AP Calculus Exam – Last minute reminders…
1.
𝑏
Distance traveled from time 𝑎 to time 𝑏 = ∫𝑎 |𝑣(𝑡)|𝑑𝑡
𝑏
Position = initial position + change in position = initial + ∫𝑎 𝑣(𝑡) 𝑑𝑡
2.
u-substitution: you can’t find the antiderivative of a product of two functions without either using your calculator or usubstitution. If you have the product of two functions, the antiderivative is NOT the anti of one times the anti of the other.
Examples:
∫ 𝑥√𝑥 2 − 1 𝑑𝑥
3.
𝑥
∫ 4𝑥 2+3 𝑑𝑥
Don’t forget to label what it is you are taking the derivative of during FRQ. If you are given velocity and you need to find
acceleration…you need to actually write 𝑣(𝑡) = or 𝑥 ′ (𝑡) = or 𝑠 ′ (𝑡) = . If you just find the derivative, but don’t indicate
anything else, you won’t get points. For example:
Say you are given 𝑣(𝑡) = cos⁡(𝜋𝑡) and you are asked to determine if the particle is speeding up or slowing down at time 𝑡 =
2
𝜋
𝜋
. In order to do this problem, you will need to find both 𝑣(𝑡) and 𝑎(𝑡) and determine the signs when 𝑡 = . If you just
2
write −𝜋 sin(𝜋𝑡), that is meaningless unless you label it as 𝑎(𝑡) = −𝜋sin⁡(𝜋𝑡) or label it as 𝑣 ′ (𝑡) = −𝜋sin⁡(𝜋𝑡). You will
miss out on easy points!
4.
Don’t forget units when they are needed.
5.
The notation + and – are meaningless. Need to write “positive” or “negative” or write 𝑓(𝑥) > 0 or 𝑓(𝑥) < 0.
6.
The notations LHL and RHL are meaningless. You need to write lim− 𝑓(𝑥) and lim+ 𝑓(𝑥).
7.
For a particle to “speed up”, velocity and acceleration need to be the same sign. The particle can have negative
acceleration and still be speeding up as long as the velocity is also negative.
8.
𝑥→𝑎
𝑑2𝑦
𝑑𝑥 2
𝑥→𝑎
means the second derivative
9.
If you see the words “separation of variables” or “differential equation”…you need to separate the variables – get all the
matching variables and dx’s or dy’s or dt’s on each side of the equation. Even if you don’t know how to solve the
antiderivatives from here, keep going. Take an antiderivative and go on with the problem, you will still get points if you use
+C and try to solve for the +C using the initial conditions.
10.
KEEP MANY DECIMAL PLACES UNTIL YOU ARE FINISHED TAKING THE CALCULATIONS. WHEN YOU ARE DONE TAKING THE
CALCULATIONS, THEN ROUND TO 3 DECIMAL PLACES. IF ROUNDING IS NEEDED, YOU MUST ROUND TO THREE DECIMAL
PLACES. (This is especially important with bounds of integration.)
11.
Finding derivatives of inverse functions. If you are given a function 𝑓(𝑥) and 𝑔(𝑥)=𝑓 −1 (𝑥) (meaning 𝑔(𝑥) is the inverse of
(𝑥) ) and you need to find the derivative of 𝑔(𝑥), you can do that by applying the derived formula (which comes from using
implicit differentiation):
𝑔′ (𝑥) =
1
𝑓 ′ (𝑦)
Also, note that if you are given 𝑓(𝑎) = 𝑏, this means that 𝑔(𝑏) = 𝑎…should you need this info to
solve the problem.
12.
Do not use “it” to describe anything! Use 𝑓 ′ (𝑥), 𝑓(𝑥), 𝑓 ′′ (𝑥),⁡etc to be clear about what you are describing.
13.
Don’t try to find antiderivatives by hand if it is open-calcluator…chances are, it’s not an integral you should be able to take
an antiderivative of. This goes for FRQ and MC. Example of MC where this might occur:
Ex: 𝑓′(𝑥) = 𝑥 ln 𝑥 and 𝑓(2) = 3. Find 𝑓(5).
