AP CALC BC Cheat Sheet 2026

The 30 highest-yield AP CALC BC facts, distilled from real exam questions. Print it, save it as a PDF, or study it here — free, no sign-up.

1. What is the area of the region enclosed by the curves x = y² and x = y + 2? → 9/2
2. What is the length of the curve y = x^(3/2) from x = 0 to x = 4? → (8/27)(10√10 − 1)
3. Which expression equals ∫ 1/((x − 1)(x + 2)) dx? → (1/3)·ln|(x − 1)/(x + 2)| + C
4. A particle's position is r(t) = ⟨t³ − t, 2t²⟩. What is its acceleration vector at t = 2? → ⟨12, 4⟩
5. A particle moves along a line with position s(t) = t³ − 9t² + 15t for t ≥ 0. At which time is the particle speeding up? → t = 2
6. A population satisfies dP/dt = 0.03P − 0.0006P² with P(0) = 10. What is lim (t→∞) P(t)? → 50
7. Consider dy/dx = y(2 − y) with y(0) = 1. What is lim (x→∞) y? → 2
8. If dy/dx = xy and y(0) = 2, what is y(2)? → 2e²
9. What is the value of ∫ from 0 to 1 of x/(x² + 1) dx? → (1/2)·ln 2
10. Let f(x) = x³ − 3x on the interval [0, 2]. Which value of c in (0, 2) satisfies the conclusion of the Mean Value Theorem? → 2/√3
11. A function f is continuous on [1, 4] with f(1) = −3 and f(4) = 5. Which statement must be true? → f(c) = 0 for at least one c in (1, 4)
12. Euler's method with two steps of size 0.5 is used to approximate y(1) for the differential equation dy/dx = x + y with y(0) = 1. What is the approximation? → 2.5
13. The tangent line to f(x) = √x at x = 25 is used to approximate √26. What is the approximation, and is it an overestimate or an underestimate of the true value? → 5.1, overestimate
14. A left Riemann sum with 4 subintervals of equal width is used to approximate ∫ from 0 to 4 of x² dx. What is its value? → 14
15. What is lim(x→0) (1 − cos x)/x²? → 1/2
16. What is the area of the region enclosed by the graphs of y = x² and y = 2x? → 4/3
17. What is the area enclosed by one petal of the rose curve r = cos 2θ, taken from θ = −π/4 to θ = π/4? → π/8
18. Let f(x) = kx + 2 for x ≤ 2 and f(x) = x² − k for x > 2. For what value of k is f continuous at x = 2? → 2/3
19. If g(x) = x^2/(x + 1), what is g'(1)? → 3/4
20. Let f(x) = x^2 for x ≤ 1 and f(x) = 2x − 1 for x > 1. Which statement is true at x = 1? → f is differentiable at x = 1 and f'(1) = 2
21. The region bounded by y = x², the x-axis, and the line x = 2 is revolved about the y-axis. What is the volume of the solid generated? → 8π
22. The curve x^2 + xy + y^2 = 7 passes through the point (1, 2). What is dy/dx at that point? → −4/5
23. What is the value of ∫ from 0 to ∞ of e^(−2x) dx? → 1/2
24. Let f(x) = x^3. Which of the following limits equals f'(2)? → lim(h→0) [(2+h)^3 − 2^3] / h, which equals 12
25. What is the value of the derivative of f(x) = arctan(2x) at x = 1/2? → 1
26. What is the value of ∫ from 1 to e of x·ln x dx? → (e² + 1)/4
27. The Maclaurin series for arctan x is Σ_{n=0}^∞ (−1)ⁿ x^(2n+1)/(2n+1) for |x| ≤ 1. What is the sum of Σ_{n=0}^∞ (−1)ⁿ / (3ⁿ(2n + 1))? → π√3/6
28. A population satisfies dP/dt = 0.2P(1 − P/500). For what value of P is the population growing most rapidly? → 250
29. What is the value of ∫ from 0 to 1 of x·eˣ dx? → 1
30. What is d/dx [arcsin(x^2)]? → 2x / √(1 − x^4)