AP Calculus AB and BC

30 AP Calculus Bell Ringers by Unit

A bell ringer earns its five minutes by forcing a prediction or a named method before any computation, producing one number or sentence checkable against the board in two minutes. The thirty below are grouped by unit, from limits through BC series, each stating what a correct answer must contain.

Commit to an answer before you compute

The weakest calculus opener is a recall question with one right word. The strongest one makes a student commit to a prediction, a method, or a number before working anything out, because the resolution is what fixes a misconception in place. Every prompt below can be answered in five minutes with no calculator and no printing.

Each entry lists the prompt and, separately, what a correct answer has to include beyond the final number. On a topic like the chain rule, two students can both write down the right derivative while only one of them actually named the outer and inner functions first, and that naming step is the part the exam rewards on problems that look unfamiliar.

Limits (Unit 1)

Limits open the course, and the habit worth building on day one is checking direct substitution first, then reaching for a specific tool only when it fails.

Prompt	What a correct answer must include
Evaluate $\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$ without a calculator.	Factoring the numerator to $(x-3)(x+3)$ , canceling the shared factor, then substituting $x=3$ to reach $6$ , not just the final number on its own.
Let $f(x) = \frac{x-2}{\|x-2\|}$ . Does $\lim_{x \to 2} f(x)$ exist?	Checking the left-hand and right-hand limits separately ( $-1$ from the left, $1$ from the right) and concluding the two-sided limit does not exist because they disagree.
A table shows $f(x)$ at $x = 0.9, 0.99, 0.999$ and $x = 1.1, 1.01, 1.001$ , all closing in on $5$ . State $\lim_{x \to 1} f(x)$ and say why the table alone is not a proof.	Reporting the limit as $5$ , plus a sentence that a table only suggests a value and cannot rule out unusual behavior between the sampled points.
Open the secant to tangent interactive, drag the second point toward the first, and predict the number the secant slope is heading toward before you let go.	Naming that number as the derivative at the fixed point, written as $\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$ , not a numeric guess with no reasoning attached.

Derivatives basics (Unit 2)

These openers drill the rules that everything later in the course assumes are automatic: power, product, and the derivatives of sine and cosine.

Prompt	What a correct answer must include
Differentiate $f(x) = 5x^4 - 3x^2 + 7$ .	$f'(x) = 20x^3 - 6x$ , with the constant term correctly dropping to $0$ .
Which rule fires first on $f(x) = e^x \ln x$ , and what is $f'(x)$ ?	Naming the product rule (two factors multiplied, not one composed inside the other), then $f'(x) = e^x \ln x + \frac{e^x}{x}$ .
State $\frac{d}{dx}[\sin x]$ and $\frac{d}{dx}[\cos x]$ from memory, then differentiate $g(x) = 3\sin x - \cos x$ .	Recalling the sign on cosine's derivative correctly ( $-\sin x$ ), then reaching $g'(x) = 3\cos x + \sin x$ .
Open the tangent line tracer, slide the point across a curve with one dip, and predict where the tangent line goes flat before you check.	Pointing to the bottom of the curve and connecting a horizontal tangent to $f'(x) = 0$ , not just eyeballing the flattest-looking spot.

Chain rule and implicit differentiation (Unit 3)

A composition is one function fed into another, and the whole rule depends on spotting that structure before differentiating anything.

\frac{d}{dx}\big[g(h(x))\big] = g'(h(x)) \cdot h'(x)

Prompt	What a correct answer must include
Differentiate $f(x) = \cos(x^3)$ .	Naming $x^3$ as the inner function, keeping it untouched under cosine, then multiplying by its derivative to reach $f'(x) = -3x^2\sin(x^3)$ .
Differentiate $f(x) = \sin^2(4x)$ , and say out loud which function is on the outside.	Naming squaring, not sine, as the outer function, applying the chain rule twice, and reaching $f'(x) = 8\sin(4x)\cos(4x)$ .
Find $\frac{dy}{dx}$ at the point $(3,4)$ on the circle $x^2+y^2=25$ .	Differentiating both sides with respect to $x$ to get $2x+2y\frac{dy}{dx}=0$ before solving for the derivative, then reporting $\frac{dy}{dx} = -\frac{3}{4}$ .
Differentiate $y = \arcsin(2x)$ .	Matching $2x$ to $u$ in the inverse sine rule and multiplying by $u' = 2$ to reach $y' = \frac{2}{\sqrt{1-4x^2}}$ .

