Derivatives of Inverse Functions

This is my fourth year teaching calculus on some level.  Every year (until this one!) my students have really struggled with finding the derivative of inverse functions at a point, especially in the manner these questions are often phrased on AP Exams.

To me, they're some of the most straight-forward multiple choice questions the students encounter on the exam; yet, year after year they miss this question (at least on their unit tests and mock exam).

So, clearly, not as straight-forward as I thought...

This year I formalized a strategy for them in three steps.  Not all three of these steps are necessary every time; but, if my students took the time to follow all three steps, they got these questions correct.

Here are the steps:

If f and g are differentiable functions and g is the inverse of f, then to find g'(a):

1. List all points given on f as ordered pairs.
2. List the points you now know are on g (switch x and y).
3. Follow this formula: g'(a)=1/(f'(g(a)).

Let me show you with a couple examples.  Here's a question I pulled from this website.

Following the steps, we would work this question as follows:

How about one that describes f as an algebraic or numeric function, such as this FRQ from 2007:

Students could certainly start with Step 1 again and work their way down, but I encourage them--once they get comfortable--to feel free to start with Step 2 and fill in the blanks as they see fit.  Here's how I would suggest they work this problem:

That's it!

BFFs: f, f', and f''

In AP Calculus, we're currently working on applications of the derivative.  As I studied past AP Calc exams this summer, it was clear to me that students need a very firm understanding of the relationships between f, f', and f'' in order to be successful on the exam.  I've been gently guiding my students in this pursuit the entire semester (in fact, that's how they discovered derivatives of trig functions), but now we're diving in head first.  I know that this is not an easy concept to master.  Very few students "get" it right away (I didn't either at their age).  But, to me, that's what makes it super fun to teach.  Or try to teach.

So, here's what we have been doing in Calc AB to help students solidify these three relationships:

• Introduction to f, f', and f'' by matching their graphs in groups of 3-4 students.  The matching activity is very similar to this one.
• Students conceptualized what it means for the first derivative to be positive, but the second derivative to be negative (for example) by filling out these charts:

How Are f, f', And f'' Related

• Students described concisely in words through this chart:

• My still all-time favorite, Inflection via Infection
• Daily Warm Up where students have to answer about ten questions like:

1. If f is increasing then f' ______________.
2. If f has a point of inflection then f' _____________.
3. If f'' is negative then f ______________.
4. If f'' is negative then f' ______________.

And then the finale:  a nine-question clicker quiz similar to the questions above.  The students who scored less than a 50% on this quick assessment are being called into lunch next week to get further help (this was totally my colleague's idea...genius!).  What I loved about the clicker quiz was that I could post the results as soon as the kids were done and then we could talk about the questions that gave them the most trouble.

For the kids who are coming in for extra help, we have created a packet where they will be given a function and then instructed to graph the function and its first two derivatives.  Then they'll answer questions like "Where is f concave up?"  "Where is f'' positive?"  "Where is f' increasing?"  And, hopefully, they'll see that the answers to all three questions are the same.

It seems my students do fairly well when they are asked questions about what the first and second derivatives tell you about the original function.  However, they have a hard time telling you what the second derivative tells you about the first derivative.  They don't seem to make the connection that that's the same thing as asking what does the first derivative tell you about the original function (which, like I said, they can do just fine!).  For example, on the quiz, the first two questions were:

1. If f is increasing, then f' is ____________.
2. If f' is increasing, then f'' is ___________.

They did beautifully on the first question; horribly on the second.  When I asked them, "Do you see how the two questions are the same?  In each case, you've only derived once."  I got a few "Ah-ha!"'s, but I think several are still struggling to see the connection.  So, that led me to create this chart:

No words.  All symbols.  And I purposely did not call any of the functions f.  My hope is, if they can understand this flow chart, they will now be able to answer questions like #2 above.  We shall see how it goes.

What other things do you do to help students with these ever-important relationships?

Derivatives of Trig Functions

One of the things I find challenging to balance is convincing kids of mathematical truths without overwhelming them.  Sometimes, I know, there is a time and a place for a bit of hand-waving.  And, sometimes, I know, there is a time and a place for formal proofs.[1]  But I think most of the time the sweet spot is somewhere in between a formal proof and "this is how it is--just memorize these rules."

