AP Calculus AB and BC

30 AP Calculus Exit Tickets by Unit

Q: How is an exit ticket different from a bell ringer?

A bell ringer opens class and asks a student to predict or name a method before computing anything. An exit ticket closes class and asks for proof the method from that day's lesson actually landed, checked in the last two minutes while the steps are still fresh. Running both gives a read on the same misconception at the start and the end of one period.

Q: Should exit tickets be graded?

No. Grading pushes a student to protect a score instead of giving an honest attempt, and the honest wrong answer is the one that tells you where a misconception lives. Sort the stack into got it, close, and missing it, and use the missing pile to plan tomorrow's opener instead of entering a score.

Q: Do AB and BC sections use different exit tickets?

AB and BC share the first seven groups above, covering units one through eight. BC students add the eighth group, parametric and series, since those two units only appear on the BC exam.

An exit ticket earns its two minutes only if it produces one line you can sort into got it, close, or missed it without rereading, since the goal is choosing what to reteach tomorrow, not grading tonight. The thirty below are grouped by unit, limits through BC series, each stating that one required line.

Sort the stack, do not grade it

A bell ringer opens class by making a student commit to something before computing. An exit ticket closes class the other way: it asks for the one line that proves the day's method landed, written in the last two minutes while the work is still fresh. The two live at opposite ends of the same period, and a class that runs both gets a read on a misconception both before and after the lesson meets it.

Each entry below lists the prompt and, separately, the one line a correct answer has to contain. That line is rarely the final number. On a related rates problem, two students can both write down the same rate while only one of them actually differentiated the constraint equation with respect to time before substituting, and that step is the part worth checking in two minutes flat.

Limits (Unit 1)

A limit exit ticket should confirm a student reaches for algebra before a calculator, and can tell a hole apart from an asymptote.

Prompt	The one line a correct answer must include
Evaluate $\lim_{x \to 4} \frac{\sqrt{x}-2}{x-4}$ .	Multiplying by the conjugate $\sqrt{x}+2$ over itself before canceling, reaching $\frac{1}{4}$ .
A piecewise function is $f(x)=x+1$ for $x<2$ and $f(x)=3x-3$ for $x \ge 2$ . Is $f$ continuous at $x=2$ ?	Comparing the left-hand limit ( $3$ ) to $f(2)$ ( $3$ ) and confirming the two match before declaring continuity.
Find $\lim_{x \to \infty} \frac{3x^2-5}{6x^2+x}$ .	Dividing every term by $x^2$ , the highest power present, to reach $\frac{1}{2}$ rather than $0$ or infinity.
Does $f(x)=\frac{1}{x-1}$ have a removable or non-removable discontinuity at $x=1$ ?	Naming it non-removable because the one-sided limits run to opposite infinities instead of approaching one shared finite value.

Derivatives basics (Unit 2)

These tickets check the rules a student needs on reflex: rewriting a radical before the power rule, ordering the quotient rule's numerator, and recalling tangent's derivative correctly.

Prompt	The one line a correct answer must include
Differentiate $f(x)=4x^3-2\sqrt{x}$ .	Rewriting $\sqrt{x}$ as $x^{1/2}$ before applying the power rule, reaching $f'(x)=12x^2-\frac{1}{\sqrt{x}}$ .
Differentiate $f(x)=\frac{x^2}{x+1}$ with the quotient rule.	Keeping the numerator order as (derivative of top)(bottom) minus (top)(derivative of bottom), reaching $f'(x)=\frac{x^2+2x}{(x+1)^2}$ .
State $\frac{d}{dx}[\tan x]$ from memory, then differentiate $g(x)=5\tan x+2x$ .	Recalling $\frac{d}{dx}[\tan x]=\sec^2 x$ , not secant's own derivative, then reaching $g'(x)=5\sec^2 x+2$ .
Differentiate $f(x)=x^2e^x$ with the product rule.	Adding (derivative of first)(second) to (first)(derivative of second), then factoring to $f'(x)=e^x(x^2+2x)$ .

Chain rule, implicit, and inverse functions (Unit 3)

A student differentiating $y^n$ inside an implicit equation is running the chain rule on $y$ itself, which is the habit these four tickets isolate.

\frac{d}{dx}\big[y^n\big] = n\,y^{n-1}\,\frac{dy}{dx}

Prompt	The one line a correct answer must include
Differentiate $f(x)=(3x-1)^5$ .	Naming $3x-1$ as the inner function and multiplying by its derivative, $3$ , reaching $f'(x)=15(3x-1)^4$ .
Differentiate $f(x)=e^{\sin x}$ .	Leaving $e^{\sin x}$ untouched and multiplying by the derivative of the exponent, $\cos x$ .
Find $\frac{dy}{dx}$ for $x^3+y^3=6xy$ at the point $(3,3)$ .	Attaching $\frac{dy}{dx}$ to every differentiated $y$ term before isolating it, then reporting $\frac{dy}{dx}=-1$ at that point.
Differentiate $y=\arctan(3x)$ .	Matching $3x$ to $u$ in $\frac{d}{dx}[\arctan u]=\frac{u'}{1+u^2}$ , reaching $y'=\frac{3}{1+9x^2}$ .

