Version 1, August 2016
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This format is the result of tinkering with a mixed lecture format for 3 terms. As such,
it is still a work in progress and we will discuss adaptations both to the general format
as well as for individual topics throughout the term.
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As we continue to refine MATH 3, we want to standardize the course a bit more than
in the past. These templates are meant to help keep all instructors roughly on the
same page so that when we get to exams, there are few surprises for both instructors
and students. As you’ll see, this is a skeleton, there is still lots of room for your own
take on the material.
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We’ve highlighted the topics that students had the most trouble with. Consequently,
we now spend more time on these topics.
For integration topics, most students master these just fine, the lower averages
simply reflect the shorter time available to practice.
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The beginning of class 1 concerns the logistics of the course. In the last 10 minutes or
so, we can start material and this slide provides a jumping off point. It introduces a
fundamental problem that motivates differential calculus and sets up some of the
issues in defining and taking limits. Reading: Stewart sections 2.1-2.3
For Khan Academy, we’ll have the students start with the skills review – this will help
them jump into the problem sets at the right place:
https://www.khanacademy.org/math/calculus-home/differential-calculus/limitstopic/limits-skill-checks/e/skill-check--estimating-limits
https://www.khanacademy.org/math/calculus-home/differential-calculus/limitstopic/limits-skill-checks/e/using-limits
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The assignment after class 1 includes reading the sections on limits with several
questions in mind:
1. Why do we need to take limits rather than evaluating?
2. For what types of functions is taking a limit easy? difficult?
3. In finding the tangent line to a curve at a point, why do we always need to take a
limit?
At the beginning of class 2, we’ll talk about these questions and fold in review of
functions and their manipulation. For example, this is a perfect place to talk about
functional notation, relation to graphs in the plane, and the domain. This is also a
good place to set notation for points (e.g. P(1,1), Q(x,x^2)) and sets.
This first piece of group work should help them crystalize question 3 above. This set
should be relatively easy – it is pre-calculus review. But, it will help the groups solidify
and get used to working together.
Khan Academy Problems sets of this type:
Slope of secant lines:
https://www.khanacademy.org/math/differential-calculus/taking-derivatives/secant-
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line-slope-tangent/e/slope-of-secant-lines
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In the second half of lecture 2, we recollect the intuitive notion of a limit and some
ways of estimating the value. Here we take the example from the group work and try
to estimate using a table. The image on the lower right shows a number of different
potential cases for limiting behavior. This will help foreshadow continuity as well.
Here, try to sort through all the cases relatively quickly, introducing notions of left
and right handed limits. Part of this helps to make the notion of continuity fall out of
the discussion rationally.
Khan Academy Problem sets of this type:
Finding Limits Numerically:
https://www.khanacademy.org/math/calculus-home/differential-calculus/limitstopic/limits-tutorial/e/finding-limits-numerically
One-sided limits from graphs:
https://www.khanacademy.org/math/calculus-home/differential-calculus/limitstopic/calculus-estimating-limits-graph/e/one-sided-limits-from-graphs
Two-sided limits from graphs:
https://www.khanacademy.org/math/calculus-home/differential-calculus/limitstopic/calculus-estimating-limits-graph/e/two-sided-limits-from-graphs
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This example shows us two things. First, graphical evidence is not always conclusive.
Second, neither is numerical evidence. Moreover, numerical evidence might be
misleading!
A key next point - we need, somehow, to test all the points near zero at once to really
know we have the answer.
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These are not slides for class per se, but these types of problems may be
incorporated as needed. These are also appropriate to give to students as additional
practice, in review, or for other purposes. They are indicative of three difficulty levels
of problems at this stage. On an exam, a student might see these alone, or
incorporated into a more advanced problem (e.g. on continuity or differentiation).
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We introduce the formalization of the limit at the end of class 2 to set the stage for
the reading (section 2.4) and a recap at the beginning of class 3. The black line in the
𝑥 2 −1
image is 𝑥−1 evaluated for 0 < 𝑥 < 2 in increments of 0.01. There is a hole at 𝑥 = 1
(see inset) as MATLAB (and you) can’t evaluate the function there.
This slide serves to start the conversation at the end of class 2, introducing the idea of
using 𝛿 to indicate how close we are to 𝑥 and 𝜖 as a measure of the error between
the function value and the proposed limiting value. The goal is to get to a verbal
description of the limit – if 𝑥 is close to 𝑎 then 𝑓 𝑥 should be close to 𝐿 – to prompt
students to think about how to formalize this while doing the reading.