Applications of derivatives (Units 4 and 5)

This is where the derivative stops being an exercise and starts describing something: a rate, a maximum, a moment a particle stops moving.

Prompt	What a correct answer must include
Open the related rates scene, a ladder sliding down a wall, and predict whether the top slides down faster when the ladder is nearly flat on the ground or nearly upright.	Setting up $x^2+y^2=L^2$ , differentiating to $\frac{dy}{dt} = -\frac{x}{y}\frac{dx}{dt}$ , and recognizing the rate grows without bound as $y$ shrinks toward $0$ , which is the nearly flat case.
A rectangle has a perimeter of $40$ . What dimensions maximize its area?	Writing area as one variable, $A(x) = x(20-x)$ , setting $A'(x) = 20-2x = 0$ , and reporting the square, $10$ by $10$ , not just the maximum area.
$f(x) = x^2$ on $[1,4]$ . Find the value the Mean Value Theorem guarantees.	Noting that $f$ is continuous and differentiable on the interval, computing the average rate $\frac{f(4)-f(1)}{4-1} = 5$ , and solving $f'(c) = 2c = 5$ for $c = 2.5$ .
A particle's position is $s(t) = t^3 - 6t^2 + 9t$ . Find every time it is momentarily at rest.	Differentiating to velocity $v(t) = 3t^2-12t+9$ , factoring to $3(t-1)(t-3)$ , and giving both $t=1$ and $t=3$ .
$f(x) = x^3 - 3x$ . State where $f$ is increasing and where it is decreasing.	Finding $f'(x) = 3x^2-3$ , the critical numbers $x=\pm1$ , and a sign check on $f'$ across all three resulting intervals, not just the critical numbers themselves.

Integration (Unit 6)

Integration openers work best when they isolate one habit at a time: reading a sum's direction, applying the Fundamental Theorem directly, or checking a substitution before running it.

\frac{d}{dx}\int_{a}^{x} f(t)\,dt = f(x)

Prompt	What a correct answer must include
Open the Riemann sum slider on an increasing function and predict whether a left sum overestimates or underestimates the true area before dragging the subintervals up.	Stating that the left sum underestimates an increasing function, because each rectangle uses the smaller left-edge height, tied to the function's own shape rather than a memorized rule with no reason attached.
Evaluate $\frac{d}{dx}\int_{2}^{x}(t^2+1)\,dt$ .	Recognizing the Fundamental Theorem of Calculus applies directly, so the answer is $x^2+1$ with no integration performed.
Find $\int (3x^2 - 4x + 1)\,dx$ .	$x^3 - 2x^2 + x + C$ , with the constant of integration present.
For $\int 2x\cos(x^2)\,dx$ , name the substitution $u$ and confirm $du$ matches what is left over, without integrating yet.	$u=x^2$ and $du=2x\,dx$ , plus the observation that $2x\,dx$ is exactly what remains in the integral once $u$ is chosen.

Differential equations (Unit 7)

A slope field opener checks whether a student is reading the drawn slopes or just sketching a curve that looks plausible.

Prompt	What a correct answer must include
A slope field for $\frac{dy}{dx}=x-y$ is on the board. Trace the solution curve through $(0,2)$ .	Following the direction each tick mark shows at every point the curve passes through, starting from a slope of $-2$ at $(0,2)$ , rather than a smooth curve that ignores the field.
Separate the variables in $\frac{dy}{dx}=xy$ .	Moving every $y$ term to one side and every $x$ term to the other before integrating: $\frac{dy}{y}=x\,dx$ .
A population satisfies $\frac{dP}{dt}=0.05P$ with $P(0)=200$ . Write $P(t)$ .	Naming this as exponential growth and writing $P(t)=200e^{0.05t}$ , citing the general solution to $\frac{dP}{dt}=kP$ .

Applications of integrals (Unit 8)

Setup is the part worth timing. A student who can set up the right integral in ninety seconds rarely misses the evaluation.

Prompt	What a correct answer must include
Set up, without evaluating, the integral for the area between $y=x^2$ and $y=x+2$ .	Finding the intersection points $x=-1$ and $x=2$ first, then writing $\int_{-1}^{2}\left[(x+2)-x^2\right]dx$ with the line correctly on top.
Open the solid of revolution builder, rotate $y=\sqrt{x}$ from $x=0$ to $x=4$ about the x-axis, and predict disk or washer before checking the picture.	Choosing the disk method because the region already touches the axis of rotation, and writing $V=\pi\int_{0}^{4}x\,dx$ .
Find the average value of $f(x)=x^2$ on $[0,3]$ .	Including the $\frac{1}{b-a}$ factor in front of the definite integral, giving $\frac{1}{3}\int_{0}^{3}x^2\,dx = 3$ .

