Week 6
Modelling change: Derivatives

Business

The start of the week email.

⦿ Monday: Sketching derivatives; Basic rules for differentiation

Two fundamental concepts Pencast 10:59|PDF
Two tools from calculus, 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.
Summary: Sketching the derivative
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: Rules for differentiation

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 other 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 \(f(g(x))\).
Summary: Differentiation rules
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: Examples; Fine Print; Applications

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 wear out equipment or 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 intersection of two lines, or 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 prosper. 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.

Assignment

Assignment 6 is posted. Due by 1 pm Monday February 23. 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