# To dx or not dx

As exam time nears, teachers become concerned about exactly what to give credit for and what not to give credit for when grading their students’ work on past AP free-response questions.

Former Chief Reader Stephen Davis recently posted a note on the grading of a fictitious exam question showing how 2 points might have been awarded on a L’Hospital’s Rule question.  The note is interesting because it shows the details that exam leaders consider when deciding what to accept and what not; it shows the details that readers must keep in mind while grading. This type of detail with examples is given to the readers in writing for each part of every question. With hundreds of thousands of exams each year, this level of detail is necessary for fairness and consistency in scoring.

Why show your students the minimum they can get away with? How does that help them? Do your students a favor: score the review problems more stringently than the readers. If their answer is not quite right, take off some credit and help them learn how to do better. It will help them in the long run.

# Sequences and Series (Type 10)

AP Questions Type 10: Sequences and Series (BC Only)

The last BC question on the exams usually concerns sequences and series. The question may ask students to write a Taylor or Maclaurin series and to answer questions about it and its interval of convergence, or about a related series found by differentiating or integrating. The topics may appear in other free-response questions and in multiple-choice questions. Questions about the convergence of sequences may appear as multiple-choice questions. With about 8 multiple-choice questions and a full free-response question this is one of the major topics on the BC exams.

Convergence tests for series appear on both sections of the BC Calculus exam. In the multiple-choice section, students may be asked to say if a sequence or series converges or which of several series converge.

The Ratio test is used most often to determine the radius of convergence and the other tests to determine the exact interval of convergence by checking the convergence at the end points. Click here for a convergence test chart students should be familiar with; this list is also on the resource page.

Students should be familiar with and able to write a few terms and the general term of a Taylor or Maclaurin series. They may do this by finding the derivatives and constructing the coefficients from them, or they may produce the series by manipulating a known or given series. They may do this by substituting into a series, differentiating it, or integrating it.

The general form of a Taylor series is $\displaystyle \sum\limits_{{n=0}}^{\infty }{{\frac{{{{f}^{{\left( n \right)}}}\left( a \right)}}{{n!}}{{{\left( {x-a} \right)}}^{n}}}}$; if a = 0, the series is called a Maclaurin series.

What Students Should be Able to Do

• Use the various convergence tests to determine if a series converges. The test to be used is rarely given so students need to know when to use each of the common tests. For a summary of the tests click: Convergence test chart.  and the posts “What Convergence Test Should I use?” Part 1 and Part 2. In 2022 BC 6 (a) students were asked to state the condition (hypotheses) of the convergence test they were asked to use.
• Understand absolute and conditional convergence. If the series of the absolute values of the terms of a series converges, then the original series is said to be absolutely convergent (or converges absolutely). If a series is absolutely convergent, then it is convergent. If the series of absolute values diverges, then the original series may or may not converge; if it converges it is said to be conditionally convergent.
• Write the terms of a Taylor or Maclaurin series by calculating the derivatives and constructing the coefficients of each term.
• Distinguish between the Taylor series for a function and the function. DO NOT say that the Taylor polynomial is equal to the function (this will lose a point); say it is approximately equal.
• Determine a specific coefficient without writing all the previous coefficients.
• Write a series by substituting into a known series, by differentiating or integrating a known series, or by some other algebraic manipulation of a series.
• Know (from memory) the Maclaurin series for sin(x), cos(x), ex and $\displaystyle \frac{1}{{1-x}}$and be able to find other series by substituting into one of these.
• Find the radius and interval of convergence. This is usually done by using the Ratio test to find the radius and then checking the endpoints. for a geometric series, the interval of convergences is the open interval $\displaystyle -1 where r is the common ration of the series.
• Be familiar with geometric series, its radius of convergence, and be able to find the number to which it converges, $\displaystyle {{S}_{\infty }}=\frac{{{{a}_{1}}}}{{1-r}}$. Re-writing a rational expression as the sum of a geometric series and then writing the series has appeared on the exam.
• Be familiar with the harmonic and alternating harmonic series. These are often useful series for comparison.
• Use a few terms of a series to approximate the value of the function at a point in the interval of convergence.
• Determine the error bound for a convergent series (Alternating Series Error Bound or Lagrange error bound). See my posts on Error Bounds and the Lagrange Highway
• Use the coefficients (the derivatives) to determine information about the function (e.g., extreme values).

