AP Calculus AB and BC

AP Calculus Project Ideas for High School

Below are twelve AP Calculus projects that require a real measured number, not a poster. Each lists the deliverable, the duration, the AP Calculus unit it hits, a one-line rubric criterion, and the calculus involved, followed by a section on grading them without drowning in grading time.

Skip the real-world-applications poster

The default calculus project in most classrooms is a poster: find three real-world uses of derivatives, add a picture from a search engine for each one, write a caption. It teaches students something they already believed, that calculus describes real things, and it grades layout instead of a derivative. Every image can be found without opening a calculus book, so it cannot even confirm the student did the algebra.

The twelve projects below replace the picture with a measurement the student takes themselves: a stopwatch reading, a logged speed, a weighed solid, a temperature that actually cools. Each one names a deliverable, a duration, the AP Calculus unit it belongs to, one rubric line, and the calculus that has to appear in the writeup. A section on grading them fairly follows the list.

1. The piecewise pricing audit

Deliverable	Duration	Unit	Rubric criterion
A hand-drawn graph of a real piecewise pricing or tax system with every breakpoint labeled and classified	Three days	1: Limits and Continuity	Each classification is justified with the one-sided limits at that point, not with a description of how the graph looks

Students find a real piecewise system: a tax bracket schedule, a shipping cost tier, a data overage fee, a parking rate that jumps at each hour. They graph it and classify every break as a removable discontinuity, a jump discontinuity, an infinite discontinuity, or not a break at all. Tax brackets are the most common wrong guess, because most students expect a jump at each bracket edge and are surprised to find the schedules are built to stay continuous.

\lim_{x \to c^{-}} f(x) = \lim_{x \to c^{+}} f(x) = f(c)

That equation is the whole rubric in symbols: a point is continuous only when the left limit, the right limit, and the function value all agree. Students who cannot state which of the three fails at a given breakpoint have not actually classified it, no matter how confident the label on their graph looks.

2. The motion-capture rate of change

Deliverable	Duration	Unit	Rubric criterion
A short video of real motion marked at five timestamps for the average rate, plus three additional closely spaced timestamps near the instant being estimated, and a graph of average and instantaneous rate of change	One week	2: Differentiation: Definition and Fundamental Properties	The instantaneous rate comes from a difference quotient with a shrinking interval, not from reading a slope by eye

Students film something with measurable position over time: a dropped ball against a tape-measure backdrop, a bike ride with a phone's distance readout, a sprint with cones at known distances. From five timestamped positions across the filmed interval they compute an average rate of change; they then mark three more timestamps, spaced increasingly close together around one specific moment, and use them to estimate an instantaneous rate of change by shrinking the interval around that moment.

\text{average rate} = \frac{f(b) - f(a)}{b - a} \qquad \text{instantaneous rate} = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}

Real position data never lets $h$ actually reach zero, which is the point. Students report the shrinking sequence of difference quotients and explain, in their own words, why the number they converge toward is the instantaneous rate even though it was never directly measured.

3. The related rates field test

Deliverable	Duration	Unit	Rubric criterion
A measured related-rates scenario with a stopwatch and tape measure, and the calculus prediction checked against the measured rate	One week	4: Contextual Applications of Differentiation	The equation relating the two quantities is justified geometrically before it is differentiated, not assumed

A sliding ladder, a draining container, or a lengthening shadow all work, and the related rates interactive has a hands-on version that is a good rehearsal before the physical version. For the ladder, students measure the base sliding away from a wall at a timed rate and predict how fast the top is sliding down using the fixed-length constraint, then check the prediction with a second stopwatch reading.

x^2 + y^2 = L^2 \quad \Rightarrow \quad 2x\frac{dx}{dt} + 2y\frac{dy}{dt} = 0

The grading question is not whether the two numbers match exactly. Real measurement error is expected. The question is whether the student can point to the constant, $L$ , and explain why it has to stay fixed for the differentiated equation to mean anything.

4. The minimum-material container

Deliverable	Duration	Unit	Rubric criterion
A physical prototype built to the dimensions calculus predicts minimize material for a fixed volume, kept alongside one rejected alternative	Two weeks	5: Analytical Applications of Differentiation	The critical point comes from setting a derivative equal to zero, not from guessing dimensions that look efficient

Students pick a container shape (a cylinder, a box with a square base) and a fixed volume, write the objective function for the material used, substitute using the volume constraint, and solve for the dimension that minimizes it. They then build the predicted shape out of cardboard or paper alongside one deliberately worse shape at the same volume, and weigh or measure the material each one used.