In search of that happy medium, I created decks of 12 cards (6 with the graphs of the basic trig functions {orange} and 6 with the graphs of their derivatives {blue}).  I had students match them up with a partner.

Matching a function to its derivative using only graphs is new for my kids, so I knew this would be a challenge if I didn't lead them quite a bit.  However, gathering data from a graph is so heavily tested on the AP exam that I figured it wouldn't hurt to start making some connections.

After they matched them up, I followed up with these questions:

Derivative of Trig Functions Follow Up Questions

Here are the cards I made, if you're interested (thanks, Desmos!).

6 basic trig functions (enough for 16 decks):

Trig Functions Matching Cards1

6 derivatives (enough for 16 decks):

Derivative of Trig Functions Matching Cards2

[1]  Although I'm beginning to think I show proofs more for myself than my kids.

Understanding the Derivative via Strogatz

If you've never read Steven Strogatz's book The Joy of x, you should put it on your reading list.  Strogatz, in my opinion, is able to sell and teach the development of mathematics to a general audience--which is no easy task.  He's a brilliant teacher in this book and can be appreciated by both "math people" and "non-math people," educators and non-educators alike.

I have a class set of his books, and I got to put them to use for the first time this week.  I had my calculus students read the beginning of the chapter entitled "Change We Can Believe In."  Strogatz does such a great job explaining the value of a derivative in this chapter.  I gave my students an anticipation guide and explained the value of anticipating where an author is going with the material...before you read the actual material.  I think this is especially true in mathematics:  it took me a looooong time as a student to realize math textbooks could be used for more than just the problem sets.  But, when I did start to fully appreciate math texts for their entire content, I was invested in the material because I would make predictions about the proofs before reading.  If I could get through the proof without the help of the author, wohoo! (rare, but wohoo nonetheless).  If not, I had invested enough time and energy into the problem that, by golly, I was going to figure it now.  Which meant I needed to READ.

I digress.  This wasn't supposed to be a post on the value of this literacy strategy.  But there you have it anyway.

Here's the AG I gave the kids.  They did argue through a few of the statements, which is exactly what I'd hoped for.

The Joy of X-Change We Can Belive in (AG)

Students asked when they would get to read from the book again and where they could their own copy of the book...so I count this as a success.

Derivatives of Exponentials and Logs with Desmos

Here's a Desmos activity I typed up for my calc kids to find the the derivatives of exponential and logarithmic functions.  Sadly...the class set of laptops would not connect to the domain because they hadn't been used all summer, so we did this together as a class, which was not what I wanted, but what can you do?  Instead of having the kids click pause on their own screens, I had them yell "STOP" at me...so it was still entertaining.

All this to say, I don't know if this is good or not since I haven't gotten to test it out on students yet, but here it is.  Use/modify if you'd like!

Derivatives of Exp and Logs Desmos

In the "notice/wonder" section of f(x)=a^x, one student said he noticed that the derivative was proportional to the given function.  This made me a very proud momma and was a perfect segue into finding the derivative when a is different from e.  (We explored f(x)=2^x, f'(x), and g(x)=lna(2^x), and found that a=2).

Desmos also sent me this great online activity.

Introduction to Tangent Lines

I've been loving this introduction to calculus that we're doing with our Pre-Calc classes currently.  I don't know about you, but when I was in Pre-Calc, I didn't do any calculus.  Not a single thing.  I had no clue what a limit was, and certainly not a derivative. My Pre-Calc class was pretty much just trig, trig, and more trig, with a bit of "advanced" algebra thrown into the mix.  (I'm not complaining though--it was a great class, honestly...and I'm told I should be thankful that I'm young enough to even have had a class termed "Pre-Calculus.")

Anyway.  All this to say--it's darn exciting introducing kids to concepts such as limits, derivatives, and integrals because they're so powerful and beautiful...and so unlike other stuff we teach (no?).

So, a few things I'd like to share from this week.  Nothing's super original, but I did put a lot of time and energy into making them work for my students.

First:  Visualizing secant lines turning into the tangent line via Desmos.  Again, I know there are plenty of applets out there, but I couldn't find any that my students in the back of the room would be able to see.  Also, I wanted to input my own functions.  Also, I wanted to create it because it's fun and allows me to use mathematics.  So, here you go.  Slide a, change the function, change the point of interest.  Best of all, put it in projector mode so everyone can see--even the kids in the back.

