# Adapting 2021 AB 4 / BC 4

Four of nine. Continuing the series started in the last three posts, this post looks at the AP Calculus 2021 exam question AB 4 / BC 4. The series considers each question with the aim of showing ways to use the question with your class as is, or by adapting and expanding it.  Like most of the AP Exam questions there is a lot more you can ask from the stem and a lot of other calculus you can discuss.

## 2021 AB 4 / BC 4

This is a Graph Analysis Problem (type 4) and contains topics from Units 2, 4, and 6 of the current Course and Exam Description. The things that are asked in these questions should be easy for the students, however each year the scores are low. This may be because some textbooks simply do not give students problems like this. Therefore, supplementing with graph analysis questions from past exams is necessary.

There are many additional questions that can be asked based on this stem and the stems of similar problems. Usually the graph of the derivative is given and students are asked questions about the graph of the function. See Reading the Derivative’s Graph.

Some years this question is given a context, such as the graph is the velocity of a moving particle. Occasionally there is no graph and an expression for the derivative or function is given.

Here is the 2021 AB 4 / BC 4 stem:

The first thing students should do when they see $G\left( x \right)=\int_{0}^{x}{{f\left( t \right)}}dt$ is to write prominently on their answer page ${G}'\left( x \right)=f\left( x \right)$ and $\displaystyle {G}''\left( x \right)={f}'\left( t \right)$. While they may understand and use this, they must say it.

Part (a): Students were asked for the open intervals where the graph is concave up and to give a reason for their answer. (Asking for an open interval is to remove any concern about the endpoints being included or excluded, a place where textbooks differ. See Going Up.)

Discussion and ideas for adapting this question:

• Using this or similar graphs go through each of these with your class until the answers and reasons become automatic. There are quite a few other things that may be asked here based on the derivative.
• Where is the function increasing?
• Decreasing?
• Concave down, concave up?
• Where are the local extreme values?
• What are the local extreme values?
• Where are the absolute extreme values?
• What are the absolute extreme values?
• There are also integration questions that may be asked, such as finding the value of the functions at various points, such as G(1) = 2 found by using the areas of the regions. Also, questions about the local extreme values and the absolute extreme value including their values. These questions are answered by finding the areas of the regions enclosed by the derivative’s graph and the x-axis. Parts (b) and (c) do some of this.
• Choose different graphs, including one that has the derivative’s extreme value on the x­-axis. Ask what happen there.

Part (b): A new function is defined as the product of G(x) and f(x) and its derivative is to be found at a certain value of x. To use the product rule students must calculate the value of G(x) by using the area between f(x) and the x-­axis and the value of ${f}'\left( x \right)$ by reading the slope of f(x) from the graph.

Discussion and ideas for adapting this question:

• This is really practice using the product rule. Adapt the problem by making up functions using the quotient rule, the chain rule etc. Any combination of $\displaystyle G,{G}',{G}'',f,{f}',\text{ or }{f}''$ may be used. Before assigning your own problem, check that all the values can be found from the given graph.
• Different values of x may be used.

Part (c): Students are asked to find a limit. The approach is to use L’Hospital’s Rule.

Discussion and ideas for adapting this question:

• To use L’Hospital’s Rule, students must first show clearly on their paper that the limit of the numerator and denominator are both zero or +/- infinity. Saying the limit is equal to 0/0 is considered bad mathematics and will not earn this point. Each limit should be given separately on the paper, before applying L’Hospital’s Rule.
• Variations include a limit where L’Hospital’s Rule does not apply. The limit is found by substituting the values from the graph.
• Another variation is to use a different expression where L’Hospital’s Rule applies, but still needs values read from the graph.

Part (d): The question asked to find the average rate of change (slope between the endpoints) on an interval and then determine if the Mean Value Theorem guarantees a place where $\displaystyle {G}'$ equals this value. Students also must justify their answer.

Discussion and ideas for adapting this question:

• To justify their answer students must check that the hypotheses of the MVT are met and say so in their answer.
• Adapt by using a different interval where the MVT applies.
• Adapt by using an interval where the MVT does not apply and (1) the conclusion is still true, or (b) where the conclusion is false.

Next week 2021 AB 5.

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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