Allen Tsao

# How to Prepare for the AP Calculus AB Free Response Questions

Updated: May 7, 2019

I have a lot of students who are preparing for AP Calculus exams right now, so I thought I'd cover some helpful tips to prepare. The best thing you can do is practice. You want to be constantly doing previous year's free response questions.

But here's the most important tip for the free response questions (FRQ): **they are fairly predictable**. For the most part, the types of questions used on the AP Calculus Exam are similar each year. Below I included a picture of all the FRQs from the previous years. I split it into two tables so you could see somewhat more recent exams versus a longer period of time, but the trend is the same.

Your goal is to be a master of the top 7 types of questions here. That's why I organize my AP Calculus YouTube __playlists__ by these topics so that you can practice a lot of those topics that you may be struggling with.

**Rate Problems**

Rate problems usually are describing the rate of something happening in a word problem. Typical rate problems will talk about flowing in and flowing out and you're given an equation describing the rate of change. Here's an example of the 2018 free response question on rates:

You want to make sure you understand a few basic concepts to answer this question:

The integral of the rate of change is the amount. In this example, the rate r(t) is people per second who enter the elevator, so the integral of this rate is the number of people who enter the escalator.

There's usually a flow-in rate and a flow-out rate. In this problem, r(t) is the number of people who enter per second, which is the flow-in rate. The flow-out rate is 0.7 people per second because that is the rate people are leaving the escalator. The net rate of change of people on the escalator is the flow-in rate minus the flow-out rate.

**Interpreting Graphs**

This is a favorite for AP Calculus AB. I would definitely expect one of these types of questions on your AP Calculus Exam. This kind of problem requires you to find information about a function by knowing a graph of its derivative. Thus, the normal techniques for finding derivatives and integrals analytically won't be of any use. Instead, you have to understand the physical interpretation of derivatives of integrals. Here's an example from the 2018 AP Calculus exam.

In order to answer this type of question, you want to make sure you understand the following:

The "change" in the value of a function is represented as an area under the derivative. This is basically the Fundamental Theorem of Calculus.

The derivative of a function can be seen as the slope of the tangent line to the curve.

How to determine when functions are increasing/decreasing, concave up/down, and identify relative minimum, maximums and points of concavity.

**Differential Equations**

A differential equation sounds like something complicated, but it's simply an equation that has a derivative in it. Solving general differential equations is actually extremely difficult, and you could take a lot of math classes devoted solely to solving different types of differential equations. For the AP Calculus exam, the only method you need to know is called separation of variables. Here's an example from the 2016 AP Calculus exam.

Know how to draw a slope field from a differential equation

Sketch a particular solution to a differential equation given a slope field

Using separation of variables solve a differential equation, and using the initial value to find the particular solution

**Approximating Derivatives and Integrals**

For these types of problems, you will be given a table of values of a continuous function and asked to estimate the derivative and/or area under the curve for the points. These are estimates because you aren't given the exact function. Here is the problem from the 2018 AP Calculus exam.

For these problems, you are typically asked about the following topics:

Estimating derivatives using a secant line slope and interpreting the slope as a rate of change.

Estimating the area under the curve using a Riemann sum: left-hand sum, right-hand sum, midpoint sum, or trapezoid sum.

Application of intermediate value theorem or mean value theorem.

**Application of Integrals**

This includes a few different types of problems that are common:

Finding area under or between curves

Finding volumes of shapes due to cross-sectional areas or revolving a region around a line

Finding average value

Here's an example from the 2016 AP Exam.

To find the volume, you need to make sure you understand how to cut up the shape into small cross-sectional discs, find the volume of each disc, and sum them up with an integral.

**Motion**

Motion problems involve understanding the relationship between position, velocity and acceleration. In a problem like this you need to make sure you understand:

The following equations relating acceleration, velocity and position.

Speed has no direction (i.e. it is a scalar) and it is the absolute value of velocity.

Distance traveled can be found by taking the integral of speed.

An object is speeding up at a particular time of the acceleration and velocity have the same sign. An object is slowing down if they have different signs.

**Resources**

To help prepare for the exam, here are some useful resources:

Link to Collegeboard Website of Previous AP Calculus Exam Questions

__https://apcentral.collegeboard.org/courses/ap-calculus-ab/exam__

Link to AP Calculus YouTube Playlists by Topic

__https://www.bothellstemcoach.com/ap-calculus__