AP Type Questions 6
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 question about its slope field or a tangent line approximation or something else related. 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 subscore 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).
- 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.
- 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 have never asked students to actually solve a logistic equation IVP.
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.