Applications of derivatives (Units 4 and 5)

These five close a lesson on what the derivative is doing to a real quantity: a rate, an extremum, a change in concavity.

Prompt	The one line a correct answer must include
A spherical balloon's volume grows at $12\text{ in}^3/\text{s}$ . How fast is the radius changing when $r=3$ in?	Differentiating $V=\frac{4}{3}\pi r^3$ with respect to time to $\frac{dV}{dt}=4\pi r^2\frac{dr}{dt}$ before substituting, reaching $\frac{dr}{dt}=\frac{1}{3\pi}$ in/s.
Two numbers sum to $20$ . Minimize the sum of their squares.	Writing the objective in one variable before differentiating, then confirming $x=10$ is a minimum by a sign check or the second derivative, not stopping at the critical number.
$f(x)=2x^3-3x^2-12x+5$ on $[-2,3]$ . Find the absolute maximum value.	Evaluating $f$ at both critical numbers and both endpoints, the full Candidates Test, to reach a maximum value of $12$ at $x=-1$ .
A particle has $v(t)=t^2-4t+3$ and $a(t)=2t-4$ . Is it speeding up or slowing down at $t=0.5$ ?	Comparing the signs of $v(0.5)=1.25$ and $a(0.5)=-3$ ; opposite signs mean the particle is slowing down.
$f(x)=x^4-4x^3$ . Find every inflection point.	Setting $f''(x)=12x^2-24x=0$ to get $x=0,2$ , then confirming a sign change in $f''$ at each before naming both as inflection points.

Integration (Unit 6)

An integration exit ticket should show whether a student evaluates a definite integral correctly, tracks a substitution back to the original variable, and reads a Riemann sum's direction of error.

\int_{a}^{b} f'(x)\,dx = f(b) - f(a)

Prompt	The one line a correct answer must include
Evaluate $\int_0^2 (3x^2+1)\,dx$ .	Evaluating the antiderivative $x^3+x$ at both bounds and subtracting, $F(2)-F(0)=10$ , with no $+C$ left in a definite integral.
Evaluate $\int x\sqrt{x^2+1}\,dx$ using substitution.	Naming $u=x^2+1$ , adjusting for the missing factor of $\frac{1}{2}$ , and returning to $x$ at the end: $\frac{1}{3}(x^2+1)^{3/2}+C$ .
A left Riemann sum with equal subintervals estimates $\int_1^5 f(x)\,dx$ for a decreasing $f$ . Overestimate or underestimate?	Naming the left endpoint as the greatest value of a decreasing function on each subinterval, so the sum overestimates the true area.
State the identity for $\int_a^b f'(x)\,dx$ in words, then evaluate it for $f(x)=x^2$ on $[1,3]$ .	Naming it as undoing the derivative rather than restating a derivative rule, then computing $f(3)-f(1)=8$ .

Differential equations (Unit 7)

A differential equations ticket should confirm a student can check a proposed solution by substitution, separate variables cleanly, and step forward with Euler's method without skipping a step.

y_{n+1} = y_n + h \cdot f(x_n, y_n)

Prompt	The one line a correct answer must include
Verify that $y=3e^{2x}$ solves $\frac{dy}{dx}=2y$ .	Differentiating the proposed solution to $y'=6e^{2x}$ and substituting into the right side, $2y=6e^{2x}$ , to confirm both sides match.
Solve $\frac{dy}{dx}=\frac{x}{y}$ with $y(0)=3$ .	Separating to $y\,dy=x\,dx$ , integrating to $\frac{y^2}{2}=\frac{x^2}{2}+C$ , and using the initial condition to fix $C=\frac{9}{2}$ before reaching $y=\sqrt{x^2+9}$ .
Use Euler's method with step size $0.5$ to estimate $y(1)$ for $\frac{dy}{dx}=x+y$ , $y(0)=1$ .	Applying $y_{n+1}=y_n+h\cdot f(x_n,y_n)$ twice in sequence, reaching an estimate of $y(1)\approx 2.5$ .

Applications of integrals (Unit 8)

Setup is what an exit ticket should time here, since a student who sets up the right integral almost never botches the arithmetic that follows.

