AP Questions Type 7: Other topics
Any topic in the Course and Exam Description may be the subject of a free-response or multiple-choice question. The topics discussed here are not asked often enough to be classified as a type of their own. The topics listed here have been the subject of full free-response questions or major parts of them. Other topics occasionally asked are mentioned in the question list at the end of the post.
Implicitly defined relations and implicit differentiation
These questions may ask students to find the first and/or second derivative of an implicitly defined relation. Often the derivative is given, and students are required to show that it is correct. (This is because without the correct derivative the rest of the question cannot be done.) The follow-up is to answer questions about the function such as finding an extreme value, second derivative test, or find where the tangent is horizontal or vertical.
What students should know how to do
- Know how to find the first derivative of an implicit relation using the product rule, quotient rule, chain rule, etc.
- Know how to find the second derivative, including substituting for the first derivative.
- Know how to evaluate the first and second derivative by substituting both coordinates of a given point. (Note: If all that is needed is the numerical value of the derivative then the substitution is often easier done before solving for dy/dx or d2y/dx2, and as usual the arithmetic need not be done.)
- Analyze the derivative to determine where the relation has horizontal and/or vertical tangents.
- Write and work with lines tangent to the relation.
- Find extreme values. It may also be necessary to show that the point where the derivative is zero is actually on the graph and to justify the answer.
Simpler questions about implicit differentiation may appear on the multiple-choice sections of the exam.
Example:
2004 AB 4
2016 BC 4
2012 AB 27 (implicit differentiation), Multiple-choice
2021 AB 5 (a) Implicit differentiation.
2023 AB 6 implicit differentiation, find horizontal and vertical tangent line location, related rate
BC classes see Implicit differentiation of parametric equations, and A Vector’s Derivative
Related Rates
Derivatives are rates and when more than one variable is changing over time the relationships among the rates can be found by differentiating with respect to time. The time variable may not appear in the equations. These questions appear occasionally on the free-response sections; if not there, then a simpler version may appear in the multiple-choice sections. In the free-response sections they may be an entire problem, but more often appear as one or two parts of a longer question.
What students should know how to do
- Set up and solve related rate problems.
- Be familiar with the standard type of related rate situations, but also be able to adapt to different contexts.
- Know how to differentiate with respect to time. That is, find dy/dt even if there is no time variable in the given equations using any of the differentiation techniques.
- Interpret the answer in the context of the problem.
- Unit analysis.
Shorter questions on this concept also appear in the multiple-choice sections. As always, look over as many questions of this kind from past exams as you can find.
For previous posts on related rates see Related Rate Problems I and Related Rate Problems II.
Examples
2014 AB4/BC4,
2016 AB5/BC5
2019 AB 4 Related Rate
2019 AB 6
2022 AB2 (d), AB4/BC4 (d) Good example that requires using product and evaluation of an expression that include dr/dt and dh/dt.
2023 AB 6 implicit differentiation, find horizontal and vertical tangent line location, related rate
Family of Functions
A “family of functions” is defined by an equation with a parameter (sort of an extra variable). Changing the parameter gives a different but similar curve. Questions should be answered in general, that is, in terms of the parameter not a specific value of the parameter. These questions appeared on some exams long ago, may be making a comeback.
Examples:
1995 BC 5
1996 AB4/BC4
2019 BC 5
Other Topics
Free response questions (many of the BC questions are suitable for AB)
- Finding derivatives using the chain rule, the quotient rule, etc. from tables of values: 2016 AB 6 and 2015 AB 6, 2023 AB 5
- L’Hospital’s Rule 2016 BC 4, 2019 AB 3 (Don’t be fooled), 2019 AB 4(c), 2023 BC 4(c)
- Continuity and piecewise defined functions: 2012 AB 4, 2011 AB 6 and 2014 BC 5
- Arc length (BC Topic) 2014 BC 5
- Partial fractions (BC Topic) 2015 BC 5
- Improper integrals (BC topic): 2017 BC 5, 2022 BC5 (c), 2023 BC 5 (b)
Multiple-choice questions from non-secure exams:
- 2012 AB 27 (implicit differentiation), 77 (IVT), 88 (related rate)
- 2012 BC 4 (Curve length), 7 (Implicit differentiation), 11 (continuity/differentiability), 12 (Implicit differentiation), 77 (dominance), 82 (average value), 85 (related rate) , 92 (compositions)
These questions may come from any of the Units in the CED.
Updated March 12, 2021, April 1, and May 14, 2022, June 4, 2023
Teaching Calculus 
Pingback: AP Calculus Review 2023 | Teaching Calculus