Category Archives: Differential Equations

2019 CED Unit 7: Differential Equations

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Applications (Unit 8) seem to fit more logically after the opening unit on integration (Unit 6). The Course and Exam Description (CED) present differential equations first probably because the previous unit ended with techniques of antidifferentiation. My guess is that many teachers will teach Unit 8: Applications of Integration before Unit 7: Differential Equations. Therefore, for those who want to present unit 8 first, I will post unit 8 next week on December 3, 2019. That way you’ll have both for reference and can choose the order you think will work best for your students. 

Unit 7 is an introduction to the initial ideas and easy techniques related to differential equations. (CED – 2019 p. 129 – 142). These topics account for about 6 – 12% of questions on the AB exam and 6 – 9% of the BC questions.

Topics 7.1 – 7.9

Topic 7.1 Modeling Situations with Differential Equations Relating a function and its derivatives.

Topic 7.2 Verifying Solutions for Differential Equations A proposed solution to a differential equation can be checked by substituting the function and its derivative(s) into the original differential equation. There may be an infinite number of general solutions (solutions with one or more constants).

Topic 7.3 Sketching Slope Fields Slope fields are a graphical representation of a differential equation and provide information about the behavior of the solutions.

Topic 7.4 Reasoning Using Slope Fields 

Topic 7.5 Approximating Solutions Using Euler’s method (BC ONLY) A numerical approach to approximating solutions of a differential equation.

Topic 7.6 Finding General Solutions Using Separation of Variable Since this unit is only an introduction to differential equations, the method of separation of variable is the only solution method tested on the AB and BC exams.

Topic 7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables An initial condition (i.e. a point on the particular solution) allows you to evaluate the constant in the general solution and find the one solution that contains the initial condition. Also, if \displaystyle \frac{{dy}}{{dx}}=f\left( x \right) has the initial condition \displaystyle \left( {a,F\left( a \right)} \right), then the solution is \displaystyle F\left( x \right)=F\left( a \right)+\int_{a}^{x}{{f\left( x \right)dx}}. Solution may also be subject to domain restrictions. 

Topic 7.8 Exponential Models with Differential Equations Applications include linear motion and exponential growth and decay. The growth and decay model is \displaystyle \frac{{dy}}{{dt}}=ky with the initial condition\displaystyle (0,{{y}_{0}}) has the solution \displaystyle y={{y}_{0}}{{e}^{{kt}}}.

Topic 7.9 Logistic Models with Differential Equations (BC ONLY) The model of logistic growth, \displaystyle \frac{{dy}}{{dt}}=ky\left( {a-y} \right), can be solved by separating the variables and using partial fraction decomposition. This has never been tested (probably because solving requires a large amount of complicated algebra). Students are expected to know how to interpret the properties of the solution directly from the differential equation (asymptotes, carrying capacity, point where changing the fastest, etc.) and discuss what they mean in context without actually solving the equation.

Timing

The suggested time for Unit 7 is  8 – 9 classes for AB and 9 – 10 for BC of 40 – 50-minute class periods, this includes time for testing etc.

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

Differential Equations A summary of the terms and techniques about 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. 

Posts on BC Only Topics

Euler’s Method

Euler’s Method for Making Money

The Logistic Equation 

Logistic Growth – Real and Simulated

Here are links to the full list of posts discussing the ten units in the 2019 Course and Exam Description.

2019 CED – Unit 1: Limits and Continuity

2019 CED – Unit 2: Differentiation: Definition and Fundamental Properties.

2019 CED – Unit 3: Differentiation: Composite , Implicit, and Inverse Functions

2019 CED – Unit 4 Contextual Applications of the Derivative  Consider teaching Unit 5 before Unit 4

2019 – CED Unit 5 Analytical Applications of Differentiation  Consider teaching Unit 5 before Unit 4

2019 – CED Unit 6 Integration and Accumulation of Change

2019 – CED Unit 7 Differential Equations  Consider teaching after Unit 8

2019 – CED Unit 8 Applications of Integration   Consider teaching after Unit 6, before Unit 7

2019 – CED Unit 9 Parametric Equations, Polar Coordinates, and Vector-Values Functions 

2019 CED Unit 10 Infinite Sequences and Series

Type 6 Questions: Differential Equations

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Differential equations are tested every year. The actual solving of the differential equation is usually the main part of the problem, but it is 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. Large parts of the BC questions are often suitable for AB students and contribute to the AB sub-score of the BC exam.

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).
  • NEW Determine the domain restrictions on the solution of a differential equation. See this post for more on this. 
  • 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.” and
  • 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 is often included on the scoring standard, but unless it is specifically asked for in the question students do not need to include it. However, the new 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

Free-response examples:

Multiple-choice examples from non-secure exams:

  • 2012 AB 23, 25
  • 2012 BC: 12, 14, 16, 23

Schedule of review postings:

 

 

 

 

 

An Exploration in Differential Equations

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This is an exploration based on the AP Calculus question 2018 AB 6. I originally posed it for teachers last summer. This will make, I hope, a good review of many of the concepts and techniques students have learned during the year. The exploration, which will take an hour or more, includes these topics:

  • 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, C
  • 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
  • Investigating a special case or two

I also hope that in working through this exploration students will learn not so much about this particular function, but how to use the tools of algebra, calculus, and technology to fully investigate any function and to find all its foibles.

