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Understanding the Slope of the Tangent Line

03 Sep

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.

E1

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.

Screen Shot 2014-09-03 at 5.52.52 AMScreen Shot 2014-09-03 at 5.53.05 AM

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

Screen Shot 2014-09-03 at 8.10.31 AM

 Screen Shot 2014-09-03 at 5.44.59 AM Screen Shot 2014-09-03 at 5.45.16 AM

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.

E2

What happens to the tangent line at D?

E3

What happens to the tangent line at C?

E4

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

Screen Shot 2014-09-03 at 5.46.16 AM Screen Shot 2014-09-03 at 5.45.37 AM 

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.

09-03-2014 Image001 09-03-2014 Image002

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

09-03-2014 Image003 

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?

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When is the derivative undefined?

09-03-2014 Image005

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

09-03-2014 Image006

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

 

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

What do we need to calculate slope?

Two points

09-03-2014 Image008

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.

Screen Shot 2014-09-03 at 7.39.30 AMScreen Shot 2014-09-03 at 7.39.50 AM

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?

09-03-2014 Image009 09-03-2014 Image010 

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.

Screen Shot 2014-09-03 at 7.40.01 AM

And then we calculated the derivative, f’(x) using the definition for one function, f(x)=x2. And we connected what we got for f’(x) when we found f’(4) to the slope of the tangent line.

Screen Shot 2014-09-03 at 7.48.21 AM

And then the bell rang.

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3 Comments

Posted by on September 3, 2014 in Calculus

 

Tags: , , ,

3 responses to “Understanding the Slope of the Tangent Line

  1. 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

     

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