V = \pi r^2 h \;\; \text{fixed} \qquad S(r) = 2\pi r^2 + \frac{2V}{r} \qquad S'(r) = 0 \;\Rightarrow\; r = \left(\frac{V}{2\pi}\right)^{1/3}

A prototype that used less material than the calculus prediction is not a failed project. It usually means the fixed volume was not actually held constant during construction, which is its own useful finding about the gap between an equation and a real object.

5. The Riemann sum odometer check

Deliverable	Duration	Unit	Rubric criterion
A logged speed-versus-time dataset from a ride, with left, right, and trapezoidal sum estimates of distance checked against the real distance traveled	One week	6: Integration and Accumulation of Change	The student predicts overestimate or underestimate for each method before computing, then checks the prediction

A phone's GPS speed readout, logged at regular intervals during a bike ride or a car trip as a passenger, gives a real speed-versus-time table. Students compute a left Riemann sum, a right Riemann sum, and a trapezoidal estimate of total distance, then compare each to the trip's actual measured distance. The Riemann sum slider is a fast way to rehearse the three formulas before the real data arrives.

L_n = \Delta t \sum_{i=0}^{n-1} v(t_i) \qquad R_n = \Delta t \sum_{i=1}^{n} v(t_i) \qquad T_n = \frac{\Delta t}{2}\left[v(t_0) + 2v(t_1) + \cdots + 2v(t_{n-1}) + v(t_n)\right]

Because real speed data is rarely purely increasing or decreasing, the overestimate-versus-underestimate rule from a textbook graph will not hold cleanly here, and explaining where it breaks down is worth more credit than a clean answer that ignores the mess in the data.

6. The cooling curve differential equation

Deliverable	Duration	Unit	Rubric criterion
Measured temperature-versus-time data for a cooling liquid, a fitted differential equation, a hand-drawn slope field, and the solved curve checked against a held-out data point	One week	7: Differential Equations	The constant in the equation is found from the data, and the solution is checked against a point that was not used to find it

A cup of hot water and a kitchen thermometer, read every few minutes, produce a real cooling curve. Students fit Newton's law of cooling, sketch the slope field by hand from the differential equation, and separate variables to find the closed-form solution.

\frac{dT}{dt} = k(T - T_a) \qquad \Longrightarrow \qquad T(t) = T_a + (T_0 - T_a)e^{kt}

The held-out point is what separates this from a curve-fitting exercise. Students find $k$ using only the first half of their readings, then use the resulting function to predict the temperature at a later timestamp they already measured but set aside, and report how close the prediction landed.

7. The Euler's method forecast

Deliverable	Duration	Unit	Rubric criterion
An Euler's method forecast table from a measured starting value, checked against real follow-up measurements of the same system	One week	7: Differential Equations (BC)	The direction of the drift is explained with the concavity of the true solution, not just reported as an error

This one reuses the cooling data from the previous project or starts fresh with any measurable rate. Students take one early measured value and a differential equation, then walk Euler's method forward by hand or in a spreadsheet with a chosen step size, before checking each forecast point against a real measurement taken later. The Euler's method versus exact solution comparison is the reference for what the drift should look like.

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

The forecast will drift in a predictable direction depending on the concavity of the true solution, underestimating a concave-up curve and overestimating a concave-down one. A student who reports the size of the drift without naming its direction, or names the direction without connecting it to concavity, has done the arithmetic without the calculus.

8. The solid of revolution build

Deliverable	Duration	Unit	Rubric criterion
A physical or digital solid formed by rotating a chosen curve around an axis, with the exact volume computed and checked against a measured or weighed version	Two weeks	8: Applications of Integration	The choice between disk, washer, and shell is justified by whether the region touches the axis, not by which formula the student remembers first

Students choose a bounded region, revolve it around an axis, and build the resulting solid, either physically from clay or layered cardstock, or with a rendered model. The solid of revolution builder is the place to test the shape before committing to a build. The disk-versus-washer choice depends entirely on whether the region leaves a gap at the axis.

V = \pi \int_a^b \left[R(x)\right]^2 dx \qquad \text{or} \qquad V = \pi \int_a^b \left(\left[R_{\text{out}}(x)\right]^2 - \left[R_{\text{in}}(x)\right]^2\right) dx

A physical build can be checked by displacement in water or by weighing it against a known-density material and back-solving for volume. Either check is rough, and the writeup should state the expected error from the measuring method rather than treating a mismatch as a mistake in the integral.