The exploration is here in a PDF file. Here are the solutions.

As always, I appreciate your feedback and comments. Please share them with me using the reply box below.

The College Board is pleased to offer a new live online event for new and experienced AP Calculus teachers on March 5th at 7:00 PM Eastern.

I will be the presenter.

The topic will be AP Calculus: How to Review for the Exam:  In this two-hour online workshop, we will investigate techniques and hints for helping students to prepare for the AP Calculus exams. Additionally, we’ll discuss the 10 type questions that appear on the AP Calculus exams, and what students need know and to be able to do for each. Finally, we’ll examine resources for exam review.

Registration for this event is $30/members and $35/non-members. You can register for the event by following this link: http://eventreg.collegeboard.org/d/xbqbjz

 

 

 

 

 

 

Differential Equations 2

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More posts on differential equations

Good Question 2: 2002 BC 5 (2-17-2015) A differential equation that cannot be solved by separating the variables is investigated anyway. Most of this question is AB material.

A Family of Functions (2-21-2015) Further investigation of the general solution of the equation discussed above in Good Question 2. Most of this question is AB material.

The Logistic Equation (1-31-2017) An outline of the logistic differential equation, its solution, its graph, and what students need to know for the exam. BC Topic.

        Don’t miss this one -> An Exploration in Differential Equations  (6-8-2018) An exploration covering pretty much all of the ideas in differential equation based on 2018 AB 6.  The exploration is here and                                                          the solutions here..

Logistics Growth – Real and Simulated (1-24-2017) Examples of logistic growth and a simulation you can use in your class. BC Topic.

Review Notes

Type 6 Questions: Differential Equations (3-23-2018) What AB and BC students need to know about differential equations for the AP Calculus exams.

The College Board is pleased to offer a new live online event for new and experienced AP Calculus teachers on March 5th at 7:00 PM Eastern.

I will be the presenter.

The topic will be AP Calculus: How to Review for the Exam:  In this two-hour online workshop, we will investigate techniques and hints for helping students to prepare for the AP Calculus exams. Additionally, we’ll discuss the 10 type questions that appear on the AP Calculus exams, and what students need know and to be able to do for each. Finally, we’ll examine resources for exam review.

Registration for this event is $30/members and $35/non-members. You can register for the event by following this link: http://eventreg.collegeboard.org/d/xbqbjz

 

 

 

 

 

 

Differential Equations 1

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Past posts on differential equations

Differential Equations (1-5-2015) The basics and definitions.

Domain of a Differential Equation (4-7-2017) notes and examples on finding the domain of the solution of a differential equation. (Updated thru the 2018 exam.)

Slope Fields (1-9-2015) Graphical solutions: The solution is lurking in the slope field.

Euler’s Method (1-12-2015) Numerical solutions (BC only topic)

Euler’s Method for Making Money (2-25-2015) The connection between compound growth (compound interest) and Euler’s Method.

Accumulation and Differential Equations  (2-1-2013) Solving differential equations without the “+C“

 

 

 

 

 

Summer Fun

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Every Spring I have a lot of fun proofreading Audrey Weeks’ new Calculus in Motion illustrations for the most recent AP Calculus Exam questions. These illustrations run on Geometers’ Sketchpad. In addition to the exam questions Calculus in Motion (and its companion Algebra in Motion) include separate animations illustrating most of the concepts in calculus and algebra. This is a great resource for your classes.

The proofreading and the cross-country conversations with Audrey give me a chance to learn more about the questions.

This year, I really got into 2018 AB 6, the differential equation question. I wrote an exploration (or as the kids would say “worksheet”) on a function very similar to the differential equation in that question. The exploration, which is rather long, includes these topics:

  • 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, C
  • 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
  • Investigating a special case or two

I also hope that in working through this exploration students will learn not so much about this particular function, but how to use the tools of algebra, calculus, and technology to fully investigate any function and to find all its foibles.

Students will need to have studied calculus through differential equations before they start the exploration. I will repost it next January for them.

The exploration is here for you to try. Try it before you look at the solutions. It will give you something to do over the summer – well not all summer, only an hour or so.

As always, I appreciate your feedback and comments. Please share them with me using the reply box below.

There will be only occasional, very occasional, posts over the Summer. More regular posting will begin again in August. Enjoy the Explorations, and, more important, enjoy the Summer!

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Posts on Differential Equations – 2

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More posts on differential equations

Good Question 2 (2002 BC 5) and A Family of Function (Good Question 2 continued) – one of my all-time favorite AP exam questions. Parts a, c, and d are suitable for AB students. Part b is a Euler’s method question. Part d is an example of a question where the second derivative test is the only approach possible.

Accumulation and Differential Equations  and Why Muss with the “+C”?

Euler’s method for Making Money an application

The Logistic Equation  – a BC only topic

Logistic Growth – Real and Simulated  – a BC only topic

Logarithms  Defining logarithms with a differential equation