Other Problems (Type 7)

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:

Good Question 17

2004 AB 4

2016 BC 4

2012 AB 27 (implicit differentiation), Multiple-choice

2022 AB 5 (a) Implicit differentiation,

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.

Good Question 9

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

Good Question 5: 1998 AB2/BC2

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
• L’Hospital’s Rule 2016 BC 4, 2019 AB 3 (Don’t be fooled), 2019 AB 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)

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.

Revised March 12, 2021, April 1, and May 14, 2022

Other Problems (Type 7)

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 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 if 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 my appear on the multiple-choice sections of the exam.

Example:

Good Question 17

2004 AB 4

2016 BC 4

2012 AB 27 (implicit differentiation), Multiple-choice

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 some previous posts on related rate 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

Good Question 9

Family of Functions

A “family of functions” are 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 some 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

Good Question 5: 1998 AB2/BC2

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
• L’Hospital’s Rule 2016 BC 4, 2019 AB 3 (Don’t be fooled), 2019 AB 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

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 question may come from any of the Units in the  2019 CED.

Revised March 12, 2021

Type 2 Questions: Linear Motion

We continue the discussion of the various type questions on the AP Calculus Exams with linear motion questions.

“A particle (or car, or bicycle) moves on a number line ….”

These questions may give the position equation, the velocity equation (most often), or the acceleration equation of something that is moving, along with an initial condition. The questions ask for information about motion of the particle: its direction, when it changes direction, its maximum position in one direction (farthest left or right), its speed, etc.

The particle may be a “particle,” a person, car, a rocket, etc.  Particles don’t really move in this way, so the equation or graph should be considered to be a model. The question is a versatile way to test a variety of calculus concepts since the position, velocity, or acceleration may be given as an equation, a graph, or a table; be sure to use examples of all three forms during the review.

Many of the concepts related to motion problems are the same as those related to function and graph analysis (Type 3). Stress the similarities and show students how the same concepts go by different names. For example, finding when a particle is “farthest right” is the same as finding the when a function reaches its “absolute maximum value.” See my post for November 16, 2012 for a list of these corresponding terms. There is usually one free-response question and three or more multiple-choice questions on this topic.

The position, s(t), is a function of time. The relationships are:

• The velocity is the derivative of the position, ${s}'\left( t \right)=v\left( t \right)$. Velocity is has direction (indicated by its sign) and magnitude. Technically, velocity is a vector; the term “vector” will not appear on the AB exam.
• Speed is the absolute value of velocity; it is a number, not a vector. See my post for November 19, 2012.
• Acceleration is the derivative of velocity and the second derivative of position, $\displaystyle a\left( t \right)={v}'\left( t \right)={{s}'}'\left( t \right)$. It, too, has direction and magnitude and is a vector.
• Velocity is the antiderivative of the acceleration
• Position is the antiderivative of velocity.

What students should be able to do:

• Understand and use the relationships above.
• Distinguish between position at some time and the total distance traveled during the time period.
• The total distance traveled is the definite integral of the speed (absolute value of velocity) $\displaystyle \int_{a}^{b}{\left| v\left( t \right) \right|}\,dt$.
• The net distance traveled, displacement, is the definite integral of the velocity (rate of change): $\displaystyle \int_{a}^{b}{v\left( t \right)}\,dt$. Note that “displacement” has not been used preciously on AP exam, but (as per the new Course and Exam Description) may be used now. Be sure your students know this term.
• The final position is the initial position plus the displacement (definite integral of the rate of change from xa to x = t): $\displaystyle s\left( t \right)=s\left( a \right)+\int_{a}^{t}{v\left( x \right)}\,dx$ Notice that this is an accumulation function equation (Type 1).
• Initial value differential equation problems: given the velocity or acceleration with initial condition(s) find the position or velocity. These are easily handled with the accumulation equation in the bullet above.
• Find the speed at a given time. The speed is the absolute value of the velocity.
• Find average speed, velocity, or acceleration
• Determine if the speed is increasing or decreasing.
• If at some time, the velocity and acceleration have the same sign then the speed is increasing.If they have different signs the speed is decreasing.
• If the velocity graph is moving away from (towards) the t-axis the speed is increasing (decreasing). See my post for November 19, 2012.
• There is also a worksheet on speed here
• THe analytic approach to speed: A Note on Speed
• Use a difference quotient to approximate derivative.
• Riemann sum approximations.
• Units of measure.
• Interpret meaning of a derivative or a definite integral in context of the problem

Shorter questions on this concept appear in the multiple-choice sections. As always, look over as many questions of this kind from past exams as you can find.

This may be an AB or BC question. The BC topic of motion in a plane, (Type 8: parametric equations and vectors) will be discussed in a later post.

Free-response examples:

• Equation stem 2017 AB 5,
• Graph stem: 2009 AB1/BC1,
• Table stem 2015 AB 3/BC3

Multiple-choice examples from non-secure exams:

• 2012 AB 6, 16, 28, 79, 83, 89
• 2012 BC 2, 89

Updated January 31, 2019, March 12, 2021

Type 1 Questions: Rate and Accumulation

The Free-response Questions

There are ten general categories of AP Calculus free-response questions, listed below. These are usually the subject of individual free-response questions. Keep in mind that two or more types may be included in the same free-response question and are also the topics of shorter multiple-choice questions. There are links to all the types here.