9. The bounded-region art piece

Deliverable	Duration	Unit	Rubric criterion
Original artwork whose colored regions are each bounded by named functions, with the exact area of every region computed	One week	8: Applications of Integration	Each region uses the correct top-minus-bottom or right-minus-left order, justified by which curve is on top over that interval

Students design an image, a logo, or a pattern where every colored region is the area between two named functions over a stated interval, then compute the exact area of each region. A printed or drawn version lets a second check happen: covering the shape with a grid and counting squares should land close to the integral's answer.

A = \int_a^b \bigl[f(x) - g(x)\bigr]\,dx, \qquad f(x) \ge g(x) \text{ on } [a, b]

The interesting failure mode here is a region where the two curves cross partway through the interval. Students who integrate straight across without splitting at the crossing point will get a number that looks plausible and is wrong, which makes this a good project for surfacing that specific mistake before the exam does.

10. The amusement ride parametrization

Deliverable	Duration	Unit	Rubric criterion
Parametric equations for a real circular motion, with the speed of a point computed at a specific time and checked against a measured rotation period	One week	9: Parametric Equations, Polar Coordinates, and Vectors (BC)	Speed comes from the parametric speed formula, not from total distance divided by the time for a full revolution

A Ferris wheel, a merry-go-round, or even a bike pedal turning at a steady rate gives real circular motion. Students write parametric equations for a point's position, time a full rotation with a stopwatch to find the angular rate, and compute the point's speed at one specific instant rather than its average speed over the whole revolution.

x(t) = r\cos(\omega t), \quad y(t) = r\sin(\omega t) \qquad \text{speed} = \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}

Because $r$ and $\omega$ are constant here, the speed formula simplifies to a constant, $r\omega$ , and a student who expected the speed to vary around the circle should say so directly rather than quietly dropping the observation from the writeup.

11. The polar curve mosaic

Deliverable	Duration	Unit	Rubric criterion
A scale design built from one or more polar curves, with the exact area of one loop computed and compared to a measured area of the built version	One week	9: Parametric Equations, Polar Coordinates, and Vectors (BC)	The bounds of integration are the actual angles where the curve traces that one loop, not zero to $2\pi$ by default

A rose curve or a cardioid makes a workable mosaic tile, garden bed layout, or logo. Students pick a polar curve, find the exact interval of $\theta$ that traces a single petal or loop, and compute that region's polar area before scaling the design to a real built size.

A = \frac{1}{2} \int_{\alpha}^{\beta} \left[r(\theta)\right]^2 d\theta

Most errors on this project come from the bounds, not the integral itself. A rose curve with an odd number of petals retraces itself over $[0, 2\pi]$ (it fully traces within an interval of length $\pi$ ), so integrating the full $2\pi$ interval without checking for overlap double-counts area for those curves; even-petal roses need the full $[0, 2\pi]$ and do not retrace, and that check belongs in the writeup as its own sentence.

12. The Taylor polynomial error hunt

Deliverable	Duration	Unit	Rubric criterion
Taylor polynomials of increasing degree for a function without a simple closed-form value, with the measured error at several points reported against the Lagrange bound's prediction	One week	10: Infinite Sequences and Series (BC)	The measured error is reported next to the bound's predicted maximum, with a sentence on why the bound is allowed to overestimate

A function like $\sin x$ or $e^x$ , computed to many digits by a calculator, gives students a real target to approximate. They build Taylor polynomials of degree 1, 3, and 5 centered at a chosen point, evaluate each at several test values, and compare the actual difference from the calculator's value to what the Lagrange error bound predicted as the worst case.

R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x - a)^{n+1} \quad \text{for some } c \text{ between } a \text{ and } x

The bound is a ceiling, not a forecast, since it uses the worst possible value of the derivative on the interval rather than the actual one. A student who finds their measured error well under the bound has not made a mistake; they have found direct evidence for why the bound is called a bound.

Grading projects fairly

Twelve different projects with twelve different rubrics is a grading schedule nobody keeps up with past the first unit. The fix is one rubric that fits all of them, because every project above asks for the same four things in a different setting.