BC series and parametric (Units 9 and 10)

BC sections get two extra units, and both reward the same habit: write the formula in its general form before plugging anything in.

Prompt	What a correct answer must include
$x(t)=t^2$ , $y(t)=t^3$ . Find $\frac{dy}{dx}$ .	Building the ratio $\frac{dy/dt}{dx/dt}=\frac{3t^2}{2t}$ rather than differentiating $y$ with respect to $x$ directly, then simplifying to $\frac{3t}{2}$ .
Does $\sum_{n=1}^{\infty}\frac{1}{n^2}$ converge or diverge? Name the test.	Naming the p-series test, identifying $p=2$ , and concluding it converges because $p>1$ .
Write the first three nonzero terms of the Maclaurin series for $e^x$ .	Starting from the general term $\frac{x^n}{n!}$ and listing $1+x+\frac{x^2}{2}$ , in that order.

\frac{dy}{dx} = \frac{dy/dt}{dx/dt}

How to run these

The routine does more work than any single prompt does.

Same slot, same length, every class. Put the prompt on the board before the bell so students start without being told.
Do not grade them. The moment a bell ringer counts for points, students stop giving an honest attempt, and an honest wrong answer is what shows you a misconception exists.
Always resolve it before moving on. Two minutes of showing the correct steps is enough; an opener that is never checked teaches students the first five minutes do not matter.
Recycle across units. A limit opener from Unit 1 makes a strong review prompt once integration starts, since the exam mixes units freely across both sections.

For a substitute day

The five openers that link to an interactive need the page to load first, so on a day you will not be in the room, swap in a table, table-values, or vocabulary prompt from the same unit instead.

Worked examples

Worked example

The ladder problem, worked in full

A 10-foot ladder leans against a wall. The base slides away from the wall at 2 feet per second. How fast is the top sliding down when the base is 6 feet from the wall?

Name the relation. If $x$ is the base's distance from the wall and $y$ is the top's height, the ladder's length is fixed: $x^2+y^2=10^2=100$ .
Differentiate both sides with respect to time: $2x\frac{dx}{dt} + 2y\frac{dy}{dt} = 0$ .
Find the missing side. When $x=6$ , $y=\sqrt{100-36}=\sqrt{64}=8$ .
Substitute the known values, $x=6$ , $y=8$ , and $\frac{dx}{dt}=2$ , then solve for $\frac{dy}{dt}$ : $2(6)(2) + 2(8)\frac{dy}{dt} = 0$ , so $\frac{dy}{dt} = -\frac{24}{16} = -1.5$ .

The top slides down at 1.5 feet per second.

Worked example

A parametric derivative at a point

A curve is given by $x(t) = t^2$ and $y(t) = t^3$ . Find $\frac{dy}{dx}$ at $t = 2$ .

Differentiate each piece with respect to $t$ separately: $\frac{dx}{dt} = 2t$ and $\frac{dy}{dt} = 3t^2$ .
Build the ratio, since $y$ is not written directly as a function of $x$ : $\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{3t^2}{2t} = \frac{3t}{2}$ for $t \neq 0$ .
Substitute $t=2$ : $\frac{dy}{dx} = \frac{3(2)}{2} = 3$ .

$\frac{dy}{dx} = 3$ at $t=2$ .

Frequently asked questions

How is a calculus bell ringer different from a review problem?

A review problem usually asks a student to execute a method that has already been named. A bell ringer asks the student to decide which method applies, or to predict an outcome, before any computation starts. That decision step is what the exam actually tests on an unfamiliar problem, so practicing it separately from the mechanics is worth five minutes a day.

Should bell ringers count toward a grade?

No. Grading a bell ringer pushes students to protect their score instead of giving an honest attempt, and the honest wrong answer is the one that tells you where a misconception lives. Keep them ungraded and always resolve the correct approach before moving on to the lesson.

How should these change for a BC section?

AB and BC share the first seven groups, covering Units 1 through 8. BC students then add the eighth group, series and parametric, since those two units only appear on the BC exam.