My calculus students used a toothpick to explore tangent lines. I heard this idea from Paul Foerster in a workshop at some point along the way.

We started with the tangent line at A. Everyone didn’t have a horizontal tangent line, but most did. Students quickly determined that A’ would be at the relative minimum of the graph.

Is the slope of the tangent line greater at B or at C?

How do you know?

I used the TI-Nspire document on the board as we talked about their conjectures.

Move the tangent line (toothpick) all along the curve. Write down at least two observations (by yourself).

Then students discussed their observations with their teams. And then each team told the whole class one observation at a time until we had heard them all.

We discussed increasing/decreasing intervals, concavity, relative extrema, and more.

What happens to the tangent line at D?

What happens to the tangent line at C?

Students create their own function with certain requirements for tangent lines at A, B, and C.

Next we moved to Derivative Trace. The y-coordinate of Point P represents the slope of the tangent line for the each x-coordinate of P. What path does P trace? Students watched a few times. A few said that the path P followed looked like a sine curve; others said that the path P followed looked quadratic.

We used the Automatic Data capture feature of TI-Nspire to see a scatterplot of the path that P followed.

We will call that path the derivative.

What is true about the derivative when the original function is increasing?

What is true about the derivative when the original function is decreasing?

When is the derivative equal to zero?

When is the derivative undefined?

What is the derivative of f(x)=sin(x)?

What is the derivative of f(x)=e^{x}?

So how will we calculate the slope of the tangent line?

What do we need to calculate slope?

Two points

Then we can calculate the slope of the secant line.

We named the points on the secant line as (x,f(x)), (x+∆x,f(x+∆x)). Students wrote a representation for the slope of the secant line.

But we don’t really want the slope of the secant line – we want the slope of the tangent line. How can we change the secant line into the tangent line?

We want ∆x to get as small as possible.

We want ∆x to approach 0.

We want the limit as ∆x approaches 0 of the slope of the secant line.

And then we calculated the derivative, f’(x) using the definition for one function, f(x)=x^{2}. And we connected what we got for f’(x) when we found f’(4) to the slope of the tangent line.

And then the bell rang.

howardat58

September 3, 2014 at 9:59 am

Just in case some of your students cannot cope with limits, have a look at my posts on this subject

http://howardat58.wordpress.com/2014/08/19/calculus-without-tears-that-is-without-limits/

http://howardat58.wordpress.com/2014/08/21/calculus-without-limits-2/

Number 3 is there, number 4 to come, for sine and cosine

jwilson828

September 3, 2014 at 10:13 am

I’ve seen several posts recently about calculus without limits. David Bressoud has been posting on his blog at MAA: http://launchings.blogspot.com/2014/06/beyond-limit-i.html, http://launchings.blogspot.com/2014/08/beyond-limit-ii.html, http://launchings.blogspot.com/2014/09/beyond-limit-iii.html.

For now, I think that technology plays a big role with limits making sense to my students.

howardat58

September 3, 2014 at 9:05 pm

I checked these out. His approach is about seeing slopes of tangents/rates of change as approximations. It seems to be avoiding the issue. I do not see how that way you will ever end up with d/dx(x^2) = 2x

I do think that “differentiate from first principles” problems should be banned!

I am going to send you some stuff on reflections soon.