*First of nine. *One of the things many successful AP Calculus teachers do is to use past AP exam questions throughout the year. Individual multiple-choice exam questions are used as the topics they test are taught; free-response questions are adapted and expanded. There are several ways to do this:

- Assign parts of a free-response (FR) question as is as the topic it tests is taught. Later, other parts from the same stem can be assigned. Including previously assigned parts is a spiraling technique. Once students see that you are doing this, they will be more likely to keep up to date on past topics.
- Adapt and expand the questions is another way to use FR questions.

This summer I will be discussing how to do just that. Each week I will look at one of the released 2021 FR questions and suggest how to expand and adapt it. Each stem allows for many more questions than can be asked on any one exam. You have the luxury of asking other things based on the same stem.

This summer’s series of posts will take one question at a time discuss it and suggest additional questions or explorations that may be asked. I will not be presenting solutions. They are available on AP Community bulletin board here and here. I will link the posts to the scoring standards when they are published.

## 2021 AB 1 / BC 1

This is a Reimann sum and Table question (Type 5) and covers topics from Units 6 and 8 from the current Course and Exam Description. All four parts are fairly typical for this type problem. There is a little twist in part (b). The context is the density of bacteria growing in a petri dish.

Density is not listed in the Course and Exam Description. It is not covered well in many textbooks. Since it is not listed you need not teach it; exam questions referencing density have enough included information so that a student who has never seen the concept will still be able to answer the question. Keep this in mind as you look at each part; help your students see past the context and look at the calculus. More information on density see these posts Density Functions, and Good Question 15 and Good Question 16.

The stem for 2021 AB 1 / BC 1 reads:

**Part (a)**: Students were asked to estimate the value of the derivative of *f* at *r* = 2.25 and explain its meaning, including units, in the context of the problem. The expected procedure is to find the slope between the two values closest to *r* =2.25. The interpretation is the increase in density as you move away from the center. The units are milligrams per square centimeter per centimeter distant from the center .

Discussion and ideas for adapting this question:

- AP exams have always asked this question at a value exactly half-way between two values in the table. You may change this to some other place such as
*r*= 3 or*r*= 0.8. - Units of the derivative are always the units of the function divided by the units of the independent variable. Be sure your students understand this.
- The units can be correctly written as , but here is a good change to discuss what the units mean. Why does “milligrams per square centimeter per centimeter distant from the center” make more sense?
- Ask “Is there a point in the interval [2, 2.5] where the slope of the tangent line is 8? Justify your answer.” This makes use of the Mean Value Theorem.

**Part (b) :** As usual in this type of problem, students are asked to write a Riemann sum based on the intervals in the table. The difference here is that the integral being approximated, , has an “extra” factor of *r* in it.

Discussion and ideas for adapting this question:

- The question asked for a right Riemann sum. You can easily adapt this by asking for a left Riemann sum, a midpoint Riemann sum, and/or a Trapezoidal approximation.
- You may ask for a Riemann sum without the “extra” factor.
- You may find a different Riemann sum problem and include an “extra” factor in it.
- The integral is the integral for a
*radial density function*. See the Density blog post cited above, example 2. - The radial density function looks like the integral for finding volumes by the method of cylindrical shells. This is more than a coincidence. Why?

**Part (c)**: This part asked if the answer in (b) was an overestimate or an underestimate, with an explanation. For any approximation, some idea of its accuracy is important. In BC questions on power series approximations, a numerical estimate of the error bound is a common question.

Discussion and ideas for adapting this question:

- Ask the same question for a different Riemann sum (left, midpoint, trapezoid).
- The error in right and left Riemann sums estimates depend on whether the function is increasing or decreasing, and therefore on the first derivative. Midpoint and Trapezoidal approximation estimates are related to the concavity and therefore to the second derivative. See: Good Question 4)
- A visual idea helps keep all this straight. Draw sketches showing the Riemann sum rectangles or trapezoids. Whether they lie above or below the graph of the function determines whether the approximation is an overestimate or underestimate.

**Part (d)**: Typical of the Riemann sum table question is the final part with a related question based on a function and *not* based on the table.

Discussion and ideas for adapting this question:

- This is a calculator allowed question. Students should not try to do the integration by hand.
- The question asked for the average value of the function on an interval. Other question you could ask are find the rate of change (derivative) at a point, the total mass (note “extra”
*r*), the average rate of change on an interval,*etc.*

Next week 2021 AB 2.

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

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