Vol. II · Bluebook Edition

The Desmos Calculus Companion

College Board
Bluebook 2026
AB & BC

Every Desmos input, shortcut, and trick worth knowing ❦ For the calculator-required sections of AP Calculus AB and BC in Bluebook

The Big Four — the operations that win calculator-active points

01
Numerical derivative at a point
f'(2)
02
Definite integral
int02 f(x) dx
03
Solve equation / find zero
f(x) = 0 → click x-intercept
04
Find max, min, intersection
Click the dot on the graph

00 Bluebook Reality Check

What's actually true on test day — the constraints that change how you use the tool.

When Desmos is available
Calculator-required parts only:
  • Section I, Part B — last 15 MC questions
  • Section II, Part A — first 2 FRQs
Not available on no-calculator parts. Don't build habits that depend on it for hand-computable work.
Hybrid format
MC is digital in Bluebook; FRQ answers are handwritten in a paper booklet. You'll have 2 sheets of scratch paper. Plan to copy diagrams or set-ups by hand.
Bluebook Desmos only
The web and mobile Desmos apps are banned on exam day. Only the Desmos embedded in Bluebook counts. Practice in Bluebook's official AP Calc practice exam beforehand.
No copy/paste
Bluebook blocks copy-paste. Define every function once as f(x) = ..., then reuse the name. Re-typing long expressions costs time and breeds typos.
Drag the panel, save the screen
Click and drag the Desmos panel anywhere. Park it where it doesn't cover the question.
Two handhelds still allowed
You can bring up to two approved graphing calculators alongside Desmos. Use whichever is faster for the task.

01 Derivatives

All derivatives Desmos returns are numerical, not symbolic. Define a function first, then differentiate.

Derivative operator
Type d/dx — Desmos auto-formats the fraction. Then put the expression in.
d/dx (x^3 - 2x + 5)
# returns 3x² − 2 numerically as you scrub
Prime notation shortcut
After defining f(x), type f then apostrophe '. The cleanest way to evaluate a derivative at a point.
f(x) = sin(x) + x^2
f'(2) # first derivative at 2
f''(2) # second derivative at 2
Tangent line at x = a
y = f(a) + f'(a)(x - a)

02 Integrals

Definite only — Desmos cannot give you an antiderivative formula.

Definite integral
Type int to summon the ∫ symbol, then tab through bounds.
int04 (x^2 + 1) dx
Area between two curves
Top minus bottom, on the interval where they cross. Use abs( ) if signs flip.
intab (f(x) - g(x)) dx
Volume — disk / washer about the x-axis
π intab (R(x))^2 dx
π intab ((R)^2 - (r)^2) dx
Volume by known cross-sections
Square: (side)². Equilateral ▵: (√3/4)(side)². Semicircle: (π/8)(diameter)².
intab A(x) dx
Average value of f on [a, b]
(1 / (b - a)) intab f(x) dx
Improper integral — BC
Type infty as a bound. Convergent integrals evaluate; divergent ones return undefined.
int1infty 1/x^2 dx # → 1
int1infty 1/x dx # → undefined

03 Limits & Solving

Desmos has no lim operator. Estimate limits numerically or visually instead.

Estimate a limit by table
Define f, then evaluate at values squeezing the target from both sides. The outputs are the limit candidate.
f(x) = (sin x)/x
f(0.1) f(0.01) f(0.001)
f(-0.1) f(-0.01) f(-0.001)
# all approach 1 → limit is 1
Estimate a limit by graph
Plot the function, hover near the target x-value, and read the y. Holes show as open circles.
Limits at infinity
Evaluate at large magnitudes, or zoom out and inspect the horizontal asymptote.
f(1000) f(10000) f(100000)
# stable value → lim as x → ∞
Solve an equation
Just type it. Desmos shows the solution(s) as labeled points.
x^2 - 5x + 6 = 0
# click each black dot to read x = 2, x = 3
System of equations
Graph both. Click the intersection to read coordinates.
y = x^2
y = 2x + 3
Numerical "solve for x" trick
Set up g(x) = LHS - RHS, graph, click x-intercept.

04 Reading the Graph

Desmos labels key points automatically — just click them to read exact values.

Click-to-find features
  • x-intercepts (zeros) — gray dots on the x-axis
  • y-intercepts — gray dot on the y-axis
  • Local max / min — dots at peaks and valleys
  • Intersections — dot where two curves meet
  • Holes / undefined pts — open circles
Trace any function
Hover (or tap on mobile) along the curve to read (x, y) live.
Inflection points
Not auto-labeled. Define f, then graph f''(x) and click its zeros — possible inflections.
f(x) = ...
y = f''(x)
# click x-intercepts → possible inflection
Zoom & window
  • Pinch / scroll to zoom
  • Wrench icon → set exact x and y bounds
  • Toggle Radians/Degrees in wrench menu

05 Defining Functions

Standard definition
f(x) = x^2 - 4x + 1
f(3) # → -2
Restrict the domain
Append { } with the inequality.
f(x) = x^2 {0 ≤ x ≤ 5}
Piecewise function
Comma-separated cases inside braces.
f(x) = {x < 0: -x, x ≥ 0: x^2}
Save values to variables
Letters become reusable. Great for storing answers and avoiding rounding.
a = int02 f(x) dx
b = f'(3)
a + b # reuses full precision
Inverse function (numerical)
Graph x = f(y) directly — reflects across y = x.
x = y^3 + y

