Unit 2: Derivative Rules & Tangent Lines
A few notes for this unit (and those following):
 Most of the rest of my units are through PowerPoint, which I will post below. It is my goal, one day, to make guided notes. That has not happened yet :)
 My Unit 2 & Unit 3 are often combined in textbooks. However, I split up this Derivative Rules chapter into the two units for a few reasons. First, it's a pretty massive section to take in (and test on) all at once. Second, my Unit 3 (Implicit, Inverses, and Related Rates) tend to be the topics students struggle with the most in the first semester, and I like to focus on only those three topics at one time.
Day 10 – Secant and Tangent Lines
We begin our descent into derivatives with a summary of Secant and Tangent lines. Most students need a refresher on how to write the equation of a line (efficiently), so we do a bit on (h,k) form of lines. Specifically, we discuss how a secant line can be created through two points, let's say (a,f(a)) and (b,f(b)), and would take the form:
s(x) = m(xa)+f(a) or s(x)=m(xa)+f(a)
where m is found using the slope formula. Note that if you're teaching BC, you might want to rearrange your line formula to read:
s(x)=f(a) + m(xa)
Which is really the beginning of a Taylor series! Good practice from the getgo.
Then, the idea of a tangent SLOPE at a point is introduced through the concept of limits. We do NOT use the word "derivative" yet; that comes tomorrow!
Handouts:
1.) Day 10 PowerPoint in PDF format
s(x) = m(xa)+f(a) or s(x)=m(xa)+f(a)
where m is found using the slope formula. Note that if you're teaching BC, you might want to rearrange your line formula to read:
s(x)=f(a) + m(xa)
Which is really the beginning of a Taylor series! Good practice from the getgo.
Then, the idea of a tangent SLOPE at a point is introduced through the concept of limits. We do NOT use the word "derivative" yet; that comes tomorrow!
Handouts:
1.) Day 10 PowerPoint in PDF format
Day 10  Secant & Tangent Lines  
File Size:  3388 kb 
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Day 11 – Introduction of the Derivative Function
Up to this point, and even for the first half of class, we've only looked at the slope at a specific point on a function. Halfway through class, we make the big reveal of the DERIVATIVE FUNCTION! This is done (of course) using limits. Students still don't know any shortcuts so they're forced to evaluate limits the long way around. Therefore, the homework is still short (no reason to do a ridiculous amount of work that the following class will be made moot).
Handouts:
1.) Day 11 PowerPoint in PDF format
Handouts:
1.) Day 11 PowerPoint in PDF format
Day 11  Introduction of the Derivative  
File Size:  3278 kb 
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Day 12 – The Power Rule
Although students thoroughly enjoy evaluating limits (hint: they don't), at this point it's time to introduce the Power Rule for derivatives. I actually take some time to discuss the history of Calculus before the Power Rule due to all the different notations that are about to be thrown at our students. We discuss the rift in mathematics caused by the Newton/Leibniz feud and how that led to multiple notations being developed across Europe. Of course, the students are expected to know both Newton and Leibniz notation for the AP test, so let's start now!
Then, we move on to the Power Rule. This is one of the few rules we DON'T prove in class, mostly because the proof in their textbook uses the Binomial Expansion Theorem, which not all of our students have been taught. I hand out a note taker on the Power Rule and we go through a few examples together, also covering derivatives additive constants, multiplicative constants, and multiple functions adding together. Then, it's practice time!
Handouts:
1.) Day 12 PowerPoint in PDF format
2.) Power Rule Handout
Then, we move on to the Power Rule. This is one of the few rules we DON'T prove in class, mostly because the proof in their textbook uses the Binomial Expansion Theorem, which not all of our students have been taught. I hand out a note taker on the Power Rule and we go through a few examples together, also covering derivatives additive constants, multiplicative constants, and multiple functions adding together. Then, it's practice time!
Handouts:
1.) Day 12 PowerPoint in PDF format
2.) Power Rule Handout


