Orchestrating the FTC

Conversation:Explore, Prove, Apply

Brent FergusonThe Lawrenceville School, NJ

BFerguson@Lawrenceville.org

Purposes of today’s talk• To initiate some reflection on one of the ‘big ideas’

in mathematics (FTC) and to suggest that multiple, rigorous approaches will enhance understanding.

• To show some ways in which we can stimulate interdisciplinary connections with FTC, and teach a few non-mathematical lessons along the way.

• To share some tasks that bring attention to the importance of the idiosyncratic but useful notation and conventions of our discipline. This will help make the case that doing math well requires careful reading.

Speaking of reading, here’s a gem …from over a century ago

This excerpt (1911) precedes the cognitive science of today

that has only served to verify the points Whitehead makes about working memory space, etc.

Brilliant…

From Introduction to Mathematics, A. N. Whitehead, which is vol. 15 of the “Home University Library of Modern Knowledge” series (1911)

Sequence

• Some language considerations• A classical approach:

– FTC-II generated by considering Euler’s linearization method for approximating functions, then a definition of the definite integral

– proving FTC-I using that definition

• Making connections: Algebra I class, personal development, identity formation, and growth mindset

The Fundamental Theorem?Let’s look at a familiar item first…

• “The Distributive Property”: (“distribution” instead)– …of multiplication over addition?

– …of multiplication over multiplication?

– …of addition over multiplication?

– …of exponentiation over multiplication?

– …of exponentiation over addition?

– …of differentiation over addition? …over multiplication?

It makes a difference if we take the time to disabuse, to explain, to point out the use of structure and notation!

• …of Arithmetic: (unique factorization theorem) – every positive integer can be written uniquely in standard form as a product of primes.

• …of Algebra: A polynomial of degree n has n roots.

• …of Calculus: (1) integration and differentiation are inverse operations, and (2) antidifferentiation, evaluated at two inputs, can be used to calculate a definite integral.

“The Fundamental Theorem”

(1) and (2)

+ + . . . + + = for complex

where , , , etc. through all primes.

Development of the Concept

• Context for consideration: https://

www.youtube.com/watch?v=o2xmAWS4akE

• Riemann Sums, using data from helmet speedometer.

• This leads us to ask questions…– How accurate was our answer? –Was it likely an under- or overestimate?– Can we do better? How (or why not)?

• Definition time: the definite integral…

FTC-II can emerge from a linearization (Euler’s method*)… or simply by defining

Since each represents a rate at which is changing at a particular time within a time interval , that means that each product is a small bit of change in the quantity over the time interval .

Then would be an estimate for on a given interval , the total (summed) change of on when it is divided into n smaller subintervals.

Want a better estimate? Use a larger n. Want an exact calculation? Consider ‘taking it to the limit’ for larger and larger n: , or more properly…(drumroll)

FTC-II emerges…first!

Then letting means that , which (by definition of ) .

FTC-I emerges next, by proof…

If we let , the accumulated signed-area between the t-axis and on the interval , we can investigate , the rate at which changes with respect to x.

(insert here: proof by formal limit definition of derivative A’(x))

Result: . Of course, by FTC-II, , as expected … and this approach helps us with other forms.

Observations…

• Differentiation and integration are inverse functions. (This is a big deal!)

• The independent variables t & x are a bit tricky.

• Order of operations matters. Things get a little funky if you switch it around (let’s look at some examples).

Fun with the F.T.C.

(a)

(b)

(c)

(d)

(e)

(f)

Fun with the F.T.C.(a) =

(b) =

(c) =

(d) =

(e)

…why?

(f) =

A Calculus ‘poem’

The RACE CARD project (http://theracecardproject.com/) asks participants to address race by describing themselves with a 6-word (max) ‘essay.’ But they didn’t limit non-verbal symbols, so…I cheated. After my six words, I added the FTC, applied to a community of changing humans.

Fundamentally: differentiation, integration…inverses.

Together: IDENTITY.

This got me thinking…I am a changing person, and my community is changing, too. Calculus gives us a language with which to express our change in progress. Who are you?

• I am today who I was yesterday, plus some changes.

(follow-up question…at what rate am I changing, really…?)

• I am this year who I used to be, plus some changes.• My school this year is what it used to be…plus girls!

From algebra class: . Better yet, . …this one goes out to all the statisticians in the audience!

Better still, .In function notation: .

Linearization as an approximating model

But in Calculus, we can explore much more complicated functions – not just those with a constant rate of change:The linear approximation to at gives us:

…this takes us to Euler’s method.

* Euler’s method as a connection to FTC

Why approximate when you can calculate more precisely?

Then , or ...as needed!

Other practices for success in Calculus

• Math histories: students write 2-4 pages about their relationship with math over the years…a trove of information!

• Formative feedback, decoupled in time from delivery of grades.

• Required revisitation of all test problems, and modeling/practice in class: homework revision.

• Keen attention in the first weeks to the form of students’ written homework, and commentary on how and why to use the notation effectively.

• Strong exhortation to use study groups and TALK mathematics…again, this is modeled and practiced in class as a way to equip students.

• Music, videos, and bad jokes distributed liberally.

A Message on ‘Fit’

“Good teachers join self and subject and students

in the fabric of life.”

–Parker Palmer, p.11, The Courage to Teach

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Brent Ferguson, The Lawrenceville School

BFerguson@Lawrenceville.org

Taylor’s Theorem (time permitting): a proof with asynthesis of FTC, product rule, & substitution

Begin with: (NOTE: This means that ; we’ll use this below…)

Taylor’s Theorem: an ‘integrative’ proof

Recall that starting from: , we generated , from the previous page:

=…and so on: