Orchestrating the FTC
Conversation:
Explore, Prove, Apply
Brent Ferguson
The 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?
• 𝑎 ∙ 𝑏 + 𝑐 =? ≠ 𝑎 ∙ 𝑏 + 𝑎 ∙ 𝑐
– …of multiplication over multiplication?
• 𝑎 ∙ 𝑏 ∙ 𝑐 =? ≠ 𝑎 ∙ 𝑏 ∙ 𝑎 ∙ 𝑐
– …of addition over multiplication?
• 𝑎 + 𝑏 ∙ 𝑐 =? ≠ 𝑎 + 𝑏 ∙ 𝑎 + 𝑐
– …of exponentiation over multiplication?
•
𝑏∙𝑐
𝑛
=? ≠ 𝑏 𝑛 ∙ 𝑐 𝑛
– …of exponentiation over addition?
•
𝑏+𝑐
𝑛
=? ≠ 𝑏𝑛 + 𝑐 𝑛
– …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!
“The Fundamental Theorem”
• …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.
𝑁=
∞
𝑖=1
𝑝𝑖 𝑒𝑖 = 𝑝1
𝑒1
∙ 𝑝2 𝑒2 ∙ 𝑝3 𝑒3 ∙ . . . where 𝑝1 = 2, 𝑝2 = 3, 𝑝3 = 5, etc. through all primes.
𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + 𝑎𝑛−2 𝑥 𝑛−2 + . . . + 𝑎1 𝑥 + 𝑎0 = 𝑎𝑛 ∙ 𝑥 − 𝑟1 ∙ 𝑥 − 𝑟2 ∙ 𝑥 − 𝑟3 ∙ . . . ∙ 𝑥 − 𝑟𝑛
for complex 𝑟𝑖 .
(1)
𝑑 𝑥
𝑓
𝑑𝑥 𝑎
𝑡 𝑑𝑡 = 𝑓 𝑥
and
(2)
𝑏
𝑓′
𝑎
𝑥 𝑑𝑥 = 𝑓 𝑏 − 𝑓 𝑎
Development of the Concept
• Context for consideration:
• Riemann Sums, using data from helmet
speedometer.
• This leads us to ask questions…
https://www.youtube.com/watch?v=o2xmAWS4akE
– 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
𝑏
method*)… or simply by defining 𝑎 𝑓′ 𝑥 𝑑𝑥
Since each 𝑓′ 𝑡𝑖 represents a rate at which 𝑓 𝑡 is changing
at a particular time 𝑡𝑖 within a time interval ∆𝑡𝑖 , that means
that each product 𝑓′ 𝑡𝑖 ∙ ∆𝑡𝑖 is a small bit of change in the
quantity 𝑓 𝑡 over the time interval ∆𝑡𝑖 .
Then 𝑛𝑖=1 𝑓′ 𝑡𝑖 ∙ ∆𝑡𝑖 would be an estimate for ∆𝑓 𝑡 on a
given interval 𝑎, 𝑏 , the total (summed) change of 𝑓 𝑡 on
𝑎, 𝑏 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: ∞
𝑖=1 𝑓′ 𝑡𝑖 ∙ ∆𝑡𝑖 , or more properly…(drumroll)
FTC-II emerges…first!
𝑛
∆𝑓 𝑡 𝑜𝑛 [𝑎, 𝑏] = lim
𝑛→∞
𝑖=1
𝑓′ 𝑡𝑖 ∙ ∆𝑡𝑖
𝑏
Then letting 𝑎 𝑓′ 𝑡 𝑑𝑡 = lim 𝑛𝑖=1 𝑓′ 𝑡𝑖 ∙ ∆𝑡𝑖
𝑛→∞
𝑏
means that 𝑎 𝑓′ 𝑡 𝑑𝑡 = ∆𝑓 𝑡 𝑜𝑛 [𝑎, 𝑏], which
(by definition of ∆)
=𝑓 𝑏 −𝑓 𝑎 .
FTC-I emerges next, by proof…
𝑥
𝑓
𝑎
If we let 𝐴 𝑥 =
𝑡 𝑑𝑡, the accumulated
signed-area between the t-axis and 𝑦 = 𝑓 𝑡 on
the interval 𝑎, 𝑥 , we can investigate 𝐴′ 𝑥 , the
rate at which 𝐴 𝑥 changes with respect to x.
(insert here: proof by formal limit definition of derivative A’(x))
Result:
𝑑 𝑥
𝑓′
𝑎
𝑑𝑥
𝑑 𝑥
𝑓
𝑎
𝑑𝑥
𝑡 𝑑𝑡 = 𝑓 𝑥 . Of course, by FTC-II,
𝑑
𝑑𝑥
𝑡 𝑑𝑡 =
𝑓 𝑥 − 𝑓 𝑎 = 𝑓′ 𝑥 , 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)
𝑑 𝑥 2
1
𝑡 𝑠𝑖𝑛 2
𝑑𝑥 3
𝑡
(b)
𝑥 𝑑
3 𝑑𝑡
(c)
5 𝑑
3 𝑑𝑡
𝑑𝑡
𝑡 𝑠𝑖𝑛
1
𝑡2
𝑑𝑡
𝑡 2 𝑠𝑖𝑛
1
𝑡2
𝑑𝑡
2
(d)
𝑑 5 2
1
𝑡 𝑠𝑖𝑛 2
𝑑𝑥 3
𝑡
(e)
𝑑 𝑥3 2
1
𝑡 𝑠𝑖𝑛 2
𝑑𝑥 3
𝑡
(f)
𝑑 𝑥3
𝑑𝑥 3𝑙𝑛 𝑥
2
𝑡 𝑠𝑖𝑛
𝑑𝑡
𝑑𝑡
1
𝑡2
𝑑𝑡
(a)
(b)
(c)
Fun
with
the
F.T.C.