14.
Don’t spend too long on any particular problem, they are equally weighted.
15.
Answer all the MC questions, you are not additionally penalized for wrong answers (as you are on the SAT).
16.
Even if you don’t know how to do one of the first parts of a FRQ (free response question), continue on. If you answer part a
incorrectly and need to use that answer for part b, that’s okay, use it. If you use it correctly in b, you will still get credit for
that.
17.
You don’t need to waste time on simplifying. The answer to∫0 sin 𝑥 𝑑𝑥 is cos 𝑥 |𝜋0 = cos 𝜋 − cos 0 = (−1) − (1) = −2.
𝜋
You could leave this answer as cos 𝜋 − cos 0 instead of −2 if you want.
18.
For free-response questions, don’t put your answers all over the place. Put your answers in the designated areas for each
part. If you solve for something in another spot (like on a graph), it won’t be counted. Also, don’t put down multiple
answers for one free-response question.
AP Calculus AB Exam – Common Student Errors
1. Error: assuming that 𝑓 ′′ (𝑥) = 0 means there is an inflection point at (𝑥, 𝑓(𝑥)) or assuming that because there is an inflection
point at (𝑥, 𝑓(𝑥)) that 𝑓 ′′ (𝑥) = 0. Don’t assume these. What is important is that 𝑓′′(𝑥) changes sign at 𝑥.
2. Error: assuming that 𝑓(𝑥) is a maximum or minimum because 𝑓 ′ (𝑥) = 0. 𝑓 ′ (𝑥) = 0 by itself is not sufficient. What is important
is that 𝑓′ changes sign at 𝑥.
𝑏
3. Error: Volume of revolution = ∫𝑎 𝜋(𝑟𝑜 − 𝑟𝑖 )2 𝑑𝑥 . You need to keep the radii separate, think outer area minus inner area… Volume
𝑏
is equal to ∫𝑎 (𝜋𝑟𝑜2 − 𝜋𝑟𝑖2 )𝑑𝑥 .
4. Error: Separable differential equations can be solved without separating the variables. Always separate the variables my
multiplying or dividing first, before even thinking about integrating.
5. Error: Forgetting the constant of integration (+𝐶), especially in initial value problems. You need to solve for the +𝐶 using the
initial values, if they are given.
6. Error: Thinking that the grader will assume what you mean by using the word “it” in your justifications. If you are referring to
function 𝑔(𝑥), then write “𝑔(𝑥)” instead of “it”.
7. Error: Assuming that the grader will think your setup was correct just because you have the right answer. Don’t assume this. If
3
you use your calculator to find a value for ∫0 sin2 (𝜋𝑥) 𝑑𝑥 , don’t just write the answer, you also need to include the setup of your
integral. (And don’t write 𝑓𝑛𝐼𝑛𝑡(sin2 (𝜋𝑥) , 𝑥, 0,3) as your setup…you need to write it in integral form.) Likewise, if you use your
calculator to find that the derivative of 𝑓 at 3 is 0.367, then write 𝑓 ′ (3) = 0.367.
8. Error: Applying logarithmic antidifferentiation universally by doing ∫
1
𝑓(𝑥)
𝑑𝑥 = ln|𝑓(𝑥)| + 𝐶. Just because it’s
1
𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔
, the
antiderivative is not necessarily ln|𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔| + 𝐶. You might need to use u-substitution or it might be something like… the
antiderivative of
1
𝑥3
1
is − 𝑥 −2 + 𝐶 (not ln |𝑥 3 | + 𝐶).
2
9. Error: Forgetting to use the chain rule for derivatives! If it looks like a composite function, it smells like a composite function, or
tastes like a composite function, then APPLY THE CHAIN RULE. For example, the derivative of sin(𝑥 2 ) is not cos(𝑥 2 ), but rather
2𝑥 cos(𝑥 2 )
10. Error: Forgetting to use the product rule for related rates problems. Know which variables change with time!