Coming back to this at the beginning of class 3, let’s guide the students’ thinking to
the formal definition. The last question here is a key point – if a student can’t
connect the intuition to the definition, they will not be able to use the definition well.
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Flag: algebra – again emphasize that we use algebraic manipulations constantly in
calculus.
This also gives time for the students to absorb how the definition is simply codifying
“closeness” and provides another opportunity for quick review – functional notation,
absolute value, open sets, etc.
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Why are we doing this? Isn't the answer obvious? Well, maybe so, but think about
𝜋
the example with sin 𝑥 . There, the answer isn't obvious and the function is a
difficult one to deal with. By choosing a simple function to start with, we can
understand the definition and how to use it in a situation where the rest of the math the algebraic manipulations - are simpler.
Setting the stage for class 4, indicate that we will be doing and honest proof of a
claim – showing a particular function has a particular limit. Tell them about the
difficult elements – complex algebra, abstract reasoning, and abstract computation.
Encourage them to think of the last question here in terms of the first 3.
F16 note: groups generally won’t be set until week 2, so turning in group work is not
plausible at this point.
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This proof will (most likely) take up a good portion of class 4. Connecting to the
previous group work is helpful – show how this leads to the proof from the opposite
direction:
Q. How big can 𝑓(𝑥) be if 𝑥 − 𝑎 < 𝛿 for a fixed 𝛿?
If 𝛼 > 0 then the function is increasing and 𝑓 𝑥 is largest at 𝑥 = 𝑥0 + 𝛿, which is
𝛼𝑥0 + 𝛼𝛿 + 𝛽 = 𝐿 + 𝛼𝛿. Similarly the minimum is 𝐿 − 𝛼𝛿. So this 𝛿 works so long
as 𝜖 > 𝛼𝛿.
Our proof turns this around, starting with 𝜖 and working out 𝛿 but it is the same idea.
Key points:
• Algebra, yet again.
• Symbolic manipulation rather than just numbers.
• Almost every proof of this type is to manipulate (often algebraically) the 𝜖
inequalities to isolate an (x-x_0) term. From this, we can calculate 𝛿.
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In the second half class 4, return to the proof but emphasize the places where it
might fail. The purpose of this set of problems is to explore when the definition fails
to be true and, consequently, that the limit doesn’t exist.
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Note that in the last case, there is a convention – tending to an infinite value is
sometimes categorized as the limit not existing and in other texts categorized as
having the limit exist but at infinity. Our text chooses the former – the statement
lim 𝑓 𝑥 = ∞ describes how the limit fails to exist.
𝑥→𝑎
If all has gone well, this is the end of class 4. To prompt class 5, we tell the class that
we will introduce some further tools in evaluating limits – the squeeze theorem, and
algebraic simplifications. Ask them to do the reading (review sections 2.2-2.3) to try
to familiarize themselves with the techniques and to bring examples (not from the
book) where they work well.
Related Khan Academy set:
Finding limit at infinity when f(x) is unbounded:
https://www.khanacademy.org/math/calculus-home/differential-calculus/limitstopic/limits-infinity/e/limits-at-infinity-where-f-x--is-unbounded
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Understanding the definition allows us to motivate some additional techniques that
can help us find limits. The first half of class 5 is a short bit of lecture describing the
squeeze theorem and algebraic simplification, but emphasizing how the definition of
the limit allows these to work. If there is time for group work, these outlines can
serve as templates, otherwise, they are templates for examples in lecture.
For the squeeze theorem, the idea is that if we can use an 𝜖 − 𝛿 argument for 𝑓 and
𝑔, which are generally simpler, the argument carries over to ℎ(𝑥).
These next two pieces of group work demonstrate how to use these two techniques.
Khan Academy Problem set:
https://www.khanacademy.org/math/calculus-home/differential-calculus/limitstopic/squeeze-theorem/e/squeeze-theorem
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Prompt this at the end of lecture 4 and take it up at the beginning of lecture 5. As
with the squeeze theorem, algebraic simplification produces equivalent forms to the
original function away from the point in question. Thus, as before, an 𝜖 − 𝛿 pair that
works for the simplification will work for the original. Indeed we could use the
squeeze theorem to show this by taking 𝑓 𝑥 = 𝑔(𝑥) equal to the simplification.
