Intersect Prep

How to use Desmos on the AP Calculus BC exam

Updated

The AP Calculus BC exam runs in the Bluebook app, and Bluebook has the Desmos graphing calculator built in for the parts of the exam that allow a calculator. You can also bring an approved handheld calculator if you'd rather use one.

Only some of the exam allows a calculator, and the College Board expects four specific skills on those parts. This guide shows how to do each one in Desmos, what to write on your free-response paper, and how to handle parametric and polar questions.

When you can use Desmos

  • Section I, Part B: 13 multiple-choice questions in 38 minutes. Calculator allowed.
  • Section II, Part A: 2 free-response questions in 30 minutes. Calculator allowed.
  • Section I, Part A and Section II, Part B: no calculator. That's 29 multiple-choice and 4 free-response questions, and about two-thirds of your score.

So Desmos helps on a real part of the exam, but it can't replace knowing the calculus. The full breakdown is in the AP Calculus BC exam format guide.

The four skills the exam expects

The College Board says a calculator on this exam needs to do four things: graph a function in any window, find zeros to solve equations, calculate a derivative at a point, and calculate a definite integral. Every calculator question can be handled with those four. Desmos does all of them.

1. Graph in the window you need

Type the function and zoom until the interval in the question fills the screen. A small feature near the origin can vanish at the default zoom, and it can change the answer.

Zoom before you count

A particle moves along a line with velocity v(t)=e0.2tcos⁡t−1v(t) = e^{0.2t}\cos t - 1. How many times does the particle change direction on 0<t<70 < t < 7?

Graph y=e0.2xcos⁡x−1y = e^{0.2x}\cos x - 1. A particle changes direction where vv changes sign. At the default zoom you'll see the crossing near t=5.083t = 5.083, and it looks like the curve simply leaves the origin going down.

1234567-3-2-1123t ≈ 0.39t ≈ 5.083
v(t)=e0.2tcos⁡t−1v(t) = e^{0.2t}\cos t - 1

Zoom in near the origin. vv is slightly positive just after t=0t = 0 and crosses zero near t=0.39t = 0.39. That's a second sign change, so the count is two.

Answer: 22 times

2. Solve equations with zeros and intersections

Graph both sides of the equation, click a curve, then click the gray dot where they cross. Desmos shows the coordinates. Use the marked point rather than eyeballing the grid, and zoom in if you need more digits, since the exam needs three decimal places.

Area between curves

Let RR be the region enclosed by y=cos⁡xy = \cos x and y=x2y = x^2. Find the area of RR.

Graph both curves. They cross at x≈±0.824x \approx \pm 0.824. Keep at least three decimal places in the limits, and don't round the final value until the end.

-1.5-1-0.50.511.5-0.50.511.5x ≈ 0.824
y=cos⁡xy = \cos xy=x2y = x^2

Cosine is on top, so the area is ∫−0.8240.824(cos⁡x−x2) dx\int_{-0.824}^{0.824} (\cos x - x^2)\,dx. Type that into Desmos to get 1.0951.095 (rounded).

Answer: 1.0951.095

3. Find a derivative at a point

Define the function, then evaluate its derivative. Prime notation works directly: type f(x)=…f(x) = \dots on one line and f′(2)f'(2) on the next. You can also type "d/dx" to get the derivative operator.

A derivative no one wants to do by hand

Let f(x)=xesin⁡xf(x) = x e^{\sin x}. Find the instantaneous rate of change of ff at x=2x = 2.

Type f(x)=xesin⁡xf(x) = xe^{\sin x}, then f′(2)f'(2). Desmos returns about 0.416340.41634. On a free-response question, write f′(2)f'(2) and the value, with the units if there are any.

Answer: 0.4160.416

4. Calculate a definite integral

Type "int" and Desmos builds the integral sign with limits. Fill in the limits, the function, and dxdx, and the value appears on the right. Limits can be numbers, sliders, or other expressions.

An integral with no simple antiderivative

Find the value of ∫031+x3 dx\int_0^3 \sqrt{1 + x^3}\,dx.

Type the integral exactly as written. Desmos returns about 7.341417.34141.

Answer: 7.3417.341

Parametric and polar in Desmos

BC adds parametric, polar, and vector questions, and Desmos handles them directly. For the formulas behind these, see the parametric and polar guide.

  • Parametric: type an ordered pair such as (3cos⁡t+t, 2sin⁡t)(3\cos t + t,\ 2\sin t), then set the range of tt under the expression.
  • Polar: type r=2+cos⁡(3θ)r = 2 + \cos(3\theta). Desmos treats it as polar, and you can set the range of θ\theta.
  • Arc length and area: these are ordinary integrals once you've set them up, so use the integral from skill 4.

Polar area

Find the area enclosed by r=2+cos⁡(3θ)r = 2 + \cos(3\theta).

The curve traces once for 0≤θ≤2π0 \le \theta \le 2\pi, and rr is always positive. The area is 12∫02π(2+cos⁡(3θ))2 dθ\frac{1}{2}\int_0^{2\pi} (2 + \cos(3\theta))^2\,d\theta. Desmos returns about 14.1371714.13717. Replace θ\theta with xx in the integral if that's quicker to type.

Answer: 14.13714.137

What to write on your paper

Free-response answers are handwritten, and readers can't see your screen. The College Board requires you to write the setup, meaning the equation you solved or the derivative or integral you evaluated. A number with nothing behind it won't earn full credit.

  • Write math, not keystrokes. Write ∫031+x3 dx=7.341\int_0^3 \sqrt{1 + x^3}\,dx = 7.341, not what you typed into the calculator.
  • Give three decimal places. Unless the question says otherwise, a decimal answer must be correct to three places. Don't round until the final step.
  • Reuse values instead of retyping them. Give an integral or a function a name in Desmos and refer to it by name, so rounding doesn't creep in.

There's more on this in how to earn free-response points.

Keep going

Practice on exam-style questions

Intersect Prep has 420 multiple-choice questions across all 10 units, with a full solution for each one. Create a free account to try a full lesson with the calculator. One payment of $49 unlocks the course for 6 months.

Questions

Is Desmos allowed on the AP Calculus BC exam?
Yes, on the calculator parts. Bluebook has Desmos built in for Section I Part B and Section II Part A. It isn't available on the no-calculator parts.
Can I bring my own calculator instead?
Yes, as long as it's on the College Board's approved list. Check the current list before exam day.
How accurate do decimal answers need to be?
Unless a question says otherwise, three places after the decimal point. Keep more digits while you work and round only at the end.
Can Desmos find an antiderivative for me?
No. Desmos calculates the value of a definite integral but doesn't give symbolic antiderivatives, and most of the exam has no calculator anyway, so you still need those methods by hand.

Desmos is a trademark of Desmos Studio PBC.