Unit 3: Implicit, Inverses, & Related Rates
This miniunit is really a continuation of Unit 2, and in most textbooks (ours included), this is NOT a separate chapter. However, the idea behind this miniunit is to really focus in on three topics that students seriously struggle with: implicit differentiation, derivatives of inverses, and related rates. This unit also culminates with a project instead of a test. There are two reasons behind this: (a) My test scores on this unit were TERRIBLE the few times I did a test and (b) I think it's nice for the students to switch things up!
Day 27 – Implicit Differentiation
Implicit differentiation is really no different than regular differentiation. In fact, when taking the derivative of something like y = sin(x), we say the derivative is y' = cos(x) * 1, simplifying to y' = cos(x). Notice that we actually took the derivative of both sides, with y being one side and sin(x)*1 being the other. The derivative of y is y' on the left side (no surprise here), and the derivative of x is 1 on the right side (still no surprise) . However, the students still seem confused when y and x are on the same side, and they are asked to differentiate. For example: x + y = 3 > 1 + y' = 0. Why is the derivative of y confusing? Mostly, the Chain Rule messes with their heads...I think. Either way, fairly straight forward lecture and practice today. More to come next class.
Handouts:
1.) Day 27 PowerPoint in PDF format
Handouts:
1.) Day 27 PowerPoint in PDF format
Day 27  Implicit Differentiation  
File Size:  2661 kb 
File Type: 
Day 28 – More Implicit Differentiation
The first part of class is used to answer homework questions (there are always many). The second half is dedicated towards practicing multiple choice and free response questions. The homework is ALL of the multiple choice on the first page, as well as the FIRST FRQ on the second page. Strangely, I've also created workedout solutions for the multiple choice, and those are available here.
Handouts:
1.) Multiple Choice and FRQ Practice
2.) Multiple Choice Solutions
3.) Day 28  Implicit Practice PowerPoint in PDF format
Handouts:
1.) Multiple Choice and FRQ Practice
2.) Multiple Choice Solutions
3.) Day 28  Implicit Practice PowerPoint in PDF format



Day 29 – Even More Implicit Practice
We begin today by bringing out the rubric for the FRQ students finished for homework. Once students grade their own, they receive three actual student responses from that FRQ. Students then grade those three student responses according to the rubric. This has a few purposes. First, it allows students to see what earns (or doesn't earn) points on the AP exam. Second, they practice the skill of reading other people's work, gaining insight into other possible ways to work the same problem.
Next, students get to practice a timed FRQ. There are 6 FRQs on the actual AP exam over 90 minutes, giving 15 minutes (ish) per FRQ. So, 15 minutes on the clock, and students work the last FRQ from last class (2004, Question 4). At the end of the 15 minutes, students grade their own FRQ. Then, 2 more student responses to grade. No homework tonight, though!
Handouts:
1.) Rubrics/Student handouts for FRQs
2.) Day 29  PowerPoint in PDF format
Next, students get to practice a timed FRQ. There are 6 FRQs on the actual AP exam over 90 minutes, giving 15 minutes (ish) per FRQ. So, 15 minutes on the clock, and students work the last FRQ from last class (2004, Question 4). At the end of the 15 minutes, students grade their own FRQ. Then, 2 more student responses to grade. No homework tonight, though!
Handouts:
1.) Rubrics/Student handouts for FRQs
2.) Day 29  PowerPoint in PDF format


