Determine behaviors of a function based on the derivative of that function.
I can use the sign of a function’s first derivative to determine intervals when the function is increasing or decreasing.
I can locate relative extrema of a function by determining when a derivative changes sign.
Quick Lesson Plan
In this lesson, we create some motivation for the first derivative test with a stock market game. Students keep track of the change in value (derivative) of the stock as well as the current value and make predictions about the best time to “exit” the game (a.k.a. sell stock). They learn through play that the maximum of a function occurs when the derivative switches from positive to negative.
To begin the game, you may want to remind students of the #1 rule of stock investments: buy low and sell high. Choose a volunteer to be player 1 and explain the rules of the game. Every player’s starting value is $10. For each day of the game, you (the teacher) will give them the change in the value of the stock. A recorder keeps track of this on the board and all students also keep track on their lesson page. Player 1 then decides if they want to keep playing or exit the game. Player 1 will likely play all 10 days since there are not many patterns to notice yet.
Player 2 is now up to play. The same rules apply, although this student may have noticed some patterns from player 1, and may choose to leave the game on day 5. If a student exits the game before all 10 days are completed, have students use a different color to finish the table and record the values they would have gotten.
We suggest being as dramatic as possible when revealing the changes in stock value. Our students tend to be at the edge of their seat. They want to know if they made a good decision or not!
Player 3 will probably be surprised that their stock value is decreasing right away! They will likely hang in the game until day 7, thinking their stock will decrease in value again after the day of no change. Revealing the change in value on days 8-10 reveals a key results: just because a derivative has a value of 0, doesn’t mean it is necessarily a maximum or minimum. It’s possible the stock increases, has no change, and then increases again. I refer to Player 3 by name whenever we do a problem where the critical point is neither a maximum or a minimum (“just like what happened with Daniel’s stock!”)
As soon as the game is done, assign students to complete questions 1-4 on their page.
Key takeaways from the stock market game:
--Pay attention to when the derivative is 0! These are important (critical) values! Whenever students see max/min problems, they should always know to set the derivative equal to 0 (or see where it is undefined). This is an entry point that makes these types of questions accessible to all students.
--A relative maximum occurs when the derivative is equal to 0 (or undefined) AND changes from positive to negative. This type of justification is critical on the AP Calc FRQ questions.
--Absolute maximums can occur when there is a relative maximum OR at the endpoints. Player 3 would have reached their highest stock value on day 10! The candidates test will be explored in greater depth in the next lesson but this is an appropriate preview.
--Although the value of real stocks does not change so predictably, many functions do! If a function’s derivative is continuous it must pass through 0 before switching from positive to negative values or from negative to positive values, thus giving us important information about when we’ve reached a maximum or minimum.
When debriefing the game, question students about why the stock value is not the greatest when the change in value (derivative) is the greatest, since this can be a common misconception. Explain the idea that even if there are only tiny gains made, the value of the stock is still increasing, and thus better for the stockholder.
Sign charts as the sole justification of relative extreme values has not been deemed sufficient to earn points on free response questions. Please review the article “Sign Charts in AP Calculus Exams,” available on the AP Central site. Students must present evidence of calculus knowledge by declaring a change in the sign of the first derivative: the First Derivative Test. For example: g(x) has a relative minimum at x = 3 where g’(x) changes from negative to positive. See 2016 AB 3a, 2015 AB 1bc, 1998 AB2, and 1987 AB 4. Other explanations will suffice after students explore the Second Derivative Test.
Here are several important details often neglected by students which have been highlighted in this activity. Use “Playing the Stock Market” to emphasize that the behavior of the first derivative over an interval must be examined before students claim a relative max or a relative min at a critical point. As the activity illustrates, a derivative value of zero does not always indicate relative extrema! Continue to encourage investigations at end points of closed intervals when searching for absolute (global) extrema, even though the Candidate Test has not been formally introduced.