Then, equality holds throughout.
Khan Academy problem sets:
Two-sided limits using algebra:
https://www.khanacademy.org/math/calculus-home/differential-calculus/limitstopic/algebraic-limits/e/two-sided-limits-using-algebra
Two-sided limits using advanced algebra:
https://www.khanacademy.org/math/calculus-home/differential-calculus/limitstopic/algebraic-limits/e/limits_2
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We mean these problems to push students beyond a basic understanding. The first is
simply difficult as it requires not just algebra but an estimation. The second is not as
hard technically, but requires logically reworking the definition. So, provide hints like
the inequality most likely to be useful in the first problem and/or the restatement of
the definition of the limit that indicates when a particular choice of L is not the limit.
These questions are not really meant for in-class work – they take too much thought
and some trial and error. I’ll likely talk about them with the class and then assign
them as one of the group homework sets.
Before this point in the class, use your judgement. Should you do a few more
examples of different types first?
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This is a nice application to show as it 1) demonstrates a different, natural, way limits
arise and 2) connects to a hard (and historical) problem – computing the value of 𝜋.
It also gives a sense of the necessity of the generality of the definition of the limit –
we need to be thinking about all possible types of functions, not just the quotients
that arise in the tangent line computation. This can be worked into class when (if?) it
fits or might be a nice topic for an xhour.
Start with a circle of diameter 1, then we can define 𝜋 as the ratio of the
circumference and diameter which, in this case, is simply the circumference. The
point of the picture is that the circumference of the circle of radius one is “obviously”
between the perimeter of the green and red 2𝑛 -gons. We can compute the
perimeter of the red n-gon by looking at the side length of the largest red triangle
(right), which is a right triangle with hypotenuse 1 and angle 𝜃. Then, 𝑑𝑛 = sin 𝜃
and the perimeter is 𝑃 𝑛 = 2𝑛 𝑠𝑖𝑛
𝜋
𝑛
.
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At the end of lecture 5, we set up continuity as a logical extension of finding limits,
namely an answer to the question, When is the limit of a function at a point simply
the value at that point? Refer back to the formal proof for linear function as an
example where evaluation works just fine. The reading is Stewart section 2.4.
One method of introduction is graphical. Looking at the graph above, we can identify
lots of places where the limit is simply the value, and a few where it is not. I’ll spend
some time sorting through these to the left (or on the board) bringing in concepts like
one-sided limits, the domain and range, etc.
The reading for the next class is on continuity – formal definitions and properties,
including continuity of the composition of functions, etc. While the reading is
somewhat far-ranging, the questions the students should focus on for the next class:
1. What makes a function continuous?
2. How can a function fail to be continuous? How is this related to our work on
limits?
3. Can you formalize the classification of discontinuities?
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Khan Academy Problem set:
Continuity:
https://www.khanacademy.org/math/calculus-home/differential-calculus/limitstopic/continuity-limits/e/continuity
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Begin class 6 with a presentation of the definition on the right and a couple of
examples drawn from the limit examples you’ve done in class previously. Connect up
to the questions for the reading.
Possible choices:
2𝑥
for 𝑥 ≥ 0
. Is f continuous at x=0? If so why? If not, what part(s)
𝑥 + 1 for 𝑥 < 0
of the definition is/are violated?
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For what value of c is 𝑔 𝑥 = ቊ3𝑥 − 𝑥 + 1 for 𝑥 ≥ 3 continuous?
10𝑥 + 𝑐
for 𝑥 < 3
Let 𝑓 𝑥 = ቊ
Is the function
𝑥 2 −1
𝑥−1
continuous at 𝑥 = 1?
Use the examples to introduce/review jump discontinuities, infinite discontinuities,
removable discontinuities, and one-sided continuity.
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This is a nice, relatively open-ended problem. It incorporates the algebraic
manipulations from the earlier work on limits and guides students to think about the
problem from several angles and to break it up into pieces.
#4 is a good problem to have student present at the board. It allow you to link back
lots of the pieces to earlier definitions, ideas, and techniques, and provides a great
opportunity to coach them on clear and complete writing of mathematical solutions.
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After considering the definition, we will likely have a short lecture portion on the
implications of continuity and some base cases. A simple example of using the IVT
will set up the next part of the group work.
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In the second half of class 6, before we embark into derivatives, we’ll provide a set of
exercises to help demonstrate the IVT.
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