Honestly, if you ask anyone who sat through the May 2018 administration of the Advanced Placement exams, they probably still have nightmares about the escalator. Or maybe the grain silo. The AP Calc AB 2018 FRQ wasn't necessarily the hardest set of Free Response Questions ever released by the College Board, but it was incredibly "wordy." It forced students to step out of the comfort of pure symbols and into the messy world of modeling.
Calculus isn't just about taking a derivative. It’s about understanding what that derivative means when people are getting on and off a moving staircase.
The Escalator Problem: A Lesson in Rate In and Rate Out
Question 1 of the 2018 set is a classic. You have people entering a line for an escalator at a rate $r(t) = 44(\frac{t}{100})^3 (1 - \frac{t}{300})^7$. Then, they exit at a constant rate of 0.7 persons per second. This is the "Rate In/Rate Out" archetype that the College Board loves.
Most students can handle the integration. You integrate $r(t)$ from 0 to 300 to find the total number of people who entered. That's fine. But the part that caught people off guard was the nuance of "how many people are in line at time $t$?" You have to account for the initial 20 people already there. If you forget the initial condition, the whole house of cards falls.
What’s interesting here is how the math reflects reality. The rate $r(t)$ is a high-degree polynomial. It starts slow, peaks, and then tapers off as the 300-second window closes. In a real-world setting, like a subway station after a train arrives, this is exactly how crowds behave. The 2018 exam used this to test if students understood the Net Change Theorem. Basically, the number of people in line $L(t)$ is:
$$L(t) = 20 + \int_{0}^{t} r(s) ds - 0.7t$$
If $L(t)$ ever hits zero, the formula changes because you can't have negative people in a line. That’s the kind of "gotcha" that separates a 4 from a 5.
Particle Motion and the Frustration of Question 2
Then came the particles. In the AP Calc AB 2018 FRQ, Question 2 gave us two particles, $P$ and $Q$, moving along the x-axis. Particle $P$ had its velocity defined by $v_P(t) = \sin(t^{1.5})$, while Particle $Q$ was just a position function.
Why do students hate this?
Because it requires juggling two different frameworks at once. You’re looking at one particle through the lens of its velocity and the other through its position. To find if they are moving toward or away from each other, you have to look at both their positions and the signs of their velocities.
It’s tedious.
It’s also where many students lose points on "justification." You can't just say "they are moving together." You have to show that at time $t=1$, $x_P(1) < x_Q(1)$ and $v_P(1) > 0$ while $v_Q(1) < 0$. It’s about building a legal case for your answer using calculus as evidence.
That Grain Silo and the Chain Rule
Question 4 moved into the "No Calculator" section, and it brought the infamous leaking grain silo. This was a related rates problem, but it was disguised within a differential equation.
The height of the grain $h$ changes over time according to $\frac{dh}{dt} = -\frac{1}{10}\sqrt{h}$.
It looks simple. It isn't. Part (c) of this question asked for the second derivative, $\frac{d^2h}{dt^2}$. To do this correctly, you must use implicit differentiation.
$$\frac{d^2h}{dt^2} = -\frac{1}{10} \cdot \frac{1}{2\sqrt{h}} \cdot \frac{dh}{dt}$$
Most students forgot to multiply by $\frac{dh}{dt}$ at the end. They forgot the chain rule. Without that substitution, you can't get the numerical value for the second derivative. This specific mistake is why the mean score on Question 4 was significantly lower than teachers expected. It’s a classic "calc trap."
The Table Problem: Approximations and the MVT
Question 3 provided a table of values for a function $f$ and its derivative $f'$. This is standard fare, yet the 2018 version felt particularly clinical. It asked for a Right Riemann Sum to estimate an integral.
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Riemann sums are just rectangles.
But when the intervals in the table aren't uniform—when the "width" of your rectangles changes from 2 to 3 to 5—students who rely on "formulas" instead of "drawing the picture" fail. The 2018 exam was a direct attack on students who memorize instead of understand.
The question also poked at the Mean Value Theorem (MVT). It asked if there was a time $c$ such that $f'(c) = 0$. To answer this, you had to check if $f(a) = f(b)$ for some interval, applying Rolle’s Theorem (a specific case of MVT). If you didn't explicitly state that $f$ is continuous and differentiable, you didn't get full credit. The College Board is picky. They want the "if-then" logic clearly laid out.
Why 2018 Still Matters for Current Students
If you're studying for the AP exam today, the AP Calc AB 2018 FRQ is your best training ground. Why? Because it represents the "modern" style of the exam. The questions aren't just "solve this equation." They are "explain why this model makes sense."
The 2018 exam was the first year where the emphasis on "Mathematical Practices" really became visible.
- Communication: You had to write sentences.
- Modeling: You had to link the math to physical objects like trees and escalators.
- Technology: You had to use the graphing calculator efficiently in the first two questions.
Take Question 6, the differential equation $\frac{dy}{dx} = \frac{1}{3}x(y-2)^2$. Solving this requires separation of variables. You put all the $y$'s on one side and the $x$'s on the other. If you don't separate the variables, you get zero out of five points for that part. Zero. It’s brutal, but it teaches you the stakes of the procedure.
Breaking Down the Scoring Rubrics
The Chief Reader’s report for 2018 noted that students struggled heavily with the "Average Value" vs. "Average Rate of Change."
Average Value of a function: $\frac{1}{b-a} \int_{a}^{b} f(x) dx$
Average Rate of Change: $\frac{f(b) - f(a)}{b-a}$
In the 2018 context, specifically with the tree growth problem (Question 4), people were mixing these up constantly. They were calculating the average height when the question asked for the average rate of height increase. It's a one-word difference that changes the entire calculus operation.
Actionable Prep Steps
If you are going to tackle these problems for practice, don't just look at the answers. Do this:
- Time Yourself: Give yourself 15 minutes per question. The 2018 questions are long; you need to feel the pressure of the clock.
- Write the "Because": Every time you find a relative maximum or minimum, write: "f has a relative max at x=c because f' changes from positive to negative." This was worth 1 point on almost every 2018 FRQ.
- Check Units: In 2018, units were required for several answers (like feet per second or people). If you miss the units, you miss the point.
- Re-do Question 6: If you can solve that differential equation without looking at your notes, you're in a great spot for any upcoming AP exam. Separation of variables is the single most valuable skill in the FRQ section.
The 2018 exam wasn't about being a math genius. It was about being a careful reader. Those who slowed down and actually looked at what the variables represented—whether it was the height of a tree or the number of people in a queue—were the ones who walked away with a 5.
Keep your notation clean. Don't say "it." Don't say "the graph increases." Say "$f(x)$ increases." Precision is the only way to beat the 2018 rubric.
Next Steps for Success
Download the 2018 Scoring Guidelines from the College Board website and grade your own work with a red pen. Specifically, look at Question 4 and see if you remembered to include the constant of integration ($+C$) when solving the differential equation. Most students forget it on their first try, which automatically caps their score for that problem. Once you've mastered the 2018 set, move on to the 2021 FRQs to see how the "wordiness" of the problems has continued to evolve.