06 Useful Built-ins

Common functions
sqrt(x) abs(x) |x|
ln(x) log(x) # log = base 10
log_2(x) # type log then _ for base
e^x exp(x)
sin, cos, tan, sec, csc, cot
arcsin, arccos, arctan
Constants
pi # → π
e # → 2.71828...
infty # type "infty" or "infinity" → ∞
Subscripts & superscripts
_ for subscript, ^ for superscript. Use them for variable names: x_1, x_2, Δx.
Verify algebra by overlay
Suspect your simplification is wrong? Graph both. If the curves perfectly overlap on the visible window, they're equivalent.
y = (x^2 - 1)/(x - 1)
y = x + 1
# overlap (with hole at x=1) → equivalent
Decimal ↔ fraction
Click the small icon at the left of an expression line to toggle a decimal answer to a fraction.

07 Sums & Riemann

Sigma notation makes Riemann sums one-line operations — an AP Calc favorite.

Summation
Type sum for Σ. Set lower index, upper bound, and the expression.
sumn=110 n^2
# → 385
Left Riemann sum (n rectangles on [a,b])
a = 0
b = 4
n = 8
d = (b - a)/n
sumi=0n-1 f(a + i·d) · d
Right Riemann sum
sumi=1n f(a + i·d) · d
Midpoint rule
sumi=0n-1 f(a + (i + 0.5)·d) · d
Trapezoidal rule
Average of left and right sums.
(d/2) · (f(a) + 2·sumi=1n-1 f(a + i·d) + f(b))

08 Tables & Lists

Make a table
Click the + button (top-left) → table. Or type a list.
List notation
L = [1, 2, 3, 5, 8, 13]
mean(L) total(L)
L^2 # element-wise → [1,4,9,...]
Apply a function to a list
x_1 = [0, 1, 2, 3, 4]
y_1 = f(x_1) # list of outputs
Sliders for parameters
Just type a = 1, then click "add slider" in the prompt that appears.

09 Parametric — BC

Plot a parametric curve
Enter as a point with parameter t. Set t-range in the wrench.
(t·cos(t), t·sin(t))
Speed — |v(t)|
sqrt((dx/dt)^2 + (dy/dt)^2)
Arc length on [a, b]
intab sqrt((x'(t))^2 + (y'(t))^2) dt
Total distance traveled
Same as arc length integral — always use the absolute-value (square root) form, not displacement.
Position from velocity
x(T) = x_0 + int0T v_x(t) dt

10 Polar — BC

Plot a polar curve
Wrench icon → Grid → switch to polar. Then write r as a function of θ.
r = 1 + cos(theta)
r = 2 sin(3 theta)
Area inside r = f(θ)
(1/2) intαβ (f(theta))^2 dtheta
Area between two polar curves
(1/2) intαβ ((R)^2 - (r)^2) dtheta
Find polar intersections
Graph both curves. Click each intersection. Watch for extra crossings at the pole.

11 Series & Approximations — BC

Partial sum with slider
Make N a slider; watch the partial sum converge.
N = 10
S = sumn=0N (-1)^n / (2n+1)
Taylor polynomial — write out each term
No f(k) indexing in Desmos. Prime notation only goes a few apostrophes deep, so type each term explicitly.
a = 0 # center
T(x) = f(a) + f'(a)(x-a)
    + f''(a)/2 · (x-a)^2
    + f'''(a)/6 · (x-a)^3
    + f''''(a)/24 · (x-a)^4
Known Maclaurin series via sigma
When the coefficient pattern is closed-form, the sum works fine — no derivative indexing needed.
N = 10
E(x) = sumn=0N x^n / n! # e^x
S(x) = sumn=0N (-1)^n x^(2n+1)/(2n+1)! # sin x
C(x) = sumn=0N (-1)^n x^(2n)/(2n)! # cos x
Logistic / differential equation slope field
Use a list-based grid. For dy/dx = f(x,y), plot many tiny segments centered on grid points.
(X, Y) for X=[-5...5], Y=[-5...5]
# draw segments with slope f(X,Y)

12 Exam-Day Tips

  • Radians. Calculus is in radians. Desmos defaults there — confirm in the wrench menu.
  • Three decimal places. AP requires answers truncated or rounded to 3 places. Compute to full precision in Desmos, then round only at the very end.
  • Store, don't retype. Save intermediates to a, b, c — preserves precision and beats Bluebook's no-paste rule.
  • Show the setup on paper. FRQs are graded by hand. Write the integral, equation, or formula before reading the value off Desmos. A bare numerical answer earns 0.
  • Don't cite calculator keys. Justify with math (e.g., "f''(3) = -2.275 < 0, so f is concave down"), never "I used the d/dx button."
  • Parentheses everywhere. Especially around exponents and arguments. e^(2x) ≠ e^2x.
  • Implicit multiplication watch. 2x is fine, but if you've defined x elsewhere or used a function name, x(x+1) may parse as a function call. Use x·(x+1) when unsure.
  • No CAS. Desmos returns numbers, not formulas. For symbolic "Find f'(x)" answers, you still differentiate by hand.
  • Check the window. A "no solution" might just be off-screen. Zoom out, or set bounds in the wrench menu.
  • Bring a backup. Two handheld graphing calculators are allowed. If Desmos misbehaves — or your laptop hiccups — you still have a tool.
  • Hand methods still matter. Be ready to find max/min by setting f'(x) = 0 on paper for the no-calculator parts.