Man, 2019 was a weird year for math. If you were sitting in a high school gym that May, staring down the 2019 AP Calculus BC FRQ booklet, you probably remember the fish. Specifically, the fish entering and leaving a lake. It sounds simple. It wasn't. While the BC exam is notorious for throwing series and polar coordinates at you, it's often the "simple" rate-in/rate-out problems that trip up even the kids headed for engineering degrees.
College Board has a specific way of breaking hearts. They don't just ask you to derive a function; they ask you to interpret the soul of the function. In 2019, the Free Response Questions (FRQs) were a brutal mix of classic accumulation and some truly "out there" polar area calculations. Honestly, if you didn't have your calculator settings perfectly tuned for Question 2, you were basically toast before you even hit the Taylor series in the back of the packet.
Why Question 1 and the Fish Still Haunt Students
Let’s talk about that fish. $E(t)$ and $L(t)$. One rate represents fish entering the lake, the other represents fish leaving. It’s a classic accumulation model. You’ve seen it with water in a tank or people in a line. But the 2019 twist involved a multi-step justification for the absolute minimum.
Most students can find the derivative. They can set $E(t) - L(t) = 0$. But the 2019 graders were looking for the "Candidates Test." If you didn't check the endpoints—$t=0$ and $t=8$—you lost points. It's a silly mistake. It's also the most common one. People forget that in the world of the 2019 AP Calculus BC FRQ, a local minimum isn't enough. You need the global one.
You also had to explain, in plain English, what an integral meant in context. If you said "the amount of fish," you might have been okay, but the nuance of "the total number of fish that entered the lake from time 0 to time 5" is what secured the full point. Precision matters more than speed here.
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The Polar Curve That Ruined GPAs
Question 2 was a polar nightmare involving two curves: $r = 3$ and $r = 3 - 2\sin(2\theta)$. Looking back at the scoring distributions, this is where the "5s" were separated from the "4s."
Polar area is inherently messy. You're dealing with $\frac{1}{2} \int \alpha^{\beta} r^2 d\theta$. In 2019, the trick was identifying the intersection points correctly. If your calculator was in degrees instead of radians? Game over. I've seen brilliant students lose half the points on this page just because they didn't realize the shaded region required subtracting one integral from another rather than just finding the area of one.
It’s about symmetry. The graph looked like a strange sort of flower. Many kids tried to calculate the whole thing at once. Smart testers calculated one leaf and multiplied. But you had to be careful—the bounds for $\theta$ aren't always $0$ to $2\pi$. In this specific 2019 prompt, the curve $r = 3 - 2\sin(2\theta)$ actually stays within certain bounds that make the "inner" and "outer" loops tricky to distinguish.
The Mean Value Theorem and the Hidden Trap
Question 3 gave us a table for $h(t)$, representing the height of a tree. This is the "easy" point-grabber, right? Not exactly. Part (c) asked about the Mean Value Theorem (MVT).
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To use MVT, you have to explicitly state that the function is continuous and differentiable. You can't just skip to the math. If you didn't write "Since $h(t)$ is differentiable, it is also continuous," the graders stopped reading. It feels like bureaucracy. It is bureaucracy. But it’s how the 2019 AP Calculus BC FRQ was designed to filter for conceptual depth.
Dealing with the Series Beast (Question 6)
The final boss of any BC exam is the Taylor Series. In 2019, they gave us a function $f$ with a Maclaurin series that looked like this:
$$f(x) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1} (x-1)^n}{n \cdot 5^n}$$
Wait, that's actually the series for Question 6, but specifically focusing on the ratio test for the radius of convergence. You had to find the interval of convergence. This involves the limit as $n$ goes to infinity of the absolute value of the $(n+1)$ term over the $n$ term.
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Most people got the radius, which was 5. The real pain? Checking the endpoints. You had to plug $x = -4$ and $x = 6$ back into the original series. One becomes an alternating harmonic series (converges), and the other becomes a standard harmonic series (diverges). If you used a parenthesis where you should have used a bracket in your final interval, you just threw away a point.
What We Learned from the 2019 Scoring Guidelines
The Chief Reader’s report for 2019 was eye-opening. They noted that students are getting much better at the "calculator" parts but are failing at the "justification" parts.
For instance, on Question 4 (the cylindrical barrel problem), students struggled with related rates when the variable wasn't just $x$ or $y$. The barrel had a height $h$ and a radius $r$, but the radius was constant! Many students tried to use the product rule on $V = \pi r^2 h$ when they could have treated $\pi r^2$ as a simple constant. It's a classic case of overthinking the math and underthinking the geometry.
Practical Steps for Mastering Similar FRQs
If you’re practicing with the 2019 AP Calculus BC FRQ today, don't just do the problems. Grade yourself like a jerk. Use the official College Board rubric.
- Check your units. If the problem mentions "feet per hour," your answer better have units or you're losing points on part (d).
- The "Second Derivative Test" is your friend. When justifying a local max or min, don't just say "the graph goes up then down." Use the sign of $f''(x)$.
- Write out your setups. Even if you use a calculator to get the number, the integral setup itself is usually worth 1-2 points. If you write the answer only and it's wrong, you get a zero. If the setup is right, you're still in the game.
- Memorize your Taylor Series. You should know $e^x$, $\sin(x)$, $\cos(x)$, and $\frac{1}{1-x}$ like your own phone number. The 2019 exam showed that they love to manipulate these common series rather than making you build one from scratch.
The 2019 exam wasn't the hardest ever—that's probably still 2016 or some of the early 2000s papers—but it was a "precision" exam. It rewarded students who were tidy. It punished the "shaky" thinkers who knew the calculus but forgot the algebra.
Your Next Move
Grab a timer. Set it for 15 minutes. Try Question 4 from the 2019 set without looking at the solutions. It’s a separation of variables problem involving a differential equation. If you can get through the natural log integration and solve for the constant $C$ without making a sign error, you’re in the top 15% of testers. Once you've finished, cross-reference your work specifically with the "Scoring Guidelines" PDF available on the College Board website to see exactly where you would have lost points for notation.