14 Key Concepts for AP Calculus AB AP Exams Spring 2021
The Princeton Review
Hello, my friends!
Welcome back!
I’m Rob Franek, Editor-in-Chief at The Princeton Review, and today we’re sharing 14 very
important Calculus AB concepts for those of you preparing to take that AP exam in the
near future.
AP Calculus is divided into two types: AB and BC.
So, before we jump in, let’s quickly discuss the differences between Calculus AB and Calculus
BC.
The first key difference is that Calc AB is supposed to be equivalent to one semester
of college calculus, while Calc BC is equivalent to a full year.
In truth, however, AB calculus covers closer to three-quarters of a year of college calculus!
The second key difference is that Calculus BC covers more theoretical aspects of calculus
and is, objectively speaking, much harder than Calculus AB.
The AB exam usually tests more straightforward problems, and you’ll find that the problems
aren’t as tricky or as varied as the problems you’ll find on the BC exam.
That’s not to say that prepping for the AB exam will be a walk in the park!
Both tests focus on the basics of differential and integral calculus, and both will prove
to be difficult mainly due to the breadth of concepts that they cover.
The standard Calculus AB exam is 3 hours and 15 minutes long with a total of 51 questions.
It is divided into two sections, with a multiple-choice and open-ended section, and both sections
include calculator and non-calculator parts.
Now that we’ve reviewed the basics, let’s take a look at the key concepts you’ll need
to tackle for this exam!
#1 – Limits, Asymptotes, and Continuity: In order to understand calculus, you need
to understand limits.
Limits are the foundation of important definitions and theorems that will be tested throughout
the exam, so you should be prepared to evaluate limits not only via equations, but also via
graphical, symbolic, numerical, and verbal representations.
You should understand asymptotes graphically and be able to compare the growth rates of
different types of functions.
You’ll also need to be able to determine asymptotes and limits at infinity.
The exam will also test you on continuity.
You should be able to test the continuity of a function in terms of limits, and you
should understand continuous functions graphically.
You should understand the Intermediate Value Theorem, as well as the four types of discontinuities—jump,
point, essential, and removable.
#2 – The Definition of the Derivative: Every problem you’ll encounter in differential
calculus makes use of the derivative (hence the name).
By test day you should be able to find a derivative by finding the limit of the difference quotient.
Hint: the difference quotient is a special form of the slope formula!
You should also know the relationship between differentiability and continuity.
That is, if a function is differentiable at a point, it’s continuous there.
But if a function is continuous at a point, it’s not necessarily differentiable there.
#3 – Basic Differentiation: The exam will test your mastery of basic differentiation
rules.
So, you should understand the applications of the Power Rule, the Product Rule, the Quotient
Rule, and the Chain Rule.
The exam will also test your ability to find Higher-Order Derivatives by taking the derivative
of a function more than once.
The derivative of the first derivative is called the second derivative.
The derivative of the second derivative is called the third derivative, and so on.
You should be clear on these concepts by exam day.
You should also know how to relate the graph of a function to the graph of its derivative,
and vice versa.
And, finally, you should understand the relationship between the sign of a derivative and whether
the function is increasing or decreasing – a positive derivative means it’s increasing,
and a negative derivative means it’s decreasing!
#4 – Implicit Differentiation: You should expect to be tested on your ability to use
implicit differentiation.
In cases when you’re asked to find the derivative of an equation, but can’t isolate y in terms
of x – meaning you can’t differentiate this equation explicitly, you’ll need to
know how to take the derivative implicitly.
While implicit differentiation is one of the simpler techniques you need to learn in calculus,
it can initially give many students trouble.
So, be sure to try your hand at plenty of practice problems!
#5 – Basic Applications of the Derivative: Once you understand the fundamentals of basic
and implicit differentiation, it is important to understand the basic applications of the
derivative.
This includes the ability to find the slope of a curve at a point, and the tangent and
normal lines to a curve at a point.
This also means being able to use local linear approximation and differentials to estimate
the tangent line to a curve at a point.
You can also expect to see questions related to the Mean Value Theorem for Derivatives
and Rolle’s Theorem on the exam.
So, be sure to review the applications of both theorems in detail.
#6 – Maxima, Minima, and Curve Sketching: Another key concept—and one of the most
common applications of the derivative—is to find a maximum or minimum value of a function.
The maximum or minimum of a function occurs at a point where the derivative of a function
is zero.
These values can be called extreme values, optimal values, or critical points.
(It’s a truism in calculus that if you don’t know what to do, take the derivative and set
it equal to zero.)
You should also be familiar with the Extreme Value Theorem, which explains that if the
function is continuous in an interval, then you must alway have a maximum or minimum somewhere
in the interval, or else the curve is flat.
You should also know how to sketch a curve.
This means being able to (1) identify the x and y intercepts, (2) find any horizontal
and/or vertical asymptotes, (3) determine the critical points of the function, (4) find
the location of the maxima and minima, and (5) determine the points of inflection.
#7 – Motion: You’re likely to see some basic problems about motion on the exam, specifically
on related rates and the relationship between the velocity and acceleration of a particle.
For related rates problems, the idea is pretty simple – you’ll be given an equation relating
two or more variables.
These variables will change with respect to time, and you’ll use derivatives to determine
how the rates of change are related.
