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## A New Function

09 Feb

One of the NCTM Principles to Actions mathematics Teaching Practices is support productive struggle in learning mathematics. In the executive summary, we read “Effective teaching of mathematics consistently provides students, individually and collectively, with opportunities and supports to engage in productive struggle as they grapple with mathematical ideas and relationships.”

In calculus, we started the semester with a unit on Transcendental Functions. On the first day, students figured out everything they could about F(x).

What is F’(x)?

What is F’’(x)?

What is F(1)?

Where is F(x) increasing, decreasing?

Where is F(x) concave up, concave down?

What is the domain for F(x)? the range?

Then they sketched a graph of F(x) from what they figured out, and determined that F(x)=ln(x), and F’(ln(x))=1/x.

(I found the suggestion for students coming up with F(x)=ln(x) by thinking through these questions somewhere else. But I don’t remember where, and I can’t find it anymore.)

So the next day, I asked them to differentiate y=log(2x).

I had not given them any “formula” for differentiating logarithmic functions. They had only figured out that the derivative of ln(x) was 1/x.

I sent the question to them as a Quick Poll to watch their progress.

I watched for a long time.

I saw and I heard productive struggle.

And eventually, their struggle turned into success.

We can cover so many more examples when we don’t give students time to grapple with mathematical ideas and relationships. But how effective are the examples without the productive struggle?

Ultimately, are my students better off having struggled to think through change of base to get to the derivative of log(2x) using what they already know about the derivative of ln(x)? Or would they have been better off with me giving them the textbook way to calculate the derivative of logb(x)?

I’m hoping for the former, as the journey continues …

4 Comments

Posted by on February 9, 2015 in Calculus

### 4 responses to “A New Function”

1. February 9, 2015 at 7:35 pm

Feet on the ground says “Where in the real world does anybody want to differentiate log(x) (base 10). Engineers, scientists and mathematicians always use natural logarithms. The base 10 stuff is a hangover from the days of log tables (I was there!). I guess it’s like long division, something to put in the exam.
On the other hand, I would be tempted to convert y=log(x) into x=10^y = e^(ln(10)y)
Then 1 = e^(ln(10)y).ln(10).y’ = x.ln(10).y’
and so the derivative y’ = 1/(x.ln(10))

The most important thing to me is a^x = e^(x.ln(a))

I didn’t understand the contents of the last box. Where did the u come from?

I’ve spent too much time programming computers, this all on the line stuff is screaming out for * , and this type face can’t even do that properly!

• February 11, 2015 at 9:06 am

Of course it should be log base a of u … That’s what happens when I retype equations. Thanks for point out the typo. As far as who will use it, a lot of what I teach is driven by what is on the AP exam. We do enough “when somebody really uses this” examples that my students are okay with having to know a few extras.

2. February 10, 2015 at 6:55 am

You may have read my post on calculus without limits 5. If not you might find it interesting:
https://howardat58.wordpress.com/2014/10/15/calculus-without-limits-5-log-and-exp/

• February 11, 2015 at 9:06 am

Thank you!