• Type 1 questions – Rate and accumulation questions
• Type 2 questions – Linear motion problems
• Type 3 questions – Graph analysis problems
• Type 4 questions – Area and volume problems
• Type 5 questions – Table and Riemann sum questions
• Type 6 questions – Differential equation questions
• Type 7 questions – miscellaneous
• Type 8 questions – Parametric and vector questions (BC topic)
• Type 9 questions – Polar equations
• Type 10 questions – Sequences and Series

AP Type Questions 1: Rate and Accumulation

These questions are often in context with a lot of words describing a situation in which some things are changing. There are usually two rates acting in opposite ways (sometimes called in-out question). Students are asked about the change that the rates produce over some time interval either separately or together.

The rates are often fairly complicated functions. If they are on the calculator allowed section, students should store the functions in the equation editor of their calculator and use their calculator to do any graphing,  integration, or differentiation that may be necessary.

The main idea is that over the time interval [a, b] the integral of a rate of change is the net amount of change

$\displaystyle \int_{a}^{b}{{f}'\left( t \right)dt}=f\left( b \right)-f\left( a \right)$

If the question asks for an amount, look around for a rate to integrate.

The final (accumulated) amount is the initial amount plus the accumulated change:

$\displaystyle f\left( x \right)=f\left( {{x}_{0}} \right)+\int_{{{x}_{0}}}^{x}{{f}'\left( t \right)}\,dt$,

where ${{x}_{0}}$ is the initial time, and  $f\left( {{x}_{0}} \right)$ is the initial amount. Since this is one of the main interpretations of the definite integral the concept may come up in a variety of situations.

What students should be able to do:

• Be ready to read and apply; often these problems contain a lot of writing which needs to be carefully read.
• Recognize that rate = derivative.
• Recognize a rate from the units given without the words “rate” or “derivative.”
• Find the change in an amount by integrating the rate. The integral of a rate of change gives the amount of change (FTC):

$\displaystyle \int_{a}^{b}{{f}'\left( t \right)dt}=f\left( b \right)-f\left( a \right)$.

• Find the final amount by adding the initial amount to the amount found by integrating the rate. If $x={{x}_{0}}$ is the initial time, and $f\left( {{x}_{0}} \right)$  is the initial amount, then final accumulated amount is

$\displaystyle f\left( x \right)=f\left( {{x}_{0}} \right)+\int_{{{x}_{0}}}^{x}{{f}'\left( t \right)}\,dt$,

• Understand the question. It is often not necessary to as much computation as it seems at first.
• Use FTC to differentiate a function defined by an integral.
• Explain the meaning of a derivative or its value in terms of the context of the problem. The explanation should contain (1) what it represents, (2) its units, and (3) how numerical argument applies in context.
• Explain the meaning of a definite integral or its value in terms of the context of the problem. The explanation should contain (1) what it represents, (2) its units, and (3) how the limits of integration apply in context.
• Store functions in their calculator recall them to do computations on their calculator.
• If the rates are given in a table, be ready to approximate an integral using a Riemann sum or by trapezoids.
• Do a max/min or increasing/decreasing analysis.

Shorter questions on this concept appear in the multiple-choice sections. As always, look over as many questions of this kind from past exams as you can find.

Typical free-response examples:

Typical multiple-choice examples from non-secure exams:

• 2012 AB 8, 81, 89
• 2012 BC 8 (same as AB 8)

Updated January 31, 2019, March 12, 2021

March

The AP Calculus exams this year are on Tuesday morning May 5, 2015. Most of you will be finishing your new work this month and getting ready to review. As usual I like to stay a little ahead of where you are so you have time to consider what is offered here.

To help you plan ahead, below are links to previous posts specifically on reviewing for the AP Calculus exams and on the type questions that appear on the free-response sections of the exams.

I suggest you review by topic spending 1 – 3 days on each type so that students can see the things that are asked for and the different ways they are asked. Most of the questions include topics taught at different times during the year; students are not used to this. By considering each type separately students will learn how to pull together what they have been studying all year.

Many of the same ideas are tested in smaller “chunks” on the multiple-choice sections, so looking at the type should help with not only free-response questions but many of the multiple-choice questions as well. You may also find multiple-choice questions for each of the types and assign a few of them along with the corresponding free-response type.

Ideas for Reviewing for the AP Calculus Exams

Calculator Use on the AP Exams (AB & BC)

Interpreting Graphs AP Type Questions 1

The Rate/Accumulation Question AP Type Question 2

Area and Volume Questions AP Type Question 3

Motion on a Line AP Type Question 4

The Table Question AP Type Question 5

Differential Equations AP Type Question 6

Implicit Relations and Related Rates AP Type Question 7

Parametric and Vector Equations AP Type Question 8 (BC)

Polar Curves AP Type Question 9 (BC)

Sequences and Series AP Type Question 10 (BC)

I’ll be traveling this month to do some workshops and will not be posting too much new until I return.