Almost every AP exam will also have a question on position, velocity, or acceleration.
If you have a function that gives you the position of an object – usually called a
particle – at a specified time, then the derivative of that function with respect to
time is the velocity of the object, and the second derivative is the acceleration.
Be sure to familiarize yourself with these concepts by test day.
#8 – Exponential, Trigonometric, and Logarithmic Functions: Remember all of the trigonometric,
logarithmic, exponential functions you studied in your precalculus class?
If not, be sure to give yourself a refresher!
Because the AP exam will test your ability to find the derivative and integral of trig
functions, logarithmic functions, and exponential functions.
Note that a lot of these derivatives and integrals can (and should!) be memorized — doing so
will make it easier to apply the Product, Quotient, and Chain Rules on these questions.
#9 – The Derivative of an Inverse Function: The College Board will occasionally ask a
question about finding the derivative of an inverse function.
So, you’ll need to know how to find the derivative of a function’s inverse at a
particular point by taking the reciprocal of the derivative at that point’s corresponding
y-value.
#10 — Integrals: It is integral for you to know about integrals.
Dad jokes aside, so far we’ve covered differential calculus and what you need to know about the
derivative, but the exam will also test you on the reverse – how to take an integral.
You will need to understand the different purposes an integral serves as the antiderivative.
An antiderivative is a derivative in reverse.
You should be able to apply the rules you learned with derivatives to integrals.
For example, you should know how to reverse the Power Rule for antiderivatives and how
to use the integration corollary of the Chain Rule (called u-substitution), when the integrand
is a composite function.
#11 — Definite Integrals: The AP exam will also test you on definite integrals.
So, you’ll need to know how integration is used to find the area under a curve, including
how to use rectangles to approximate the area between the curve and the x-axis, how to find
a Riemann sum, and how to use the Fundamental Theorem of Calculus and the Trapezoid Rule.
Next, you'll need to understand how the Mean Value Theorem for Integrals, otherwise known
as MVTI, applies to integrals.
The exam will likely test your knowledge of the MVTI by asking you to find the average
value of a function.
While we already mentioned the Fundamental Theorem of Calculus, it’s important to know
that it has two parts.
There’s the First Fundamental Theorem of Calculus, used in evaluating definite integrals,
and then there’s the Second Fundamental Theorem of Calculus, which tells us how to
find the derivative of an integral.
Aim to understand both parts of this theorem.
Lastly, the exam will also have problems that deal with accumulation functions.
They’re called accumulation functions because the value of the integral is the area under
the curve, from the constant to the value x.
As x gets bigger, so does the area.
Hence, it accumulates.
Be sure to understand how these functions work.
#12 – The Area Between Two Curves: In addition to understanding the area between a curve
and the x-axis, the AP exam will also test your ability to find the area between two
curves.
This includes the ability to find the area by slicing up the region both vertically and
horizontally.
If you encounter difficulty with this concept, know that you’re not alone!
Students often have terrible trouble setting up these problems.
Fortunately, the College Board asks relatively simple versions of these problems on the exam.
That said, be sure to allocate time to practice this concept.
#13 – The Volume of a Solid of Revolution: Folks, this concept is widely seen as one
of the most difficult on the exam.
And while there is always a volume question on the exam, the good news is that you’re
almost never asked to evaluate the integral—you usually only have to set it up.
You’ll need to know how to take the region between two curves and rotate it around a
line to find the volume of the region.
There are two methods to find the volume of the region – the washers method and the
cylindrical shells method.
The latter method is more advanced and isn't likely to appear on the AP Exam; however,
it’s still worth familiarizing yourself with both techniques.
Volumes of Solids of Revolution require you to sketch the region carefully and to decide
whether it’ll be easier to slice the region vertically or horizontally.
Once you figure out the slices’ boundaries and the limits of integration, it’s just
a matter of plugging into the integral.
With enough practice, finding the volumes will become easier!
And folks, once you’ve conquered this topic, you’ll be ready for anything!
There is another type of volume that you need to be able to find, known as volumes of solids
with known cross-sections.
In this case, you will be given an object where you know the shape of the base and where
perpendicular cross-sections are all the same regular, planar geometric shape.
You’ll need to know how to find the side of the cross-section in terms of y.
#14 – Differential Equations: Last but certainly not least, you’ll need to familiarize yourself
with differential equations.
While there are many types of differential equations, only a small number of them appear
on the Calculus AB exam.
As such, you only need a basic introduction to the topic by test day.
First, you’ll need to understand the method known as separation of variables.
If given an equation in which the derivative of a function is equal to some other function,
you should be able to determine the original function by integrating both sides of the
equation and then solving for the constant term.
(And don’t forget about that big C!)
Next, you’ll need to understand the idea behind slope fields, also known as direction
fields.
You will be given a differential equation, but not the equation itself, and will need
to know how to make a graphical representation of the slope of a function at various points
in the plane.
And if this concept sounds daunting, please rest assured that you will only be asked to
sketch the simplest ones on the exam.
Phew!
That’s a lot of Calculus — but know that with practice and preparation, you can totally
handle it.
Check out our AP Prep series.
The Premium edition of AP Calculus AB gives you access to 7 full-length practice tests!
That’s all for today!
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