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Derivative Rules

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.

Learning Objective 2.1C: Calculate Derivatives

Essential Knowledge 2.1C2: Specific rules can be used to calculate derivatives for classes of functions, including polynomial, rational, power, exponential, logarithmic, trigonometric, and inverse trigonometric.

Essential Knowledge 2.1C3: Sums, differences, products, and quotients of functions can be differentiated using derivative rules.

Mathematical Practice for AP Calculus (MPAC) 1: Reasoning with definitions and theorems

Students can: develop conjectures based on exploration with technology.

 

How do you provide students the opportunity to develop conjectures?

After determining the derivative of a few quadratic functions using the definition, we use our TI-Nspire Computer Algebra System (CAS) software to explore derivatives.

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We use the power rule to make conjectures about the product rule. (I think that I saw this suggestion in a Mathematics Teacher magazine in the early 90s, but I can’t find the reference now.)

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We know what the derivative should be, because we know the derivative of x^5. How could we use f, f ‘, g, and g ‘ to get to what we know is the derivative from the power rule?

Once students made conjectures about the product rule, we formalized the rule.

I asked students to predict the derivative of f(x)=sin(3x). As expected, many thought that it would be f ’(x)=cos(3x). When we looked at the graph of the derivative of f(x), students realized that f ‘(x)=3cos(3x).

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We used CAS to explore the chain rule (power and composite) in more detail.

Students practiced “Notice and Note”. Several generalized the chain power rule before I asked.

Once students knew the chain rule, we used the chain rule to derive the quotient rule.

And so the journey providing opportunities for students to make sense of rules instead of just telling them rules continues …

 
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Posted by on September 14, 2016 in Calculus, Derivatives

 

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An Infinite Number of Rectangles

We have started our unit on the definite integral for a few years now with Lin McMullin’s The Old Pump.

I love watching students work without yet having developed Riemann Sums. Many use areas of rectangles to approximate the amount of water in the tank, but even then, they don’t all do it the same way.

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That work leads us to developing the idea of estimating area between a curve and the x-axis using Riemann Sums and the Trapezoidal Rule. And then we are finally ready to determine the exact area between a curve and the x-axis using a Riemann Sum with an infinite number of rectangles.

We practice reason abstractly and quantitatively throughout these lessons.

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Once we’ve thought about numerical approximations for area between a curve and the x-axis, we spend some time writing a Riemann Sum to represent area and evaluating its limit as the number of rectangles approaches ∞. I want them to be able to go backwards, too. So we start with a limit, and I ask them what definite integral will have the same value.

Which is apparently not as difficult as the groans suggested when I first gave it to them.

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But we are always working on our Mathematical Flexibility, and while I was pleased that everyone can get a definite integral, I was disappointed that they all did it the same way. Jill Gough has provided us with a leveled learning progression for Mathematical Flexibility.

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Can you write another definite integral for which the area can be calculated using the given limit?

It took a while. But students made progress. Some made use of the symmetry of the graph of y=x2 to write a second integral.

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Some figured out that translating the parabola and the limits of integration one unit to the right would result in a region that has the same area.

Those were the types of answers I was expecting. But I also got answers I wasn’t expecting.

Some of my students were on the path to Level 4 of reason abstractly and quantitatively, beginning to generalize the idea of translating the parabola and the limits of integration c units to the right, resulting in a region that has the same area. They didn’t quite make it, as their limits were shifted to the right c but their parabola shifted to the left c. I was still impressed by their jump to Level 4, finding connections between pathways.

Our TI-Nspire CAS software let us check our results and helped us attend to precision.

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And so the journey continues … learning more from my students and our technology every day about mathematical flexibility.

 
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Posted by on February 11, 2015 in Calculus

 

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