*Six of nine. *Continuing the current series of posts, this post looks at the AB Calculus 2021 exam question AB 6. Like most of the AP Exam questions, there is a lot more you can ask based on the stem of this question and a lot of other calculus you can discuss. This series of post offers suggestions as to how to adapt, expand, and use this question to help your students dig deeper and learn more.

## 2021 AB 6

This is a standard Differential Equation (Type 6) question and contains topics mainly from Unit 7 (Differential equations) and a little from Unit 3 (implicit differentiation) of the current Course and Exam Description. A differential equation with an initial condition is given in a context. The main part is the solution of the initial value question with three short other questions included.

The stem for 2021 AB 6 is:

**Part (a): **A slope field in the first quadrant with no scale on either axis is given. Students are asked to sketch the solution curve starting at the initial condition, the point (0, 0). (I prefer this kind of slope field question to those where students are given a few points and asked to sketch the slope field through them. No one draws slope field by hand; slope field drawn by computers are used to study the approximate shape of the solution and determine its interesting properties as is done here and in part (b)). When drawing slope fields, the sketch should extend to from one border to another and contain the initial condition point.

Discussion and ideas for adapting this question:

- Have student sketch solution through one or more different points. Copy the slope field and add the initial condition point somewhere else.
- Add an initial point above the horizontal asymptote.
- Compare and contrast the solutions drawn through several points.
- Ask what the horizontal segments (at
*y*= 12) tells you in the context of the problem.

**Part (b): **Students are given the limit at infinity for the as yet unknow solution and asked to interpret it in the context of the problem including units of measure.

Discussion and ideas for adapting this question:

- Discuss why this is so.
- Discuss how to determine the units of the function from the given information.
- Discuss how to determine the units of the derivative from the given information.
- Discuss how to determine the units of the derivative from the units of the function.
- Discuss how to determine the units of the function from the units of the derivative.
- Discuss whether the interpretation of the limit makes sense in the context of the question.

**Part (c): **Students are asked to solve the initial value question using the method of separation of variables.

Discussion and ideas for adapting this question:

- Since separation of variables is the only method for solving a differential equation that students are responsible for knowing, there is not much you can do to adapt or change this question.
- The initial condition may be substituted immediately after the integration is done the “+
*C*” is attached, or it may be done later after the expression is solved for*y*. Show students both method and discuss which is more efficient and which makes more sense to them. - Removing the absolute value signs is another place that may confuse students. While some textbooks suggest using a “ ± “ sign and deciding sign which to use later, the better way is to decide as soon as possible. Ask yourself is the expression enclose by the absolute value signs positive or negative near (at) the initial value. If positive, then the absolute value is replaced by the same expression (as in this question); if negative, then the expression is replaced by its opposite. Then complete the question from there.

**Part (d):** This part needs careful reading. Students are asked, for a slightly different differential equation, if the *rate of change* in the amount of medicine is increasing or decreasing at a given time. Therefore, students must find the rate of change of the rate of change (the given derivative): the derivative of the derivative (i.e., the second derivative of the function). This requires *implicit differentiation* of the derivative using the quotient rule.

Discussion and ideas for adapting this question:

- The second derivative has the first derivative as one of its factors. Students may (automatically) substitute the first derivative before simplifying or evaluating. This correct, but unnecessarily long. Show the students how to find and substitute the
*value*of the first derivative along with the other numbers. - Do as little arithmetic as possible. You need only determine if the second derivative is positive or negative.
- Discuss the meaning of the answer in the context of the problem.

Next week 2021 BC 2

I would be happy to hear your ideas for other ways to use this question. Please use the reply box below to share your ideas.