Second:  We had an extra day built-in for tangent lines, so during collaboration, I asked if we could create a packet that introduces the kids to how to draw those lines exactly.  And how does the algebra relate to the geometry?  My department head and I discussed the objectives, and then she miraculously turned our words into this beauty:

Tangent Line to a Curve

Third:  This Warm Up that I rather like (Day 3 of Tangents):

Fourth:  I used these sites so the kids could get some practice visualizing what the derivative function would look like without taking the time to actually find it algebraically.  I love exercises like this because they truly require deeper thinking.  You can't bs your way through them.

• Some multiple choice practice
• Derivative plotter

The Importance of Change

I was at a car parts store the other day, and I saw a sun shade for your windshield that advertised this:

I had to double-take.

A change of 44F is the same as a change of 7C?  Now, I'm not as savvy at converting from Celsius to Fahrenheit as Kate Nowak, but even I knew this was some faulty converting by a marketing department.

No.  I take that back.  I admit, they did the conversion fairly well.  44F is indeed (approximately) equal to 7C.  But a change of 44F and a change of 7C?  That's a little different.

Couldn't they just have tried an example?  One example.  That's all I ask.  Like, take two numbers whose change is 7, say 0C and 7C.  Covert those to Fahrenheit (32F and 45F, respectively), and you'd see right away--that's not a 44 degree change.

Check for the reasonableness of your answer, as I tell my students.

So, what is a change of 7C equal to in Fahrenheit?  Let's do a little algebra!

We know to get from Celsius to Fahrenheit we can use:  

So, let's take two temperatures, in Celsius and call them C1 and C2.  Then their change, in Fahrenheit, can be calculated through the following:

In other words, the change in Fahrenheit, is equal to 9/5 times the change in Celsius.[1]  Which means that a change of 7C is only equal to a change of 12.6F, a far cry from 44F, in my opinion.  And this verifies the consistency of our example that we tried earlier.  Which is good since this is a linear function we're talking about.  No change in slope here.

So, which is it?  Does this sun shade keep our vehicles up to 7C (12.6F) cooler, or 44F cooler?  Maybe we just get to choose.


[1]  This can be changed from an algebra problem to a calculus problem rather nicely.  I think this would be a lovely introduction to using differentials as approximations for actual change (or in this case, since the function is linear, the differential will be equal to the true  change).

Step 1:  Give the kids the function for F with respect C.
Step 2:  Have them calculate dF when dC=7. [dF=F'*dC=12.6]
Step 3:  Show them the  picture at the top.
Step 4:  Watch them pee their pants (they're math nerds too, right?).

I heart these two calc problems

These were both AP Calculus sample questions for 2012.  And I just really like them:

More thoughts on the Chain Rule

I posted about how teaching the Chain Rule was a lot harder than I thought it would be.  I'm still convinced this is at least partially due my students' lack of understanding/recognizing a composite function.  For example, just the other day we needed to simplify the expression



And a student (one of my top students, I might add) suggested we "divide out an ln."

Hold up.

Let's ignore the fact that dividing by any number other than 1 would change the expression.

Dividing by ln?  So...somehow there's not a connection that ln is meaningless without an argument.  "Dividing by ln" is akin to "dividing by √ " or "dividing by cos."  An empty square root or an empty cosine doesn't have any kind of value, and really doesn't mean a thing.

I was further disturbed when I gave my Business Calculus students a function like



and was told that in order to find the derivative, we should use the Product Rule.

Wha...?

Do students view ln as some sort of constant?  Like e?  It seems maybe so if the function above is thought of as a product and if we can indeed "divide out ln."

I decided we needed to revisit the Chain Rule.

I started by showing a slide that had a composite function at the top and four expressions beneath it such as:

I asked my students to tell me why we needed to use the Chain Rule and then to identify the derivative of the outside function (holding the inside) and the derivative of the inside function.  The next slide highlighted the former in red and the latter in blue:

We did several of these.  Then we concluded with other types of functions to test if they knew when to use what rules.

I think I will start with this type of presentation the next time I teach the Chain Rule.  Giving the students a limited amount of options to start out with seemed to worked fairly nicely.