Day 13 – Transcendental Functions & Higher Order Derivatives
When I say "Transcendental Functions", really we only learn the derivative of e^x today. However, I like the Numberphile video on Transcendental Functions, so we spend some time watching that. Then, we derive the derivative of e^x using a fact from the opener. Our final conversation is on "Higher Order Derivatives", which goes something along the lines of: keep taking the derivative until you're supposed to stop. We do touch a bit further on notation as Leibniz notation can be tricky for some of them. "Why is the d squared?" is a common question : )
Handouts:
1.) Day 13 PowerPoint in PDF Format
Handouts:
1.) Day 13 PowerPoint in PDF Format
Day 13  Transcendental Functions PowerPoint  
File Size:  2675 kb 
File Type: 
Day 14 – The Product Rule + e^kx
We begin by deriving the product rule for derivatives. It's important for students to be able to follow a proof from start to finish, so we prove most every derivative rule as we learn them. Additionally, we extend the product rule to more than two functions just for fun. Class ends by proving that the derivative of e^(kx) is ke^(kx). Note that we still don't know the Chain Rule, so this is probably the best way to move forward with e^x.
Handouts:
1.) Day 14 PowerPoint in PDF Format
Handouts:
1.) Day 14 PowerPoint in PDF Format
Day 14  The Product Rule  
File Size:  3287 kb 
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Day 15 – The Quotient Rule
Today is split into two parts. The first half or so of class is spent deriving and applying the quotient rule. The second half is a work day, finishing up the assignment from the night before if necessary, and working through the new assignment for tonight.
Handouts:
1.) Day 15 PowerPoint in PDF format
Handouts:
1.) Day 15 PowerPoint in PDF format
Day 15  The Quotient Rule  
File Size:  2487 kb 
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Day 16 – Derivatives of Trig Functions
Trig functions are pretty easy derivatives to prove, especially once you figure out sine and cosine. The trick to sine is remembering your sum to product rules (which no one ever does). Once the students prove sine, cosine is fairly quick. Then, we do two proofs for secant and tangent, and I let the students know the proofs for the other two trig functions will be on their homework!
Handouts:
1.) Day 16 PowerPoint in PDF format
Handouts:
1.) Day 16 PowerPoint in PDF format
Day 16  Trig Functions  
File Size:  2778 kb 
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Day 17 – The Chain Rule
At last, we come to the dreaded Chain Rule. Of course, the Chain Rule has been there all along...but students haven't realized it! I think it's extremely important for students to know they are ALWAYS using the Chain Rule, but their problems have been cherry picked so that they didn't know it was happening. We discuss both versions of the chain rule (Newton vs. Leibniz back at it!), and then play a whiteboard game in groups to help everyone practice.
Handouts:
1.) Day 17 PowerPoint in PDF format
Handouts:
1.) Day 17 PowerPoint in PDF format
Day 17  The Chain Rule  
File Size:  3185 kb 
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Day 18 – More e^x & Tricky Practice Problems
We quickly revisit e^x, combining our knowledge of the parent function and the Chain Rule to be able to take the derivative of e to the power of any function. Then, students are given 6 challenging problems to work on. The class is split into six groups, with each group getting a different problem. They have approximately 6 minutes to solve the problem (note: answers are provided so really students are just showing work), and then they pass the problem along (even if it's not finished). These six problems are taken from previous (released) multiple choice questions, and tend to be quite tricky!
Handouts:
1.) Day 18 PowerPoint in PDF format
2.) Day 18  6 Tricky Problems
Handouts:
1.) Day 18 PowerPoint in PDF format
2.) Day 18  6 Tricky Problems


Day 19 – Natural Logs
Class begins with a short "memory" quiz of all the topics students should know so far. Think IVT, continuity at a point, derivative rules, etc. Students have 10 minutes to complete the quiz, and then they selfgrade. We move from there onto our main topic for the day, which is the derivative of the Natural Log function. After deriving the formula for Natural Log, we do a few practice problems. Note that although Natural Log is an easy function with which to deal, one trick the AP exam likes to throw the way of students is by incorporating absolute value, so we practice this a little!
Handouts:
1.) Memory Quiz
2.) Day 19 PowerPoint in PDF Format
Handouts:
1.) Memory Quiz
2.) Day 19 PowerPoint in PDF Format


Day 20 – General Logs & Exponentials
Although we often see natural log and e^x as functions on the AP Exam, it is rarely asked of students to take the derivative of other log functions (think different bases) and other exponentials (2^x, 3^x, etc.). My guess is that they are trying to have pity on the poor AP students who actually have to do a fair amount of memorization for the AP exam. Of course, CollegeBoard does mention that students should be comfortable with these derivatives, so we learn them! Both the formulas for logarithms and exponentials are derived, and students take a bit of time to practice a few problems. I think it is interesting to see how e^x and ln(x) connect to their "parent" formulas, and why they end up looking so nice, so we have a good conversation regarding those points.
Handouts:
1.) Day 20 PowerPoint in PDF format
Handouts:
1.) Day 20 PowerPoint in PDF format
Day 20  Logs & Exponentials  
File Size:  2584 kb 
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Day 21 – Differentiability of Functions
Differentiability is a pretty interesting topic. Visually, it's fairly easy to see when a function is "not differentiable" (doesn't have a slope at a point): look for a sharp point (called a cusp), if the graph goes perfectly vertical, or if there are any holes (can't have a slope without a point). However, there has been a push recently for students to PROVE a lack of differentiability using the limit definition of the derivative (see, in particular, the 2017 released FRQs). This gives us a nice tie in to continuity at a point, and allows students additional practice with the definition of the derivative (a great connection to tomorrow's lesson!). Usually, differentiability is asked once on the multiple choice, and lately, somewhere on the Free Response. I have a handout for students to practice, as I think our book is a little weak in this area. There are also some extra derivative practice problems here.
Handouts:
1.) Differentiability & Derivatives Practice
2.) Day 21 PowerPoint in PDF Format
Handouts:
1.) Differentiability & Derivatives Practice
2.) Day 21 PowerPoint in PDF Format
Differentiability & Derivatives Practice  
File Size:  219 kb 
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Day 22 – Recognizing The 3 Definitions of the Derivative
The last topic of this unit ALWAYS rears its ugly head on the Multiple Choice section on the AP Exam, and just does the most FANTASTIC job at confusing our students. Students do pretty well on practicing the definition of the derivative on the handout that I've included below. HOWEVER, what messes them up is they begin to believe that any time they see a limit, that means they should take the derivative! Hello: remember Unit 1, where we did tons of limits that had nothing to do with the derivative (a word we didn't even know then)?! So, the challenge of today is twofold: first, recognize THREE different (looking) definitions of the derivative, and second, don't forget that not all limits are derivatives! Note: I even took the time to make a key for this  go me! The key is at the end of the assignment.
Handouts:
1.) Day 22 PowerPoint in PDF form
2.) Definition of the Derivative Practice
Handouts:
1.) Day 22 PowerPoint in PDF form
2.) Definition of the Derivative Practice


