*Eight of nine*. We continue our study of the 2021 free-response questions. We will look at ways to adapt, expand, and explore this question to help students better understand it and look at other questions that can be asked based on a similar stem.

## 2021 BC 5

This is a Differential Equation (Type 6) with a Sequence and Series (Type 10) question included. It contains topics from Units 7 and 10 of the current Course and Exam Description (CED). It is not unusual for AP Calculus exam question to include several of the types in my classification and from several of the units from the CED (here units 7 and 10). In addition, the usual solving an initial value problem and a Euler’s Method approximation are included.

The stem for 2021 BC 5 is:

**Part (a)**: Students were asked to write the second-degree Taylor polynomial for the function centered at *x* = 1 and then use it to approximate *f*(2). Students should stop after substituting 2 into their polynomial; no arithmetic or simplification is required, and a simplifying mistake will lose a point.

Discussion and ideas for adapting this question:

- Ask students to find an expression for the second derivative (implicit differentiation).
- Verify that
- Ask students to find the third-degree polynomial and use it to approximate
*f*(2)

**Part (b)**: Students were required to approximate *f*(2) using Euler’s method with two steps of equal size.

Discussion and ideas for adapting this question:

- After you solve the equation in part (c), ask students to compare the approximations from parts (a) and (b) with the exact value. Neither approximation is very close to the exact value. Discuss why this is so. Consider the slope of the graph near
*x*= 2. - Find a more accurate approximation using 3, 4, or more smaller steps. There are graphing calculator programs that will do the arithmetic. Do not hesitate to use them. Students have already shown they know how to do a Euler’s Method approximation; the point is to understand the situation.

**Part (c)**: Finding the solution of the differential equation by separating the variables is expected in this kind of question. The added twist is that the method of integration by parts is necessary to find one of the antiderivatives.

Discussion and ideas for adapting this question:

- Be sure not to skip over removing the absolute value signs. The most efficient way is to realize that at (and near) the initial condition
*y*> 0, so |*y*| =*y*. What do you do if*y*< 0? - There is not much you can do differently here. One thing is to change the initial condition. Try a negative value such as
*f*(1) = –4. - As suggested in (b), compare, and discuss the approximations with the exact value.

Next week we will conclude this series of posts with a look at 2021 BC 6.

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.