Overview

Topic 2.7 is being investigated over the course of two days: derivatives of sine and cosine on day 1 and then derivatives of e^x and ln x on day 2. Graphical representations are the focus today in order to provide students a firm visual foundation of these very important derivatives. In this lesson, students use toothpick tangents to estimate the magnitude and sign of the slope at various x-values. 

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Teaching Tips

Remind students that estimating the slope of the sine graph is sufficient for the first activity. Students are given reference values which can be used to estimate further slopes. We expect that students will make sense of the symmetry of the graph to estimate these values. On page 2, some of these reference values are taken away as we expect students to make some connections about the slopes at these points. We ask students to plot these slope values in a different color and connect them with a smooth curve, thus emphasizing that these slope values are outputs of a function: the derivative function applied to y = sin x. Developing these formulas is of enormous value for students and builds deep conceptual understanding. Instead of us simply lecturing about a new formula to memorize, students again create the derivative rule for cos x.

We recommend pausing to debrief after page 1 to make sure everyone agrees on the derivative of sin x. This also helps ensure that no group falls too far behind or rushes too far ahead. Have a student sketch their slopes graph on the board.

On page 2, the final questions asks students about the slope of the tangent line at 7pi/6, which is not one on their table, but knowing the formula for the derivative of cos x makes this a straight-forward problem. Demonstrate on the slopes graph that at x=7pi/6, that does in fact match up with a y-value of 1/2.

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Exam Insights

Knowledge of sine and cosine values (one of the algebra concepts we have to review often over the course of the year!) is imperative not only for the non-calculator parts of the AP Test, but for speed and efficiency on the calculator-allowed sections. Our students are required to know the exact values for the usual angles in the first quadrant and how to reflect these values into the other three quadrants. There are lots of tips, tricks, and strategies for accomplishing this. Whatever works best is what your students should implement!!

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Student Misconceptions

Question #6 is a surprise for many students who believe the derivative of the cosine function will simply revert back to the sine function. What they produce is, of course, the opposite of what they expect! Soon, students will have motivation to remember that the derivatives of the “co” trig functions require a negative sign.