There are still definitely some issues.  But I now have a better understanding of what needs to be emphasized  in terms of composite functions.  I will try to make them more densely populated in my algebra and pre-calculus classes from now on.

If you're interested, here's the slideshow we worked through:

Test 3 Review

M&Ms and the Population of Afghanistan

In Business Calc, we're currently studying exponential growth and decay.  I'm rather excited about this since it's something we study in College Algebra, too, and I feel--because it's material I've taught before--that I can expand a bit.  I'm learning that it's really, really hard to expand (i.e., go beyond an absolutely dazzling lecture *cough*) when I'm teaching a class for the first time.  I sort of feel like I did my first semester as a TA:  I just hope I don't screw something up too terribly. But--you gotta start somewhere, right?!

Anyway.  Back to exponential growth/decay.  In College Algebra, when we study exponential functions, I have my students model the decay of an M&M population.  I had planned to do this with my Business Calc class as well.  Then Bowman Dickson posted places to find awesome data, which made me want to use data the UN has on the world's populations instead.

The question:  How to relate M&M's to population growth or decay?

The answer:  I'm not entirely sure.  Here's what we did though...

We started out with the M&M project as in College Algebra.  Each team found the exponential regression (in the form y=ab^x) and the r^2 value for their data.  We talked about the meaning of a and b.  Then I asked them to convert their regressions to the form P(t)=P_0e^(kt), which turned out to be very close to the trendlines Excel found (yay!).  We talked about what k would mean if it this were a real population and how it's related to the derivative.

Now the challenge:  I asked them to do the same types of calculations for an actual population, using data from the UN.  They were on their own for this project, which may or may not have been a great idea.  Below is what they had to go off of.  I focused mainly on finding the exponential regression on a TI as well as understanding growth/decay rates.  But there's much, much more to do here (Bowman does a week-long project!).

And here's the project!

Population Growth or Decay Instructions

Derivative Cards

Two weeks down of summer classes, six to go.

These past two weeks have reminded me how hard it is to teach a class for the first time.  I continually feel like I'm coming up short because I compare my performance in my summer classes to my performance in a typical 16-week class that I've taught many times.  I know that's not a fair judgment, but I still do it.  I also know you can't learn how to teach a course well if you don't ever teach it that first time.  Still, I feel all my energy is spent just trying to get half-way decent lectures ready, to keep up with homework questions, and to write tests.  I don't have much time or energy to provide learning experiences outside of the typical lecture.  And I hate that.

That said, I'm learning more about the teaching and learning of calculus every day.  And I think that's pretty priceless.  Also, I think I've been given a set of unusually patient and gracious students this summer.  I get thanked about every other day for doing what I do.  And for someone who needs pats on the back, that is the biggest reward I could get.

That's my vent.  Now for some calculus...

Every semester I have my students write an introduction about themselves.  In addition to hobbies and life goals, I ask them to tell me why they're in the class, their math background, and their current feelings towards mathematics.  In doing so this semester, I learned that many of my students have had calculus before.  So, before we ever talked about the Power Rule, I would ask something like, "How can we find the derivative y=3x-2?"  I would, of course, get an eager, "Well, I learned how to find derivatives another way, and you just take the exponent and multiply it by 3 and then reduce the exponent by one...so the derivative is 3."  Everyone loves the Power Rule.

Apparently, we still need to
work on writing "lim as h
approaches 0."

"You're absolutely right, and we'll talk about the Power Rule soon.  But can you think of the geometric definition of a derivative and tell me what 3 corresponds to in the linear function?"

All this to say, I wanted to introduce the Power Rule differently, somehow.

I split the class into eight groups and gave each a "Derivative Card."  On the top it had "Find f '(x) when f(x)=..."  I used four basic functions:  f(x)=x^2, x^3, 1/x, 1/x^2.  Each function had its own color.  When the students were done, I asked them to find the other group with the same color and see if they got the same answer.  I loved this because the students were doing all the work.  We then created a table on the board using what the groups just found.  I started with f(x)=x (which I didn't give to any group).  The table looked something like this:

It's nothing new, but the students came up with it themselves, which is the great part.  They were able to generalize the rule no problem too.  Which makes my heart very happy.

Update 6/26/2014
Here are the cards from above in electronic format:

Power Rule Cards