Prompt	The one line a correct answer must include
Set up, without evaluating, the area between $y=\sqrt{x}$ and $y=\frac{x}{2}$ from their left intersection to their right one.	Finding the intersections $x=0$ and $x=4$ first, then testing a point to confirm $\sqrt{x}$ is on top: $\int_0^4\left(\sqrt{x}-\frac{x}{2}\right)dx$ .
A solid's cross sections perpendicular to the $x$ -axis are squares with side $s(x)=x$ , from $x=0$ to $x=2$ . Set up the volume integral.	Squaring the side length before integrating, since a square cross section's area is $s(x)^2$ : $V=\int_0^2 x^2\,dx$ .
Find the average value of $f(x)=\cos x$ on $[0,\pi/2]$ .	Dividing by the interval length $\frac{\pi}{2}$ outside the integral, reaching an average value of $\frac{2}{\pi}$ .

BC parametric and series (Units 9 and 10)

BC sections pick up two extra units, and both reward the same habit as everything above: build the general formula first, then substitute.

Prompt	The one line a correct answer must include
$x(t)=\cos t$ , $y(t)=\sin t$ . Find $\frac{dy}{dx}$ at $t=\pi/4$ .	Building the ratio $\frac{dy/dt}{dx/dt}$ instead of differentiating $y$ directly with respect to $x$ , reaching $-1$ at $t=\pi/4$ .
Does $\sum_{n=1}^{\infty}\frac{n}{n+1}$ converge or diverge? Name the test.	Checking that the terms approach $1$ , not $0$ , and naming the nth-term test as the reason the series diverges.
Write the Maclaurin series for $\cos x$ through the $x^4$ term.	Using only even powers with alternating signs and factorial denominators, reaching $1-\frac{x^2}{2}+\frac{x^4}{24}$ .

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

Reading a stack in five minutes

Reading thirty tickets cold takes twenty minutes and tells you nothing you can use tomorrow morning. Read for the one required line only, and the same stack takes five.

Make three piles as you go, not after: got it, close, missing it. One pass, no second reading, no marks on the paper yet.
Look only for the required line, not the full derivation. A ticket that reaches the right line with a small arithmetic slip after it still counts as got it.
Watch the missing pile's size, not any one paper in it. A single miss is a student; a third of the stack missing the same line is tomorrow's opener.
Skip the partial pile's detail work. It exists to separate got it from missing it, not to assign partial credit, since these stay ungraded.

When the missing pile is large

If more than a third of the stack lands in missing it, reteach the one required line at the start of the next class before assigning new practice. A pile that size is the lesson telling you it did not land, not a batch of careless students.

Worked examples

Worked example

The folium exit ticket, worked in full

Find $\frac{dy}{dx}$ for $x^3+y^3=6xy$ at the point $(3,3)$ .

Confirm the point lies on the curve: $3^3+3^3=27+27=54$ , and $6(3)(3)=54$ , so $(3,3)$ works.
Differentiate both sides with respect to $x$ , attaching $\frac{dy}{dx}$ to every $y$ term through the chain rule: $3x^2+3y^2\frac{dy}{dx}=6y+6x\frac{dy}{dx}$ .
Collect every $\frac{dy}{dx}$ term on one side: $3y^2\frac{dy}{dx}-6x\frac{dy}{dx}=6y-3x^2$ , then factor and divide by $3$ : $\frac{dy}{dx}=\frac{2y-x^2}{y^2-2x}$ .
Substitute $x=3$ , $y=3$ : $\frac{dy}{dx}=\frac{2(3)-3^2}{3^2-2(3)}=\frac{6-9}{9-6}=\frac{-3}{3}=-1$ .

$\frac{dy}{dx}=-1$ at $(3,3)$ .

Worked example

Two steps of Euler's method, worked in full

Use Euler's method with step size $0.5$ to estimate $y(1)$ if $\frac{dy}{dx}=x+y$ and $y(0)=1$ .

Start at $x_0=0$ , $y_0=1$ . The slope there is $f(x_0,y_0)=0+1=1$ .
Step forward: $y_1=y_0+h\cdot f(x_0,y_0)=1+0.5(1)=1.5$ at $x_1=0.5$ .
Find the slope at the new point: $f(x_1,y_1)=0.5+1.5=2$ .
Step forward again: $y_2=y_1+h\cdot f(x_1,y_1)=1.5+0.5(2)=2.5$ at $x_2=1.0$ .

$y(1)\approx 2.5$ .

Frequently asked questions

How is an exit ticket different from a bell ringer?

A bell ringer opens class and asks a student to predict or name a method before computing anything. An exit ticket closes class and asks for proof the method from that day's lesson actually landed, checked in the last two minutes while the steps are still fresh. Running both gives a read on the same misconception at the start and the end of one period.

Should exit tickets be graded?

No. Grading pushes a student to protect a score instead of giving an honest attempt, and the honest wrong answer is the one that tells you where a misconception lives. Sort the stack into got it, close, and missing it, and use the missing pile to plan tomorrow's opener instead of entering a score.

Do AB and BC sections use different exit tickets?

AB and BC share the first seven groups above, covering units one through eight. BC students add the eighth group, parametric and series, since those two units only appear on the BC exam.