Grade the setup, the calculus step, the answer, and the connection to the measured reality as four separate points. A student can lose the connection point while getting every derivative right, and that distinction is worth preserving in the grade.
Require one number that came from an actual measurement, not a value pulled from a textbook or generated by an online tool. It is the cheapest check against a project done entirely from memory, and it is also what makes a project a project instead of a worksheet.
Do not require a poster, a video with production values, or a formatted report. Presentation time scales with class size and grading time, and it tests none of the calculus above.
Let a mismatch between the calculus prediction and the real measurement earn full marks when the student explains where the mismatch came from. A ladder that slides slower than predicted because of friction is a correct related-rates model applied to an imperfect world, not a wrong answer.

The four-point rubric in one line

Setup, calculus step, answer, and connection to the real measurement, one point each, applied identically across all twelve projects.

Unit content these projects attach to lives in the unit pages, the discussion prompts that pair with them are in the AP Calculus discussion questions, and a five-minute opener that previews the relevant idea before a project's due date is in the bell ringers.

Worked examples

Worked example

Grading the minimum-material container end to end

A student builds a cylinder at $r = 4$ cm for a required volume of $500$ cubic cm, but their calculus predicted $r \approx 4.3$ cm.

Check the setup point first: did the student correctly write the constraint $V = \pi r^2 h$ and solve it for $h$ before substituting into the surface area formula? If so, award the setup point regardless of what the final number turns out to be.
Check the calculus step: did they take $S'(r)$ , set it equal to zero, and solve algebraically, rather than guessing a value of $r$ that looked reasonable? A correct derivative with an arithmetic slip in solving still earns this point if the method is visible.
Check the answer point against their own equation, not against $4.3$ directly. If their algebra genuinely produces $r \approx 4.3$ cm, that point is earned even if the number differs from a value another student or the teacher got with different rounding.
Check the connection point last: does the writeup explain why the built cylinder came out at $4$ cm instead of $4.3$ ? Cardstock thickness, measurement rounding, and a base that was cut before the final height was fixed are all valid explanations.

A student who nails three of the four points and gives a specific, checkable reason for the fourth has demonstrated more calculus understanding than one who built a cylinder at exactly $4.3$ cm without being able to say why that number came out of the equation. The rubric should reflect that.

Worked example

Reading a mismatch on the cooling curve project

A student's fitted equation predicts a temperature of $61$ degrees at the held-out timestamp, but the thermometer read $58$ degrees.

Confirm $k$ was fit using only the earlier readings, not the held-out point itself. If the held-out point leaked into the fit, the check is meaningless and the project needs to be redone with a real separation.
Ask the student whether $58$ is below or above what the model predicted, and whether the room's ambient temperature, $T_a$ , might have been measured incorrectly rather than the cooling constant $k$ .
Have the student re-run the solved equation with a slightly different $T_a$ , using the room's actual measured temperature if it was not recorded precisely the first time, and see whether the gap closes.
If the gap does not close, ask whether the liquid was disturbed, refilled, or moved during the measurement window, since Newton's law of cooling assumes no outside heat is added.

A three-degree gap between a predicted $61$ and a measured $58$ is normal for a kitchen-thermometer dataset, and the useful output of the project is identifying which assumption of the model, a constant ambient temperature, most likely explains it, not achieving an exact match.

Frequently asked questions

How many of these twelve should one class actually assign in a year?

Three or four spread across the year fits most pacing guides without crowding out review time before the exam. Pick one from limits or differentiation early in the year, one from integration or differential equations mid-year, and save any BC-only projects, like the parametric ride, the polar mosaic, or the Taylor polynomial hunt, for after that content has actually been taught.

Can an AB class use the BC-only projects with modifications?

Not the parametric, polar, or series projects as written, since AB does not cover that content and the exam will not test it. The related rates, optimization, Riemann sum, cooling curve, solid of revolution, and area projects are all AB-appropriate without any changes.

What if a student cannot get real measured data, for example during remote learning?

Publicly available real datasets substitute well: a city's recorded hourly temperature for the cooling-style projects, a transit agency's published speed or schedule data for the Riemann sum project. The requirement is real data the student did not generate themselves for the assignment, not necessarily data they personally measured with a device.

How much should a project count toward the grade compared to a unit test?

Weighting it below a unit test, commonly in the range of a quiz grade, keeps the incentives honest. A project weighted as heavily as a test invites students to lean on a family member or an online tool for the calculus, while a lighter weight keeps it a genuine chance to apply a unit's ideas without becoming a grade-determining event.