Need extra practice for your AP Calculus Unit 7 review? Outlined are the topics and differential equations practice problems aligned with College Board’s curriculum to study for a Unit 7 test.
As College Board outlines, the topics to review for Unit 7 are:
· Modeling Situations with Differential Equations
This is essentially taking words and converting the relationship to an equation. Recall from past math classes that a directly proportional relationship is of the form kx, and an inversely proportional relationship is of the form k/x.
While this is not a common question on the AP exam, it leads us into the topics below, Exponential Models or Logistic Models.
1. Write a differential equation to describe the relationship: The rate of change of a radioactive substance R with respect to t is proportional to the amount of the substance present at any given time.
· Verifying Solutions for Differential Equations
For this section, you’re often given dy/dx and asked to find a particular solution through a given point. To go from dy/dx back to y, you must take the antiderivative. In taking the antiderivative, don’t forget to add C since it’s an indefinite integral. Use the given point to find C.
2. Find the particular solution to dy/dx= sin(2x) through (0, 4).
· Sketching Slope Fields
The two common ways this section is assessed on the AP exam is first by matching the differential equation to its slope field in the multiple choice section, or second, by drawing a slope field given a differential equation in the free response section.
To fill out a slope field, plug in each coordinate into dy/dx. The result will determine the slope of the small tangent line segment you draw at that point. To differentiate between positive slopes, for example, make sure a slope of 3 is steeper than a slope of 1. Remember a slope of 0 is a horizontal tangent, while a slope that’s undefined (from dividing by 0) would be a vertical tangent.
To match a differential equation to its slope field, you can follow the same idea as above, choosing a coordinate and plugging it into dy/dx and seeing if it makes sense for the slope field. This might take several iterations to determine the correct differential equation. A tip is to choose a coordinate that has a negative slope on the slope field. That often cancels out more of the multiple choice answers.
Another tip/method is to set all the differential equations equal to 0 and solve. For example, if you had dy/dx=x-y, I would set x-y=0 and solve for y: y=x. Along that diagonal line, if the slope field has horizontal tangents, that would be the correct solution. If dy/dx=2x, setting that to 0 would give x=0. Along the vertical line x=0, or the y-axis, the correct solution will have horizontal tangents.
For a practice problem of this section, see the free response question from AP Calculus AB 2010 Form B below. Part (a) is this section and part (b) is the next section.
· Reasoning using Slope Fields
This section includes questions about slope fields like “describe all points in the xy-plane with negative slopes” or “explain why the slope field could not match the given differential equation.” For the first problem, I would see when dy/dx is negative (if it’s a complicated expression, you may need to factor and set each part to 0 first). For the second question, try to find a counterexample with conflicting information such as “The slope at (1,2) is positive on the slope field but plugging (1,2) into dy/dx equals 0”
3.
Part C is from an upcoming section Finding Particular Solutions, but since that is arguably the most important section of this chapter, I included this for extra practice.
· Approximating Solutions using Euler’s Method (BC only)
There are two ways to apply Euler's method, either using tangent line approximations or a table that organizes the point, slope (dy/dx), change in x, and change in y. If you look at scoring guidelines for an AP Calculus free response question, they’ll typically use the tangent line method. I admit, when I started teaching AP Calculus, I used the table method since my textbook used that method. Now, I recommend using tangent lines since it’s one less thing to memorize. We’ve been doing tangent line equations for months now so hopefully you feel confident in that!
4. Let y=f(x) be the solution to the differential equation dy/dx= x+y with the initial condition f(1)=2. What is the approximation for f(2) using Euler’s method with 2 steps of equal length starting at x=1?
· Finding General Solutions using Separation of Variables
Most differential equations we encounter in AP Calculus have dy/dx in terms of both x and y. To get back to the solution y=f(x), we must take an antiderivative. But we can’t take an antiderivative with x and y together. We must first separate the variables. Then integrate, or take the antiderivative, of both sides. Don’t forget the +C for indefinite integrals! The general solution will leave the C, as the solution is the family of curves with the same slope. The final step is to get y by itself.
