CA II.5 Derivative of the Power Function
Purpose: To generalize from developing the derivative of x^2, x^3 etc to the general power rule and then practice applying the power rule.
Classroom Procedure: Students work on the first page by themselves after you have emphasized that they should only be using the graphs to sketch the derivatives - we are not expecting them to guess which functions are graphed and use any prior knowledge about the derivatives of these functions. Then as instructor you provide the symbolic form of the functions that were graphed (f(x)=4, f(x)=x, f(x)=x^2, f(x)=x^3, f(x)=x^4. Now that we have the symbolic forms we can make corrections to the first 3 because we already know the derivatives of these types of functions but we are still just conjecturing on the last two. It is important not to give the "right answer" at this point but simply to write down the various conjectures. They should then work on Q2 in small groups to support their analytical conjectures numerically. After this it is a good idea to come together as a whole class and get the students to decide which of the conjectured functions they now agree on. After this they can either be given some time to work on 3 alone or it could be done as a whole class depending on time available. If time permits it is better if they can try Q4 by themselves/in small groups first before confirming the power rule as a class. Then they should be sent home to complete the practice problems.
Ideas this Activity Builds On: Students have already (or are able to have) reasoned to the derivative of functions of the form f(x)=4, f(x)=mx+b, and f(x)=x^2 from work in previous activities. We are building on this work and the ideas of numerical estimations of the derivative in this activity.
Introduction/Motivation of the Activity: One aspect of mathematics is the recognition of patterns that can save us work in the long run. For certain types of functions we can recognize a pattern in the derivative functions and write that down as a "rule". Here we will begin our work on that.
Need to Establish by the End of Activity/Wrap-Up: It is important that students walk away knowing and being able to apply the power rule. In addition we are beginning a process of investigating patterns in the derivatives of functions that we will continue in later activities so it is important that students see the value in participating in the building of the rule not just in the end product. This helps their understanding of the rule and their ability to apply it.
Additional Notes: The synthesis questions for this activity introduce the second derivative. It is a great idea to have students take home the synthesis questions and work on them for a night but then it is important to follow these synthesis questions up with some in class time where the results that they should come to are discussed. For instance the connection between the second derivative and concavity and the reasoning for this connection through the first derivative. For example one such reasoning is when a function is concave up it might be increasing at an increasing rate or decreasing at an increasing rate - the common piece of this is the increasing rate so what should be true of the first derivative - it should be increasing - therefore what should be true of the second derivative - it should be positive.