1
1
𝑑 𝑥 2
𝑡 𝑠𝑖𝑛 2
𝑑𝑥 3
𝑡
𝑥 𝑑
1
2
𝑡 𝑠𝑖𝑛 2
3 𝑑𝑡
𝑡
5 𝑑
1
2
𝑡 𝑠𝑖𝑛 2
3 𝑑𝑡
𝑡
(d)
𝑑 5 2
1
𝑡 𝑠𝑖𝑛 2
𝑑𝑥 3
𝑡
(e)
𝑑 𝑥3 2
1
𝑡 𝑠𝑖𝑛 2
𝑑𝑥 3
𝑡
𝑑 𝑥3 2
1
𝑡 𝑠𝑖𝑛 2
3
𝑑𝑥
𝑡
(f)
𝑑 𝑥3
𝑑𝑥 3𝑙𝑛 𝑥
2
𝑡 𝑠𝑖𝑛
𝑑𝑡 = 𝑥 2 𝑠𝑖𝑛
𝑥2
𝑥 2 𝑠𝑖𝑛
1
𝑥2
𝑑𝑡 = 25𝑠𝑖𝑛
1
25
𝑑𝑡 =
𝑑𝑡 =
𝑑
𝑑𝑥
− 9𝑠𝑖𝑛
1
9
− 9𝑠𝑖𝑛
1
9
𝑘 =0
𝑑𝑡 ≠
𝑥 6 𝑠𝑖𝑛
1
𝑥6
∙ 6𝑥 5
𝑑𝑡 =
𝑥 6 𝑠𝑖𝑛
1
𝑥6
∙ 3𝑥 2 …why?
1
𝑥6
∙ 3𝑥 2 − 9𝑙𝑛2 𝑥 𝑠𝑖𝑛
1
𝑡2
𝑑𝑡 = 𝑥 6 𝑠𝑖𝑛
1
9𝑙𝑛2 𝑥
∙
3
𝑥
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 nonverbal 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.
𝑑
𝑑𝑡
𝑡
𝐻𝑢𝑚𝑎𝑛𝑖 𝑛 𝑑𝑛 = 𝐻𝑢𝑚𝑎𝑛𝑖(𝑡)
𝑛𝑜𝑤
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.
11/6
𝐵𝑟𝑒𝑛𝑡 ′ 𝑡 𝑑𝑡
𝐵𝑟𝑒𝑛𝑡 𝑁𝑜𝑣 6𝑡ℎ = 𝐵𝑟𝑒𝑛𝑡 𝑆𝑒𝑝𝑡 1𝑠𝑡 +
9/1
(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!
2015
𝐿𝑣𝑖𝑙𝑙𝑒 ′ 𝑡 𝑑𝑡
𝐿𝑣𝑖𝑙𝑙𝑒 2015 = 𝐿𝑣𝑖𝑙𝑙𝑒 1987 +
1987
From algebra class: 𝑦 = 𝑚𝑥 + 𝑏.
Better yet, 𝑦 = 𝑎 + 𝑏𝑥.
…this one goes out to all the statisticians in the audience!
Better still, 𝑦 = 𝑦0 + 𝑚 𝑥 − 𝑥0 .
In function notation: 𝑓 𝑏 = 𝑓 𝑎 + 𝑚 ∙ ∆𝑥.
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 𝑓 𝑡 at 𝑥 = 𝑎 gives us:
𝑑𝑦
𝑓 𝑏 ≈𝑓 𝑎 +
|𝑥=𝑎 ∙ ∆𝑥
𝑑𝑥
…this takes us to Euler’s method.
* Euler’s method as a connection to FTC
𝑓 𝑏 ≈𝑓 𝑎 +
≈𝑓 𝑎 +
≈𝑓 𝑎
+
𝑑𝑦
|𝑥=𝑎
𝑑𝑥
𝑑𝑦
|𝑥=𝑎
𝑑𝑥
𝑑𝑦
|
𝑑𝑥 𝑥=𝑎
∙ ∆𝑥1 +
∙ ∆𝑥
∙ ∆𝑥1 +
𝑑𝑦
|
𝑑𝑥 𝑥=𝑎2
𝑑𝑦
|𝑥=𝑎2
𝑑𝑥
∙ ∆𝑥2
∙ ∆𝑥2 + . . . +
𝑑𝑦
|
𝑑𝑥 𝑥=𝑎𝑛
∙ ∆𝑥𝑛
Why approximate when you can calculate more precisely?