This list is quite long, but only a few of these items can be asked in any given year. The series question on the free-response section is usually quite straightforward. Topics and convergence tests may appear on the multiple-choice section. As I have suggested before, look at and work as many past exam questions to get an idea of what is asked, how it is a sked, and the difficulty of the questions. Click on Power Series in the “Posts by Topic” list on the right side of the screen to see previous posts on Power Series or any other topic you are interested in.

Free-response questions:

• 2004 BC 6 (An alternate approach, not tried by anyone, is to start with $\displaystyle \sin \left( {5x+\tfrac{\pi }{4}} \right)=\sin \left( {5x} \right)\cos \left( {\tfrac{\pi }{4}} \right)+\cos \left( {5x} \right)\sin \left( {\tfrac{\pi }{4}} \right)$). See Good Question 16
• 2011 BC 6 (Lagrange error bound)
• 2016 BC 6
• 2017 BC 6
• 2019 BC 6
• 2021 BC 5 (a)
• 2021 BC 6 – note that in (a) students were required to state the conditions of the convergence test they were asked to use.
• 2022 BC 6 – Ratio test, interval of conversion with endpoint analysis, Alternating series error bound, series for derivative, geometric series.

Multiple-choice questions from non-secure exams:

• 2008 BC 4, 12, 16, 20, 23, 79, 82, 84
• 2012 BC 5, 9, 13, 17, 22, 27, 79, 90

These questions come from Unit 10 of the CED.

Revised March 12, 2021, April 12, 16, and May 14, 2022

# Polar Equation Questions (Type 9)

### AP Questions Type 9: Polar Equations (BC Only)

Ideally, as with parametric and vector functions, polar curves should be introduced and covered thoroughly in a pre-calculus course. Questions on the BC exams have been concerned only with calculus ideas related to polar curves. Students have not been asked to know the names of the various curves (rose curves, limaçons, etc.). The graphs are usually given in the stem of the problem; students are expected to be able to determine which is which if more than one is given. Students should know how to graph polar curves on their calculator, and the simplest by hand. Intersection(s) of two graphs may be given or easy to find.

What students should know how to do:

• Calculate the coordinates of a point on the graph,
• Find the intersection of two graphs (e.g. to use as limits of integration).
• Find the area enclosed by a graph or graphs: $\displaystyle A=\frac{1}{2}\int_{{{{\theta }_{1}}}}^{{{{\theta }_{2}}}}{{{{{\left( {r\left( \theta \right)} \right)}}^{2}}d\theta }}$
• Use the formulas $\displaystyle x\left( \theta \right)=r\left( \theta \right)\cos \left( \theta \right)\text{ and }y\left( \theta \right)=r\left( \theta \right)\sin \left( \theta \right)$  to convert from polar to parametric form,
• Calculate $\displaystyle \frac{{dy}}{{d\theta }}$ and $\displaystyle \frac{{dx}}{{d\theta }}$ (Hint: use the product rule on the equations in the previous bullet).
• Discuss the motion of a particle moving on the graph by discussing the meaning of $\displaystyle \frac{{dr}}{{d\theta }}$ (motion towards or away from the pole), $\displaystyle \frac{{dr}}{{d\theta }}$ (motion in the vertical direction), and/or $\displaystyle \frac{{dx}}{{d\theta }}$ (motion in the horizontal direction).
• Find the slope at a point on the graph, $\displaystyle \frac{{dx}}{{dx}}=\frac{{dy/d\theta }}{{dx/d\theta }}$

When this topic appears on the free-response section of the exam there is no Parametric/vector motion question and vice versa. When not on the free-response section there are one or more multiple-choice questions on polar equations.

