Differential Equations (Type 6)

Posted on March 29, 2022 by Lin McMullin

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 1, 6, 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)

Posted on March 25, 2022 by Lin McMullin

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

Area and Volume Problems (Type 4)

Posted on March 22, 2022 by Lin McMullin

AP Type Questions 4: Area and Volume

Given equations that define a region in the plane students are asked to find its area, the volume of the solid formed when the region is revolved around a line, and/or the region is used as a base of a solid with regular cross-sections. This standard application of the integral has appeared every year since year one (1969) on the AB exam and almost every year on the BC exam. You can be fairly sure that if a free-response question on areas and volumes does not appear, the topic will be tested on the multiple-choice section.

What students should be able to do:

Find the intersection(s) of the graphs and use them as limits of integration (calculator equation solving). Write the equation followed by the solution; showing work is not required. Usually, no credit is earned until the solution is used in context (e.g., as a limit of integration). Students should know how to store and recall these values to save time and avoid copy errors.
Find the area of the region between the graph and the x-axis or between two graphs.
Find the volume when the region is revolved around a line, not necessarily an axis or an edge of the region, by the disk/washer method. See “Subtract the Hole from the Whole”
The cylindrical shell method will never be necessary for a question on the AP exams but is eligible for full credit if properly used.
Find the volume of a solid with regular cross-sections whose base is the region between the curves. For an interesting variation on this idea see 2009 AB 4(b)
Find the equation of a vertical line that divides the region in half (area or volume). This involves setting up an integral equation where the limit is the variable for which the equation is solved.
For BC only – find the area of a region bounded by polar curves: $\displaystyle A=\frac{1}{2}{{\int_{{{{\theta }_{1}}}}^{{{{\theta }_{2}}}}{{\left( {r\left( \theta \right)} \right)}}}^{2}}d\theta$
For BC only – Find perimeter using arc length integral

If this question appears on the calculator active section, it is expected that the definite integrals will be evaluated on a calculator. Students should write the definite integral with limits on their paper and put its value after it. It is not required to give the antiderivative and if a student gives an incorrect antiderivative, they will lose credit even if the final answer is (somehow) correct.

There is a calculator program available that will give the set-up and not just the answer so recently this question has been on the no calculator allowed section. (The good news is that in this case the integrals will be easy, or they will be set-up-but-do-not-integrate questions.)

Occasionally, other type questions have been included as a part of this question. See 2016 AB5/BC5 which included an average value question and a related rate question along with finding the volume.

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 this subject see January 9, 11, 2013 and “Subtract the Hole from the Whole” of December 6, 2016.

The Area and Volume question covers topics from Unit 6 of the CED .

Free-response questions:

Variations: 2009 AB 4, Don’t overlook this one, especially part (b)
2016 AB5/BC5,
2017 AB 1 (using a table),
2018 AB 5 – average rate of change, L’Hospital’s Rule
2019 AB 5
Perimeter parametric curves 2011 BC 3 and 2014 BC 5
Area in polar form 2017 BC 5, 2018 BC 5, 20129 BC 2
2021 AB 4/ BC 4
2022 AB2 – area, volume, inc/dec analysis, and related rate.

Multiple-choice questions from non-secure exams:

2008 AB 83 (Use absolute value),
2012 AB 10, 92
2012 BC 87, 92 (Polar area)

Revised March 12, 2012, March 22, 2022

Graph Analysis Questions (Type 3)

Posted on March 18, 2022 by Lin McMullin

AP Questions Type 3: Graph Analysis

The long name is “Here’s the graph of the derivative, tell me things about the function.”

Students are given either the equation of the derivative of a function or a graph identified as the derivative of a function with no equation is given. It is not expected that students will write the equation of the function from the graph (although this may be possible); rather, students are expected to determine key features of the function directly from the graph of the derivative. They may be asked for the location of extreme values, intervals where the function is increasing or decreasing, concavity, etc. They may be asked for function values at points. They will be asked to justify their conclusions.

The graph may be given in context and students will be asked about that context. The graph may be identified as the velocity of a moving object and questions will be asked about the motion. See Linear Motion Problems (Type 2)

Less often the function’s graph may be given, and students will be asked about its derivatives.