Then 𝑓 𝑏 = 𝑓 𝑎 + lim
or 𝑓 𝑏 = 𝑓 𝑎
𝑛→∞
𝑏
+ 𝑎 𝑓′
𝑛
𝑖=1 𝑓′
𝑎𝑖 ∙ ∆𝑥𝑖 ,
𝑡 𝑑𝑡...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 a
synthesis of FTC, product rule, & substitution
Begin with:
𝑓 𝑥 =𝑓 𝑎 +
𝑥
𝑓′
𝑎
𝑡 𝑑𝑡
𝑥
(NOTE: This means that 𝑥 ∙ 𝑓 𝑥 = 𝑥 ∙ 𝑓 𝑎 + 𝑥 ∙ 𝑎 𝑓′ 𝑡 𝑑𝑡 ; we’ll use this below…)
= 𝑓 𝑎 + 𝑡 ∙ 𝑓′ 𝑡 |𝑡=𝑎
𝑡=𝑥
−
𝑥
𝑡
𝑎
∙ 𝑓 ′′ 𝑡 𝑑𝑡
𝑥
= 𝑓 𝑎 + 𝑥 ∙ 𝑓 𝑥 − 𝑎 ∙ 𝑓′ 𝑎 − 𝑎 𝑡 ∙ 𝑓′′ 𝑡 𝑑𝑡
𝒙
𝑥
= 𝑓 𝑎 + 𝑥 ∙ 𝑓′ 𝑎 + 𝒙 ∙ 𝒂 𝒇′′ 𝒕 𝒅𝒕 − 𝒂 ∙ 𝒇′ 𝒂 − 𝑎 𝑡 ∙ 𝑓′′ 𝑡 𝑑𝑡
′
=𝑓 𝑎 +𝑥∙
𝑓′
𝑎 −𝒂∙
=𝑓 𝑎 + 𝑥−𝑎 ∙
𝑓′
𝒇′
𝑎 +
𝒂 +
𝑥
𝑎
𝒙
𝒙
𝒂
∙ 𝒇′′ 𝒕 𝒅𝒕 −
𝑥 − 𝑡 ∙ 𝑓′′ 𝑡 𝑑𝑡
𝑥
𝑡
𝑎
∙ 𝑓′′ 𝑡 𝑑𝑡
Taylor’s Theorem: an ‘integrative’ proof
𝑥
𝑓′
𝑎
Recall that starting from: 𝑓 𝑥 = 𝑓 𝑎 +
𝑡 𝑑𝑡, we generated , from the previous page:
𝑥
𝑓 𝑥 = 𝑓 𝑎 + 𝑥 − 𝑎 ∙ 𝑓′ 𝑎 +
𝑥 − 𝑡 ∙ 𝑓′′ 𝑡 𝑑𝑡
𝑎
𝑥
𝑓′′
𝑎
= 𝑓 𝑎 + 𝑓′ 𝑎 ∙ 𝑥 − 𝑎 +
=𝑓 𝑎 +
𝑓′
−𝒇′′
𝑎 ∙ 𝑥−𝑎 +
′
=𝑓 𝑎 + 𝑓 𝑎 ∙ 𝑥−𝑎 +
=𝑓
𝑎
=𝑓 𝑎
+ 𝑓′ 𝑎 ∙ 𝑥 − 𝑎
+
+ 𝑓′ 𝑎 ∙ 𝑥 − 𝑎
𝑓′′ 𝑎
2
+
=…and so on: 𝒇 𝒙 =
𝒕 ∙
𝒇′′ 𝒂
𝟐
𝑥−𝑎
𝑓 ′′ 𝑎
2
2
𝒙−𝒕
𝒙−𝒂
𝒏 𝒇 𝒂
𝒊=𝟎 𝒊!
𝟐
2
+∙
𝒕=𝒙
𝟐
𝒕=𝒂
𝑥
𝑓′′′
𝑎
+
−𝒇′′′ 𝒕 ∙
+
𝑥−𝑎
(𝒊)
𝟏
𝟐
𝑡 ∙ 𝑥 − 𝑡 𝑑𝑡
𝟏𝟏
𝒙−𝒕
𝟐𝟑
𝒇′′′ 𝒂
𝟑!
𝒙−𝒂
𝒙−𝒂 𝒊+
𝑥1
𝑎 2
+
2
∙ 𝑓′′′ 𝑡 𝑑𝑡
1
𝑡 ∙ 2 𝑥 − 𝑡 2 𝑑𝑡
𝒕=𝒙
𝟑
𝒕=𝒂
𝟑
𝑥−𝑡
+
𝒙 (𝒏+𝟏)
𝒇
𝒂
+
𝑥1
𝑎 3!
𝑥 (4)
𝑓
𝑎
𝒕 ∙
𝟏
𝒏!
𝑥−𝑡
𝑡 ∙
1
3!
3
∙ 𝑓 (4) 𝑡 𝑑𝑡
𝑥 − 𝑡 3 𝑑𝑡
𝒙 − 𝒕 𝒏 𝒅𝒕