Overview

This lesson, combining Topics 7.1 and 7.2, offers a low-stress introduction to Unit 7 as students review the familiar concept of differentiation. Students analyze the temperature of a cup of coffee (are we the only ones obsessed with coffee??) using verbal and graphical interpretations of a differential equation. 
Important vocabulary is introduced in the Important Ideas section. 

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Teaching Tips

If your students do not recognize a difference in the structure of today’s focus derivative, be sure to point out that dC/dt is written in terms of the function C, not in terms of the independent variable, t. This is an important departure from most of our work so far. The rate at which the temperature of the coffee is changing depends on the current temperature of the coffee --- but also on the difference between the coffee and the ambient temperature (70°F in our example). 

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You may choose to have the class explore what type of function contains the original function in its derivative. Likely, at least a few students will remember that f(x) = ex is the correct response.  Some students may even posit that a derivative of the form dy/dx = ky will return an antiderivative that contains an exponential function. 

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After a bit of discussion about Question 5, most students were able to show an equivalence between the given dC/dt expression and their derivative of C(t). Question 6 got the most heated debate in the classroom, but Mrs. Montgomery doesn’t put milk in her coffee, so she left the proof of the conjecture to Ms. Stecher!

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Exam Insights

Topics 7.1 and 7.2 are preparing students to work with slope fields and separation of variables on the AP Test. Students who are comfortable with the concepts and vocabulary in these lessons will transition more easily to creating/interpreting slope fields and solving integrals by separation of variables.