What students should be able to do:

Read information about the function from the graph of the derivative. This may be approached by derivative techniques or by antiderivative techniques.
Find and justify where the function is increasing or decreasing.
Find and justify extreme values (1^st and 2^nd derivative tests, Closed interval test a/k/a Candidates’ test).
Find and justify points of inflection.
Find slopes (second derivatives, acceleration) from the graph.
Write an equation of a tangent line.
Evaluate Riemann sums from geometry of the graph only. This usually involves familiar shapes such as triangles or semicircles.
FTC: Evaluate integral from the area of regions on the graph.
FTC: The function, g(x), may be defined by an integral where the given graph is the graph of the integrand, f(t), so students should know that if,

$\displaystyle g\left( x \right)=g\left( a \right)+\int_{a}^{x}{{f\left( t \right)dt}}$ , then $\displaystyle {g}'\left( x \right)=f\left( x \right)$ and $\displaystyle {g}''\left( x \right)={f}'\left( x \right)$ .

In this case, students should write $\displaystyle {g}'\left( x \right)=f\left( x \right)$ on their answer paper, so it is clear to the reader that they understand this.

Not only must students be able to identify these things, but they are usually asked to justify their answer and reasoning. See Writing on the AP Exams for more on justifying and explaining answers.

There are numerous ideas and concepts that can be tested with this type of question. The type appears on the multiple-choice exams as well as the free-response. Between multiple-choice and free-response this topic may account for 15% or more of the points available on recent tests. It is very important that students are familiar with all the ins and outs of this situation.

As with other questions, the topics tested come from the entire year’s work, not just a single unit. In my opinion many textbooks do not do a good job with integrating these topics, so be sure to use as many actual AP Exam questions as possible. Study past exams: look them over and see the different things that can be asked.

The Graph Analysis problem may cover topics primarily from primarily from Unit 4, Unit 5, and Unit 8 of the CED

For previous posts on this subject see October 15, 17, 19, 24, 26 (my most read post), 2012 and January 25, 28, 2013

Free-response questions:

Function given as a graph, questions about its integral (so by FTC the graph is the derivative): 2016 AB 3/BC 3, 2018 AB3
Table and graph of function given, questions about related functions: 2017 AB 6,
Derivative given as a graph: 2016 AB 3 and 2017 AB 3
Information given in a table 2014 AB 5
2021 AB 4 / BC 4
2021 AB 5 (b), (c), (d)
2022 AB3/BC3 – graph analysis, max/min

Multiple-choice questions from non-secure exam. Notice the number of questions all from the same year; this is in addition to one free-response question (~25 points on AB and ~23 points on BC out of 108 points total)

2012 AB: 2, 5, 15, 17, 21, 22, 24, 26, 76, 78, 80, 82, 83, 84, 85, 87
2012 BC 3, 11, 12, 15, 12, 18, 21, 76, 78, 80, 81, 84, 88, 89

A good activity on this topic is here. The first pages are the teacher’s copy and solution. Then there are copies for Groups A, B, and C. Divide your class into 3 or 6 or 9 groups and give one copy to each. After they complete their activity have the students compare their results with the other groups.

Revised March 12, 2021, March 18, 2022

Linear Motion (Type 2)

Posted on March 15, 2022 by Lin McMullin

AP Questions Type 2: Linear Motion

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

“A particle (or car, person, 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 on the x– or y-axis as a function of time, along with an initial condition. The questions ask for information about the 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 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 when a function reaches its “absolute maximum value.” See my post for Motion Problems: Same Thing, Different Context 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 $\displaystyle {s}'\left( t \right)=v\left( t \right)$ . Velocity 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 Speed.
Acceleration is the derivative of velocity and the second derivative of position, $\displaystyle {{s}'}'\left( t \right)={v}'\left( t \right)=a\left( t \right)$ It, too, has direction and magnitude and is a vector.
Velocity is the antiderivative of 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) $\displaystyle \int_{a}^{b}{{\left| {v\left( t \right)} \right|dt}}$ .
Be sure your students understand the term displacement; it is the net distance traveled or distance between the initial position and the final position. Displacement is the definite integral of the velocity (rate of change): $\displaystyle \int_{a}^{b}{{v\left( t \right)dt}}$
The final position is the initial position plus the displacement (definite integral of the rate of change from x= a 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 but may also be handled as an initial value problem.
Find the speed at a given time. Speed is the absolute value of velocity.
Find average speed, velocity, or acceleration
Determine if the speed is increasing or decreasing.
- When the velocity and acceleration have the same sign, the speed increases. When they have different signs, the speed decreases.
- If the velocity graph is moving away from (towards) the t-axis the speed is increasing (decreasing). See the post on Speed.
- There is also a worksheet on speed here
- The analytic approach to speed: A Note on Speed
Use a difference quotient to approximate the derivative (velocity or acceleration) from a table. Be sure the work shows a quotient.
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.

The Linear Motion problem may cover topics primarily from primarily from Unit 4, and also from Unit 3, Unit 5, Unit 6, and Unit 8 (for BC) of the CED.