This question typically covers topics from Unit 9 of the CED.

Free-response questions:

• 2013 BC 2
• 2014 BC 2
• 2017 BC 2
• 2018 BC 5
• 2019 AB 2

Multiple-choice questions from non-secure exams:

• 2008 BC 26
• 2012 BC 26, 91

Other posts on Polar Equations

Polar Basics

Polar Equations for AP Calculus

Extreme Polar Conditions

Polar Equations (Review 2018)

Revised March 12, 2021, April 8, 2022

# Parametric and Vector Equations (Type 8)

### AP Questions Type 8: Parametric and Vector Equations (BC Only)

The parametric/vector equation questions only concern motion in a plane. Other topics, such as dot product and cross product, are not tested.

In the plane, the position of a moving object as a function of time, t, can be specified by a pair of parametric equations $\displaystyle x=x\left( t \right)\text{ and }y=y\left( t \right)$ or the equivalent vector $\displaystyle \left\langle {x\left( t \right),y\left( t \right)} \right\rangle$. The path is the curve traced by the parametric equations or the tips of the position vector. .

The velocity of the movement in the x- and y-direction is given by the vector $\displaystyle \left\langle {{x}'\left( t \right),{y}'\left( t \right)} \right\rangle$. The vector sum of the components gives the direction of motion. Attached to the tip of the position vector this vector is tangent to the path pointing in the direction of motion.

The length of this vector is the speed of the moving object. Speed = $\displaystyle \sqrt{{{{{\left( {{x}'\left( t \right)} \right)}}^{2}}+{{{\left( {{y}'\left( t \right)} \right)}}^{2}}}}$. (Notice that this is the same as the speed of a particle moving on the number line with one less parameter: On the number line speed $\displaystyle =\left| {v\left( t \right)} \right|=\sqrt{{{{{\left( {{x}'\left( t \right)} \right)}}^{2}}}}$.)

The acceleration is given by the vector $\displaystyle \left\langle {{x}''\left( t \right),{y}''\left( t \right)} \right\rangle$.

What students should know how to do:

• Vectors may be written using parentheses, ( ), or pointed brackets, $\displaystyle \left\langle {} \right\rangle$, or even $\displaystyle \vec{i},\vec{j}$ form. The pointed brackets seem to be the most popular right now, but all common notations are allowed and will be recognized by readers.
• Find the speed at time t: Speed = $\displaystyle \sqrt{{{{{\left( {{x}'\left( t \right)} \right)}}^{2}}+{{{\left( {{y}'\left( t \right)} \right)}}^{2}}}}$.
• Use the definite integral for arc length to find the distance traveled $\displaystyle \int_{a}^{b}{{\sqrt{{{{{\left( {{x}'\left( t \right)} \right)}}^{2}}+{{{\left( {{y}'\left( t \right)} \right)}}^{2}}}}}}$. Notice that this is the integral of the speed (rate times time = distance).
• The slope of the path is $\displaystyle \frac{{dy}}{{dx}}=\frac{{{y}'\left( t \right)}}{{{x}'\left( t \right)}}$. See this post for more on finding the first and second derivatives with respect to x.
• Determine when the particle is moving left or right,
• Determine when the particle is moving up or down,
• Find the extreme position (farthest left, right, up, down, or distance from the origin).
• Given the position find the velocity by differentiating.
• Given the velocity, find the acceleration by differentiating.
• Given the acceleration and the velocity at some point find the velocity by integrating.
• Given the velocity and the position at some point find the position by integrating. These are just initial value differential equation problems (IVP).
• Dot product and cross product are not tested on the BC exam, nor are other aspects.

When this topic appears on the free-response section of the exam there is no polar equation free-response question and vice versa. When not on the free-response section there are one or more multiple-choice questions on parametric equations.

