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Seeing the Definition of Derivative

Big Idea 2 from the 2016-2017 AP Calculus Curriculum Framework is Derivatives.

Enduring Understanding 2.1: The derivative of a function is defined as the limit of a difference quotient and can be determined using a variety of strategies.

Mathematical Practice for AP Calculus (MPAC) 2: Connecting Concepts

  1. relate the concept of a limit to all aspects of calculus
  2. Students can connect concepts to their visual representation with and without technology.

 

How do you introduce the definition of a derivative?

We start with the visual of a tangent line at a point and a secant line containing the point.

Definition of Derivative 1.gif

We don’t need calculus to determine the slope of the secant line. We do need calculus to determine the slope of the tangent line.

How might we use the slope of the secant line to determine the slope of the tangent line?

Definition of Derivative 2.gif

 
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Posted by on September 9, 2016 in Calculus, Derivatives

 

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Seeing the Derivative

Big Idea 2 from the 2016-2017 AP Calculus Curriculum Framework is Derivatives.

Enduring Understanding 2.1: The derivative of a function is defined as the limit of a difference quotient and can be determined using a variety of strategies.

Mathematical Practice for AP Calculus (MPAC) 2: Connecting Concepts

  1. Students can connect concepts to their visual representation with and without technology.

 

How do you introduce the concept of a derivative?

We start with visuals of the derivative (from a Getting Started with Calculus activity called Derivative Trace).

Seeing the Derivative 1.gif

Seeing the Derivative 2.gif

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Tangent Line Demonstration from Calculus Nspired has some similar ideas.

We then look at the Derivative Grapher to connect the slope of the tangent line to the graph of the derivative, changing the original function as requested by the students.

Seeing the Derivative 8.gif

We begin to develop some common language around derivatives before we formalize what is a derivative, before we formalize the definition of a derivative.

 
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Posted by on September 8, 2016 in Derivatives

 

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Seeing Limits

Big Idea 1 from the 2016-2017 AP Calculus Curriculum Framework is Limits.

Enduring Understanding 1.1: The concept of a limit can be used to understand the behavior of a function.

Mathematical Practice for AP Calculus (MPAC) 2: Connecting Concepts

Students can connect concepts to their visual representation with and without technology.

Mathematical Practice for AP Calculus (MPAC) 4: Multiple Representations

Students can associate tables, graphs, and symbolic representations of functions.

Students can develop concepts using graphical, symbolical, verbal, or numerical representations with and without technology.

 

We begin calculus with a discussion of limits. I throw a dart at the dart board and ask what just happened.

You hit the bullseye.

(I really did this year – on the second try – after missing the board completely on the first try.) But how did the dart make it to the board? It had to go half the distance to the board. And then half the distance again. And again. And again. How did the dart make it to the board?

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The other demonstration is at the recommendation of my grandfather, now 101, who taught high school mathematics for over 50 years. I take a measured piece of yarn (1 meter), cut it in half, and begin to make a pile of the “halves”. How long does the yarn measure in the pile if we put it end to end? It gets closer and closer to measuring 1 meter, but does it ever make it to 1 meter?

How do you provide opportunities for students to “see” limits?

I used to [ineffectively] wave my hands. Now we use technology. Students won’t use this type of visualization while they’re taking a test, but we find it invaluable while they’re learning.

Limit 1.gif

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Limit 5.gif

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Posted by on August 24, 2016 in Calculus, Limits & Continuity

 

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