AP Exams (Advanced Placement) AP Calculus (AB & BC) Flashcards

24 sample cards from the AP Calculus (AB & BC) deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. State the Extreme Value Theorem.

   If f is continuous on a closed interval [a,b], then f attains an absolute maximum and an absolute minimum on [a,b].

2. State the Mean Value Theorem (MVT).

   If f is continuous on [a,b] and differentiable on (a,b), then there exists c in (a,b) with f'(c) = [f(b)−f(a)]/(b−a).

3. State Rolle's Theorem.

   If f is continuous on [a,b], differentiable on (a,b), and f(a)=f(b), then there exists c in (a,b) with f'(c)=0.

4. State L'Hopital's Rule and when it applies.

   If lim f(x)/g(x) gives the indeterminate form 0/0 or ∞/∞, then lim f(x)/g(x) = lim f'(x)/g'(x), provided the latter limit exists.

5. What is an antiderivative, and why is the constant +C needed in indefinite integrals?

   F is an antiderivative of f if F'=f. The constant +C is included because any two antiderivatives of f differ by a constant, so ∫f(x)dx = F(x)+C represents the whole family.

6. State the Power Rule for integration.

   ∫xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ −1; and ∫x⁻¹ dx = ln|x| + C.

7. What is the definition of a definite integral as a Riemann sum?

   ∫ₐᵇ f(x)dx = lim(n→∞) Σ f(xᵢ*)·Δx, where Δx=(b−a)/n; it is the limit of Riemann sums (signed area under the curve).

8. Compare left, right, midpoint, and trapezoidal approximations for an increasing function on [a,b].

   For an increasing function, left sums underestimate and right sums overestimate the integral; the trapezoidal rule averages left and right; the midpoint rule uses the function value at each subinterval's center.

9. State the First Fundamental Theorem of Calculus (evaluation form).

   If F is an antiderivative of f on [a,b], then ∫ₐᵇ f(x)dx = F(b) − F(a).

10. State the Second Fundamental Theorem of Calculus (derivative of an integral).

    d/dx ∫ₐ^x f(t)dt = f(x); more generally d/dx ∫ₐ^{g(x)} f(t)dt = f(g(x))·g'(x).

11. Describe the method of u-substitution for integration.

    Choose u = inner function, compute du = u'(x)dx, rewrite the integral entirely in terms of u, integrate, then substitute back (adjusting limits if definite).

12. State the integration by parts formula.

    ∫u dv = uv − ∫v du.

13. What is the formula for the area between two curves f(x) and g(x) on [a,b]?

    Area = ∫ₐᵇ [top − bottom] dx = ∫ₐᵇ [f(x) − g(x)] dx, where f ≥ g on [a,b].

14. State the disk and washer methods for volumes of revolution.

    Disk: V = π∫ₐᵇ [R(x)]² dx. Washer: V = π∫ₐᵇ ([R_outer]² − [R_inner]²) dx, where radii are distances to the axis of revolution.

15. What is the formula for the volume of a solid with known cross-sections?

    V = ∫ₐᵇ A(x) dx, where A(x) is the area of the cross-section perpendicular to the axis at position x.

16. What is the average value of a function f on [a,b]?

    f_avg = (1/(b−a)) ∫ₐᵇ f(x) dx.

17. What does an accumulation function ∫ₐ^x f(t)dt represent, and how is net change found?

    It accumulates the signed area (net amount) of f from a to x. The net change of a quantity equals the integral of its rate: F(b)−F(a)=∫ₐᵇ F'(t)dt.

18. How do you solve a separable differential equation dy/dx = g(x)h(y)?

    Separate variables: dy/h(y) = g(x)dx, integrate both sides, add +C, then apply an initial condition to solve for C.

19. What is the exponential growth/decay model and its solution?

    dy/dt = ky has solution y = y₀·e^{kt}, where y₀ is the initial amount; k>0 is growth, k<0 is decay.

20. State the arc length formula for y=f(x) on [a,b] (BC).

    L = ∫ₐᵇ √(1 + [f'(x)]²) dx.

21. State the nth-term test for divergence and the ratio test for series (BC).

    nth-term test: if lim(n→∞) aₙ ≠ 0, the series Σaₙ diverges. Ratio test: with L = lim |aₙ₊₁/aₙ|, the series converges absolutely if L<1, diverges if L>1, and is inconclusive if L=1.

22. Give the Taylor/Maclaurin series for eˣ, sin x, and cos x (BC).

    eˣ = Σ xⁿ/n!; sin x = Σ (−1)ⁿ x^{2n+1}/(2n+1)!; cos x = Σ (−1)ⁿ x^{2n}/(2n)!. The geometric series 1/(1−x)=Σ xⁿ for |x|<1.

23. For parametric equations x(t), y(t), give dy/dx and the speed (BC).

    dy/dx = (dy/dt)/(dx/dt); speed = √[(dx/dt)² + (dy/dt)²], and arc length = ∫ √[(dx/dt)²+(dy/dt)²] dt.

24. In polar coordinates, give the conversions and the area enclosed by r=f(θ) (BC).

    x = r cos θ, y = r sin θ, r² = x²+y². Area = (1/2)∫_α^β [f(θ)]² dθ.

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AP Calculus (AB & BC) flashcards FAQ