Free-response questions:

• 2012 BC 2
• 2016 BC 2
• 2021 BC 2
• 2022 BC2 – slope of tangent line, speed, position, total distance traveled

Multiple-choice questions from non-secure exams

• 2003 BC 4, 7, 17, 84
• 2008 BC 1, 5, 28
• 2012 BC 2

This question typically covers topics from Unit 9 of the CED.

Revised March 12, 2021, April 5, 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 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

# Differential Equations (Type 6)

### AP Questions Type 6: Differential Equations

Differential equations are tested in the free-response section of the AP exams almost every year. The actual solving of the differential equation is usually the main part of the problem accompanied by a related question such as a slope field or a tangent line approximation. BC students may also be asked to approximate using Euler’s Method. Several parts of the BC questions are often suitable for AB students and contribute to the AB sub-score of the BC exam. This topic may also appear in the multiple-choice sections of the exams. What students should be able to do
• Find the general solution of a differential equation using the method of separation of variables (this is the only method tested).
• Find a particular solution using the initial condition to evaluate the constant of integration – initial value problem (IVP).
• Determine the domain restrictions on the solution of a differential equation. See this post for more on the domain of a differential equation.
• Understand that proposed solution of a differential equation is a function (not a number) and if it and its derivative are substituted into the given differential equation the resulting equation is true. This may be part of doing the problem even if solving the differential equation is not required (see 2002 BC 5 – parts a, b and d are suitable for AB)
• Growth-decay problems.
• Draw a slope field by hand.
• Sketch a particular solution on a given slope field.
• Interpret a slope field.
• Multiple-choice: Given a differential equation, identify is slope field.
• Multiple-choice: Given a slope field identify its differential equation.
• Use the given derivative to analyze a function such as finding extreme values
• For BC only: Use Euler’s Method to approximate a solution.
• For BC only: use the method of partial fractions to find the antiderivative after separating the variables.
• For BC only: understand the logistic growth model, its asymptotes, meaning, etc. The exams so far, have never asked students to actually solve a logistic equation IVP
Look at the scoring standards to learn how the solution of the differential equation is scored, and therefore, how students should present their answer. This is usually the one free-response answer with the most points riding on it. Starting in 2016 the scoring has changed slightly. The five points are now distributed this way:
• one point for separating the variables
• one point each for finding the antiderivatives
• one point for including the constant of integration and using the initial condition – that is, for writing “+ C” on the paper with one of the antiderivatives and substituting the initial condition; finding the value of C is included in the “answer point.” (In the older exams one point was earned for writing the +C and another point for using the initial condition.)
• one point for solving for y: the “answer point”, for the correct answer. This point includes all the algebra and arithmetic in the problem including solving for C.
In the past, the domain of the solution was often included on the scoring standard, but unless it was specifically asked for in the question students did not need to include it. However, the CED. lists “EK 3.5A3 Solutions to differential equations may be subject to domain restrictions.” Perhaps this will be asked in the future. For more on domain restrictions with examples see this post. 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. For some previous posts on differential equations see January 5, 2015, and for post on related subjects see November 26, 2012, January 21, 2013, February 16, 2013 The Differential Equation question covers topics in Unit 7 of the CED.
Free-response examples:
• 2019 There was no DE question in the free-response. You may assume the topic was tested in the multiple-choice sections.
• 2017 AB4/BC4,
• 2016 AB 4, BC 4, (different questions)
• 2015 AB4/BC4,
• 2013 BC 5
• and a favorite Good Question 2 and Good Question 2 Continued
• 2021 AB 6, BC 5 (b), (c)
• 2022 AB5 – sketch solution on slope field, tangent line approximation, solve separable equation.
Multiple-choice examples from non-secure exams:
• 2012 AB 23, 25
• 2012 BC: 12, 14, 16, 23