Free-response examples:

Equation stem 2017 AB 5,
Graph stem: 2009 AB1/BC1,
Table stem 2019 AB2
Equation stem 2021 AB 2
Equation stem 2022 AB6 – velocity, acceleration, position, max/min

Multiple-choice examples from non-secure exams:

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

Revised March 15, and May 11, 2022

Rate & Accumulation (Type 1)

Posted on March 11, 2022 by Lin McMullin

The Free-response Questions

There are ten general types of AP Calculus free-response questions. This and the next nine posts will discuss each of them.

NOTE: The numbers I’ve assigned to each type DO NOT correspond to the CED Unit numbers. Many AP Exam questions intentionally have parts from different Units. The CED Unit numbers will be referenced in each post.

AP Questions Type 1: Rate and Accumulation

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

The rates are often fairly complicated functions. If the question is 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 $\displaystyle {{x}_{0}}$ is the initial time, and $\displaystyle f\left( {{{x}_{0}}} \right)$ 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 words which need to be carefully read and understood.
Understand the question. It is often not necessary to do as much computation as it seems at first.
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 $\displaystyle {{x}_{0}}$ is the initial time, and $\displaystyle f\left( {{{x}_{0}}} \right)$ is the initial amount, then final accumulated amount is

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

Write an integral expression that gives the amount at a general time. BE CAREFUL, the dt must be included in the correct place. Think of the integral sign and the dt as parentheses around the integrand.
Find the average value of a function
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) what the numerical argument means in the context of the question.
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 the context of the question.
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. Also, be ready to approximate a derivative using a quotient from the numbers in the table.
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.

The Rate – Accumulation question may cover topics primarily from Unit 4, Unit 5, Unit 6 and Unit 8 of the CED.

Typical free-response examples:

2013 AB1/BC1
2015 AB1/BC1
2018 AB1/BC1
2019 AB1/BC1
2022 AB1/BC1 – includes average value, inc/dec analysis, max/min analysis
One of my favorites Good Question 6 (2002 AB 4)

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 11, 2022

Unit 9 – Parametric Equations, Polar Coordinates, and Vector-Valued Functions

Posted on January 11, 2022 by Lin McMullin

Unit 9 includes all the topics listed in the title. These are BC only topics (CED – 2019 p. 163 – 176). These topics account for about 11 – 12% of questions on the BC exam.

Comments on Prerequisites: In BC Calculus the work with parametric, vector, and polar equations is somewhat limited. I always hoped that students had studied these topics in detail in their precalculus classes and had more precalculus knowledge and experience with them than is required for the BC exam. This will help them in calculus, so see that they are included in your precalculus classes.

Topics 9.1 – 9.3 Parametric Equations

Topic 9.1: Defining and Differentiation Parametric Equations. Finding dy/dx in terms of dy/dt and dx/dt

Topic 9.2: Second Derivatives of Parametric Equations. Finding the second derivative. See Implicit Differentiation of Parametric Equations this discusses the second derivative.

Topic 9.3: Finding Arc Lengths of Curves Given by Parametric Equations.

Topics 9.4 – 9.6 Vector-Valued Functions and Motion in the plane

Topic 9.4 : Defining and Differentiating Vector-Valued Functions. Finding the second derivative. See this A Vector’s Derivatives which includes a note on second derivatives.

Topic 9.5: Integrating Vector-Valued Functions

Topic 9.6: Solving Motion Problems Using Parametric and Vector-Valued Functions. Position, Velocity, acceleration, speed, total distance traveled, and displacement extended to motion in the plane.

Topics 9.7 – 9.9 Polar Equation and Area in Polar Form.

Topic 9.7: Defining Polar Coordinate and Differentiation in Polar Form. The derivatives and their meaning.

Topic 9.8: Find the Area of a Polar Region or the Area Bounded by a Single Polar Curve

Topic 9.9: Finding the Area of the Region Bounded by Two Polar Curves. Students should know how to find the intersections of polar curves to use for the limits of integration.

Timing

The suggested time for Unit 9 is about 10 – 11 BC classes of 40 – 50-minutes, this includes time for testing etc.

Previous posts on these topics:

Parametric and Vector Equations

Implicit Differentiation of Parametric Equations

A Vector’s Derivatives

Adapting 2012 BC 2 (A parametric equation question)

Polar Curves

Polar Equations for AP Calculus

Extreme Polar Conditions

Teaching Calculus

The pleasure lies not in discovering the truth, but in searching for it.

Category Archives: Integration

Differential Equations (Type 6)

AP Questions Type 6: Differential Equations