5. Find the general solution to the differential equation: dy/dx = (y+1)cosx
· Finding Particular Solutions using Initial Conditions and Separation of Variables
This section is the same idea as the previous one, but we use the initial condition/coordinate to solve for C. I like to use the steps: separate, integrate, solve for C, solve for y. You can switch the last two steps, but I’ve found it’s usually more algebraic work if you solve for y first.
6. Find the particular solution to the differential equation: dy/dx= xy^2 through the point (-1, 3).
· Exponential Models with Differential Equations
Exponential differential equations are of the form dy/dt=ky. You can find the general solution by separating and integrating, and you’ll see that the general solution to exponential differential equations is y=y0e^(kt). You probably recognize this formula from previous math classes as continuous exponential growth or decay. Any time you see dy/dt=ky or the wording “a rate proportional to the number of y” (y varies from problem to problem), you can automatically use y=y0e^(kt).
You should be given y0, the initial amount, or if it’s not given, you can use 1 for simplicity. The first step is to solve for k (you’ll be given a y value and time to plug in). Once you get k, you can plug that into your y= equation. The last step is to plug in the remaining time or y-value and solve for the other variable.
7. During the first month of the COVID pandemic, the number of people infected increased at a rate proportional to the number of people infected at that time. If 1000 people were infected when the pandemic was first discovered, and 4000 people were infected 2 days later, how many people were infected 14 days after it was first discovered?
· Logistic Models with Differential Equations (BC only)
Logistic models are typically better models for population growth than exponential models because there are factors that limit growth, like food availability, so it doesn't make sense for a population to continue to grow without bound.
Logistic differential equations are of the form dy/dt=ky(M-y). M is the carrying capacity. If you set dy/dt = 0, that will give you horizontal tangents for y. In the case of logistic curves, that will give you the two horizontal asymptotes, at y=0 and y=M. Some teachers also give the general logistic curve formula, but you don’t need to know that for the AP exam.
There’s a variety of logistic questions that can be asked (this is ultimately the application of integration using partial fractions!), but the most common questions ask about the form of a logistic differential equation, the carrying capacity, the greatest rate of change, or a limit approaching infinity. I’ve provided one problem below, but you can get more practice on logistic models and Unit 7 at that link.
Notice on the logistic curve below that the green dot for the point of inflection, or the greatest rate of change, occurs in the middle of the two horizontal asymptotes. The easiest way to calculate this is to find half of the carrying capacity.
In this case, the carrying capacity is 6 so the greatest rate of change occurs when y=3.
8. A population P changes at a rate modeled by the logistic differential equation dP/dt=0.1P(2000-P), where t is measured in years, and P(0)=300. A) At what population does the maximum rate of change occur? B) What is the limit as t approaches infinity of P(t)?
AP Calculus Unit 7 Review
For your AP Calculus Unit 7 review, make sure you are practicing differential equation problems from multiple perspectives: given functions, tables, graphs, and word problems. If you need extra practice, check out the 7 best resources to study for AP Calculus tests. These resources have questions that match the rigor of your tests.
If you need further explanation on how to approach some of these difficult Unit 7 review questions, especially when all of the topics are mixed together, consider individual tutoring with me. I answer any questions students have, then provide practice solving my past test questions and previous AP exam questions. Getting more practice with problems given graphs, tables, and word problems will help you be prepared for in class tests and the AP exam.
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Solutions to the above problems:
1. dR/dt=kR
3.
Going back to the horizontal asymptotes being y=0 and y=M, recall that a limit approaching infinity is essentially asking about horizontal asymptotes. So your answer for part B above will either be 0 or 2000. To determine which was correct, I checked dP/dt at the given point. Logistic curves are always increasing or always decreasing so if dP/dt is positive, P is increasing so 2000 is our answers. If dP/dt was negative, P would be decreasing so 0 would be the answer.
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