### Previous posts on these topics for both AB and BC include:

Differential Equations  A summary of the terms and techniques of differential equations and the method of separation of variables Domain of a Differential Equation – On domain restrictions. Accumulation and Differential Equations  Slope Fields An Exploration in Differential Equations An exploration illustrating many of the ideas of differential equations. The exploration is here in PDF form and the solution is here. The ideas include: finding the general solution of the differential equation by separating the variables, checking the solution by substitution, using a graphing utility to explore the solutions for all values of the constant of integration, finding the solutions’ horizontal and vertical asymptotes, finding several particular solutions, finding the domains of the particular solutions, finding the extreme value of all solutions in terms of C, finding the second derivative (implicit differentiation), considering concavity, and investigating a special case or two.

### Previous Posts on BC Only Topics

Euler’s Method Euler’s Method for Making Money The Logistic Equation  Logistic Growth – Real and Simulated

Revised 2/20/2021, March 29, May 14, 2022

# Riemann Sum & Table Problems (Type 5)

### AP Questions Type 5: Riemann Sum & Table Problems

Information given in tables may be used to test a variety of ideas in calculus including analysis of functions, accumulation, theory and theorems, and position-velocity-acceleration, among others. Numbers and working with numbers are part of the Rule of Four and table problems are one way they are tested. This question often includes an equation in a latter part of the problem that refers to the same situation.

What students should be able to do:

• Find the average rate of change over an interval
• Approximate the derivative using a difference quotient. Use the two values closest to the number at which you are approximating.  This amounts to finding the slope or rate of change. Show the quotient even if you can do the arithmetic in your head and even if the denominator is 1.
• Use a left-, right-, or midpoint- Riemann sums or a trapezoidal approximation to approximate the value of a definite integral using values in the table (typically with uneven subintervals). The Trapezoidal Rule, per se, is not required; it is expected that students will add the areas of a small number of trapezoids without reference to a formula.
• Average value, average rate of change, Rolle’s theorem, the Mean Value Theorem, and the Intermediate Value Theorem. (See 2007 AB 3 – four simple parts that could be multiple-choice questions; the mean on this question was 0.96 out of a possible nine points.)
• These questions are usually presented in context and answers should be in that context. The context may be something growing (changing over time) or linear motion.
• Use the table to find a value based on the Mean Value Theorem (2018 AB 4(b)) or Intermediate Value Theorem. Also, 2018 AB 4 (d) asked a related question based on a function given by an equation.
• Unit analysis.

Dos and Don’ts

Do: Remember that you do not know what happens between the values in the table unless additional information is given. For example, do not assume that the largest number in the table is the maximum value of the function, or that the function is decreasing between two values just because a value is less than the preceding value.

Do: Show what you are doing even if you can do it in your head. If you’re finding a slope, show the quotient even if the denominator is 1.

Do Not do arithmetic: A long expression consisting entirely of numbers such as you get when doing a Riemann sum, does not need to be simplified in any way. If you simplify a correct answer incorrectly, you will lose credit.

Do Not leave expression such as R(3) – pull its numerical value from the table.

Do Not: Find a regression equation and then use that to answer parts of the question. While regression is perfectly good mathematics, regression equations are not one of the four things students may do with their calculator. Regression gives only an approximation of our function. The exam is testing whether students can work with numbers.

This question typically covers topics from Unit 6 of the CED but may include topics from Units 2, 3, and 4 as well.

Free-response examples:

• 2007 AB 3 (4 simple parts on various theorems, yet the mean score was 0.96 out of 9),
• 2017 AB 1/BC 1, and AB 6,
• 2016 AB 1/BC 1
• 2018 AB 4
• 2021 AB 1/ BC 1
• 2022 AB4/BC4 – average rate of change, IVT, Rieman sum, Related Rate (part (d) good question)

Multiple-choice questions from non-secure exams:

• 2012 AB 8, 86, 91
• 2012 BC 8, 81, 86 (81 and 86 are the same on both the AB and BC exams)

Revised March 12, 2021, March 25, 2022