Teaching Calculus

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

Adapting 2021 AB 6

Adapting 2021 BC 5

Adapting 2021 AB 6

Posted on July 27, 2021 by Lin McMullin

Six of nine. Continuing the current series of posts, this post looks at the AB Calculus 2021 exam question AB 6. Like most of the AP Exam questions, there is a lot more you can ask based on the stem of this question and a lot of other calculus you can discuss. This series of post offers suggestions as to how to adapt, expand, and use this question to help your students dig deeper and learn more.

2021 AB 6

This is a standard Differential Equation (Type 6) question and contains topics mainly from Unit 7 (Differential equations) and a little from Unit 3 (implicit differentiation) of the current Course and Exam Description. A differential equation with an initial condition is given in a context. The main part is the solution of the initial value question with three short other questions included.

The stem for 2021 AB 6 is:

Part (a): A slope field in the first quadrant with no scale on either axis is given. Students are asked to sketch the solution curve starting at the initial condition, the point (0, 0). (I prefer this kind of slope field question to those where students are given a few points and asked to sketch the slope field through them. No one draws slope field by hand; slope field drawn by computers are used to study the approximate shape of the solution and determine its interesting properties as is done here and in part (b)). When drawing slope fields, the sketch should extend to from one border to another and contain the initial condition point.

Discussion and ideas for adapting this question:

Have student sketch solution through one or more different points. Copy the slope field and add the initial condition point somewhere else.
Add an initial point above the horizontal asymptote.
Compare and contrast the solutions drawn through several points.
Ask what the horizontal segments (at y = 12) tells you in the context of the problem.

Part (b): Students are given the limit at infinity for the as yet unknow solution and asked to interpret it in the context of the problem including units of measure.

Discussion and ideas for adapting this question:

Discuss why this is so.
Discuss how to determine the units of the function from the given information.
Discuss how to determine the units of the derivative from the given information.
Discuss how to determine the units of the derivative from the units of the function.
Discuss how to determine the units of the function from the units of the derivative.
Discuss whether the interpretation of the limit makes sense in the context of the question.

Part (c): Students are asked to solve the initial value question using the method of separation of variables.

Discussion and ideas for adapting this question:

Since separation of variables is the only method for solving a differential equation that students are responsible for knowing, there is not much you can do to adapt or change this question.
The initial condition may be substituted immediately after the integration is done the “+ C” is attached, or it may be done later after the expression is solved for y. Show students both method and discuss which is more efficient and which makes more sense to them.
Removing the absolute value signs is another place that may confuse students. While some textbooks suggest using a “ ± “ sign and deciding sign which to use later, the better way is to decide as soon as possible. Ask yourself is the expression enclose by the absolute value signs positive or negative near (at) the initial value. If positive, then the absolute value is replaced by the same expression (as in this question); if negative, then the expression is replaced by its opposite. Then complete the question from there.

Part (d): This part needs careful reading. Students are asked, for a slightly different differential equation, if the rate of change in the amount of medicine is increasing or decreasing at a given time. Therefore, students must find the rate of change of the rate of change (the given derivative): the derivative of the derivative (i.e., the second derivative of the function). This requires implicit differentiation of the derivative using the quotient rule.

Discussion and ideas for adapting this question:

The second derivative has the first derivative as one of its factors. Students may (automatically) substitute the first derivative before simplifying or evaluating. This correct, but unnecessarily long. Show the students how to find and substitute the value of the first derivative along with the other numbers.
Do as little arithmetic as possible. You need only determine if the second derivative is positive or negative.
Discuss the meaning of the answer in the context of the problem.

Next week 2021 BC 2

I would be happy to hear your ideas for other ways to use this question. Please use the reply box below to share your ideas.

Differential Equations – Unit 7

Posted on December 1, 2020 by Lin McMullin

Topics 7.1 – 7.9

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

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 7.7 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 of differential equation and the method of separation of variables

Domain of a Differential Equation – On domain restrictions.

An Exploration in Differential EquationsAn 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

Differential Equations (Type 6)

Posted on March 20, 2020 by Lin McMullin

AP Questions Type 6: Differential Equations

Differential equations are tested almost 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

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 9 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 2019 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

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 equation and the method of separation of variables

Domain of a Differential Equation – On domain restrictions.

Previous Posts on BC Only Topics

Revised 2/20/2021

2019 CED Unit 7: Differential Equations

Posted on November 26, 2019 by Lin McMullin

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 7.7 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 $\displaystyle y={{y}_{0}}{{e}^{{kt}}}$ .

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.

Posts on BC Only Topics