist Warder Allee suggested that there are sizes below which populations cannot recover. He has a model!
- Example: Blood alcohol levels 📹 Pencast 19:00|📘 PDF
- You have a glass of wine and alcohol enters your bloodstream. How does it do that? Quickly? Gradually? Gradually then quickly? When does the concentration peak? All questions for derivatives to answer.
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 Covers finding maxima and minima
In Calculus Volume 1 by Gilbert Strang and
Edwin
Herman
§4.3 Covers finding maxima and minima.
All the exercises in one place
All due by 4 pm Monday March 1.
Exercise Set 7 Part 1: Problems and Solutions.
Exercise Set 7 Part 2: Quadratic models Problems and Solutions.
Exercise Set 7 Part 3: Exponential models Problems and Solutions.
Assignment
Assignment 6 is posted. Due by 4 pm Friday March 5. 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 A6_firstname.pdf, e.g.
A6_Tim.pdf
If you have to submit multiple files, add a suffix number to each, e.g.
A6_Tim_1.pdf
A6_Tim_2.pdf