Day 30 – Inverse Trig Functions
The other set of uncommon functions asked for on the AP exam are derivatives of inverse trig functions. However, they do appear (though not often), and so we touch on them twice throughout the year. This is the first time, and as per usual, we derive the derivative of inverse sine. Then, students are given the other five inverse trig derivatives. There is some practice from the textbook, and then, besides the occasional practice, these will disappear for quite some time.
Handouts:
1.) Day 30 PowerPoint in PDF format
Handouts:
1.) Day 30 PowerPoint in PDF format
Day 30  Inverse Trig Functions  
File Size:  2628 kb 
File Type: 
Day 31 – Derivative of a General Inverse Function
Every equation has an inverse (found by swapping the x and y coordinates). However, just because the original equation is a function does not mean the inverse is also a function. To compound matters, just because your original equation was solved for y does NOT mean you can solve the inverse for y! Today, we explore a formula that allows us to find the derivative (at a point) of an inverse function WITHOUT having the inverse equation. It's a nice application of implicit differentiation and ALWAYS shows up on the Multiple Choice of the AP Exam. We spend two days on this, where during the first day students try to remember various facts about inverses and we discuss some notation, finally finishing with the formula for the derivative of an inverse at a point.
Handouts:
1.) Day 31 PowerPoint in PDF format
Handouts:
1.) Day 31 PowerPoint in PDF format
Day 31  Derivatives of Inverses  
File Size:  2482 kb 
File Type: 
Day 32 – Derivative of an Inverse Practice
We revisit the formula for the derivative at a point of an inverse function. There are always a few questions on the homework worth discussing (usually 49  51 are popular). Then, students get a chance to practice with the derivative of the inverse a little more. I've included my handout for this below. I even took the time to work out solutions, which are included in the handout!
Handouts:
1.) Derivative of Inverses Practice Problems + Solutions
2.) Day 32  Inverse Practice PowerPoint in PDF format
Handouts:
1.) Derivative of Inverses Practice Problems + Solutions
2.) Day 32  Inverse Practice PowerPoint in PDF format


Day 33 – Intro to Related Rates
Related rates are really the first time we apply derivatives to anything much of the real world, yet it's the one topic students tend to fear more than all others! Over the past few years, I've tried to come up with a few problems that seem common on the AP exam and organize them into some guided notes. I especially like the way I have students set up each related rates problem:
Know:
Want:
Picture:
Equation:
Derivative:
Substitute:
This allows students to translate a problem into Calculus notation and see the progression of steps that will lead them to their answer. We go pretty slowly here, first discussing what the idea of "with respect to" means for derivative notation. Then, we discover the fact that different locations on the same object move at different speeds, which is actually pretty counterintuitive! Lastly, we do a few sample problems, leaving some easy bookwork for homework.
Handouts:
1.) Related Rates Packet
2.) Day 33  Related Rates PowerPoint in PDF format
Know:
Want:
Picture:
Equation:
Derivative:
Substitute:
This allows students to translate a problem into Calculus notation and see the progression of steps that will lead them to their answer. We go pretty slowly here, first discussing what the idea of "with respect to" means for derivative notation. Then, we discover the fact that different locations on the same object move at different speeds, which is actually pretty counterintuitive! Lastly, we do a few sample problems, leaving some easy bookwork for homework.
Handouts:
1.) Related Rates Packet
2.) Day 33  Related Rates PowerPoint in PDF format


Day 34 – More Related Rates
Now that students have the basic idea (and scaffolding) of related rates down, we move on to the more complex problems: those using the Pythagorean Formula, and others using Similar Triangles. Students finish class doing one page worth of FRQs, which are homework if not done in class.
Handouts:
1.) Day 34  PowerPoint in PDF format
Handouts:
1.) Day 34  PowerPoint in PDF format
Day 34  More Practice with Related Rates  
File Size:  1447 kb 
File Type: 
Day 35 – Unit 3 Multiple Choice Practice + Intro to "The Project"
Our last day packs in a lot. We begin by answering any homework questions. Then, students have 15 minutes to do one last FRQ (the last page of the Related Rates packet), which really brings everything we've done in this unit together. After going over the answers, we move on to a few handouts.
First, students are given some Multiple Choice Review Questions that are their homework. Second, they are given "The Project" for Unit 3. The project is a "Build Your Own" exam, where students are responsible for designing (and solving) 10 problems. Requirements are given for each problem, and students have one week to complete it. The problems they design tend to be pretty awesome, and they learn a lot along the way!
Handouts:
1.) Multiple Choice + Short Answer Review Questions
2.) Unit 3 Project Guidelines & Rubric
3.) Day 35  PowerPoint in PDF format
First, students are given some Multiple Choice Review Questions that are their homework. Second, they are given "The Project" for Unit 3. The project is a "Build Your Own" exam, where students are responsible for designing (and solving) 10 problems. Requirements are given for each problem, and students have one week to complete it. The problems they design tend to be pretty awesome, and they learn a lot along the way!
Handouts:
1.) Multiple Choice + Short Answer Review Questions
2.) Unit 3 Project Guidelines & Rubric
3.) Day 35  PowerPoint in PDF format