Day 23 – FRQ Practice
Our review for this exam gets longer every year. I tried three full days this time, and still the results were significantly lower than my first exam. Of course, that's to be expected; students learned WAY more "stuff" this unit, and it's all still swirling around their brains. Luckily, this is the most memorizing they need to do for the entire year. Most of the rest of the course is just applying what they know.
For review, I split the class into 6 groups of however many students make up my class. Each group is given ONE Free Response question (or, a piece of an FRQ as students don't know how to do most full ones yet), and they are responsible for (a) doing the question and (b) checking their work against a provided rubric. Once 15 minutes has elapsed, one person from each of the 6 groups are regrouped to form groups of 6 people, where each person is an "expert" in their one FRQ. That way, they can help their group, and students "need" me less. Whatever is not finished in class is homework.
Handouts:
1.) 6 FRQs
2.) Day 23 PowerPoint in PDF format
For review, I split the class into 6 groups of however many students make up my class. Each group is given ONE Free Response question (or, a piece of an FRQ as students don't know how to do most full ones yet), and they are responsible for (a) doing the question and (b) checking their work against a provided rubric. Once 15 minutes has elapsed, one person from each of the 6 groups are regrouped to form groups of 6 people, where each person is an "expert" in their one FRQ. That way, they can help their group, and students "need" me less. Whatever is not finished in class is homework.
Handouts:
1.) 6 FRQs
2.) Day 23 PowerPoint in PDF format


Day 24 – Calculus Keys to Success, Unit 2 + Review
There are actually three handouts for today. First, students do a little "Spot Check" to see what they're weak on for Unit 2. This is self graded after 15 minutes.
Then, students get their second set of Calculus Keys to Success. Technically, this handout is for Units 2 & 3, but as they're grouped together in the chapter, I leave it grouped together on the Keys. I (try to) remember to inform the students that they won't know B17  B20, though they ask eventually if I forget. They have about 20 minutes or so to work on this. As they begin to finish, they receive their homework, which is multiple choice practice created from past AP questions.
Handouts:
1.) Spot Check
2.) Calculus Keys to Success Units 2 & 3
3.) Multiple Choice Practice
Then, students get their second set of Calculus Keys to Success. Technically, this handout is for Units 2 & 3, but as they're grouped together in the chapter, I leave it grouped together on the Keys. I (try to) remember to inform the students that they won't know B17  B20, though they ask eventually if I forget. They have about 20 minutes or so to work on this. As they begin to finish, they receive their homework, which is multiple choice practice created from past AP questions.
Handouts:
1.) Spot Check
2.) Calculus Keys to Success Units 2 & 3
3.) Multiple Choice Practice



Day 25 – Review, part 2
This second day of review, we also warm up with a "Derivative Check", just to make sure we've filled in any gaps from yesterday's "Spot Check". Students only do part A, with part B being left for them to practice for homework. The rest of today is left for questions, finishing up old assignments/Calculus Keys to Success, etc. Get ready, students!
Handouts:
1.) Derivative Check
2.) Day 25 PowerPoint in PDF format
Handouts:
1.) Derivative Check
2.) Day 25 PowerPoint in PDF format


Day 26 – Unit 2 Test
As before, I don't post my tests for two reasons: (1) I don't need students taking a sneak peak before the exam and (2) The questions are secure questions for which CollegeBoard tells me I will be shot (figuratively? literally?) if I post them anywhere online. As before, though, here is my cover page for Unit 2. See the Unit 1 test for my explanation about why I use a cover page. This test is split a little more equally between multiple choice and free response, though still weighted slightly on the side of MC (about 60/40). No calculator is allowed, which means 2 minutes per each multiple choice question.
Handouts:
1.) Cover page for Unit 2
Handouts:
1.) Cover page for Unit 2
Day 26  Unit 2 Test Cover Sheet  
File Size:  122 kb 
File Type: 