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

AP Calculus (AB & BC) syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for AP Calculus (AB & BC) in AP Exams (Advanced Placement), not a summary of it.

1. 1

   Limits and Continuity

   4 topics

   • Understanding Limits
     • Estimating limits from graphs and tables
     • Algebraic evaluation of limits
     • One-sided limits
   • Limits Involving Infinity
     • Vertical asymptotes and infinite limits
     • Limits at infinity and horizontal asymptotes
   • Continuity
     • Continuity at a point and on intervals
     • Types of discontinuities
     • Intermediate Value Theorem
   • Squeeze Theorem and Special Limits
     • Limit of sin(x)/x
     • Removing indeterminate forms
2. 2

   Differentiation

   4 topics

   • Definition and Interpretation of the Derivative
     • Derivative as a limit of difference quotients
     • Derivative as instantaneous rate of change
     • Differentiability and continuity
   • Derivative Rules
     • Power, product, and quotient rules
     • Chain rule
     • Derivatives of trigonometric, exponential, and logarithmic functions
   • Implicit and Inverse Differentiation
     • Implicit differentiation
     • Derivatives of inverse functions
     • Inverse trigonometric derivatives
   • Higher-Order Derivatives
3. 3

   Applications of Derivatives

   6 topics

   • Related Rates
   • Linearization and Differentials
   • Analysis of Functions
     • Increasing/decreasing and the first derivative test
     • Concavity and the second derivative test
     • Inflection points and critical points
   • Optimization Problems
   • Mean Value Theorem and Rolle's Theorem
   • L'Hopital's Rule
4. 4

   Integration and the Fundamental Theorem

   4 topics

   • Antiderivatives and Indefinite Integrals
   • Riemann Sums and Definite Integrals
     • Left, right, and midpoint sums
     • Trapezoidal approximation
   • Fundamental Theorem of Calculus
     • Evaluation part
     • Accumulation functions
   • Techniques of Integration
     • u-substitution
     • Integration by parts (BC)
     • Partial fractions (BC)
5. 5

   Applications of Integration and Differential Equations

   5 topics

   • Area Between Curves
   • Volumes of Solids
     • Disk and washer methods
     • Cross-sections
   • Accumulation and Average Value
   • Differential Equations
     • Slope fields
     • Separation of variables
     • Exponential and logistic models (BC)
   • Arc Length (BC)
6. 6

   Series, Parametrics, and Polar (BC)

   4 topics

   • Sequences and Series Convergence
     • Geometric and p-series
     • Ratio, comparison, and integral tests
     • Alternating series and error bounds
   • Power Series
     • Radius and interval of convergence
     • Taylor and Maclaurin series
   • Parametric Equations and Vectors
     • Derivatives of parametric curves
     • Motion in the plane
   • Polar Coordinates
     • Polar curves and area

AP Calculus (AB & BC) flashcards for AP Exams (Advanced Placement)

19 of 51 cards from the AP Calculus (AB & BC) deck — real questions with worked answers.

1. What is the formal (epsilon-delta) definition of the limit lim(x→c) f(x) = L?

   For every ε>0 there exists δ>0 such that if 0<|x−c|<δ then |f(x)−L|<ε.

2. When does a two-sided limit lim(x→c) f(x) exist?

   It exists if and only if the left-hand limit and right-hand limit both exist and are equal: lim(x→c⁻) f(x) = lim(x→c⁺) f(x).

3. How do you determine the horizontal asymptotes of a rational function from its limits at infinity?

   Compare degrees: if deg(numerator)<deg(denominator), y=0; if equal, y = ratio of leading coefficients; if numerator degree is larger, there is no horizontal asymptote (limit is ±∞).

4. What does it mean for a function to have a vertical asymptote at x=a in terms of limits?

   At least one one-sided limit is infinite: lim(x→a⁺) f(x) = ±∞ or lim(x→a⁻) f(x) = ±∞.

5. State the three conditions for a function f to be continuous at x=c.

   (1) f(c) is defined, (2) lim(x→c) f(x) exists, and (3) lim(x→c) f(x) = f(c).

6. Name and distinguish the three main types of discontinuities.

   Removable (hole; limit exists but ≠ f(c) or f(c) undefined), jump (left and right limits differ), and infinite/essential (a one-sided limit is ±∞, as at a vertical asymptote).

7. State the Squeeze (Sandwich) Theorem.

   If g(x) ≤ f(x) ≤ h(x) near c and lim(x→c) g(x) = lim(x→c) h(x) = L, then lim(x→c) f(x) = L.

8. What are the two special trigonometric limits used with the Squeeze Theorem?

   lim(x→0) sin(x)/x = 1 and lim(x→0) (1−cos x)/x = 0.

9. State the Intermediate Value Theorem (IVT).

   If f is continuous on [a,b] and N is any value between f(a) and f(b), then there exists c in (a,b) with f(c)=N.

10. What is the limit definition of the derivative f'(x)?

    f'(x) = lim(h→0) [f(x+h) − f(x)] / h (equivalently lim(x→a) [f(x)−f(a)]/(x−a) at a point).

11. What does the derivative represent geometrically and physically?

    Geometrically it is the slope of the tangent line to the curve; physically it is the instantaneous rate of change (e.g., velocity is the derivative of position).

12. If a function is differentiable at a point, what can you conclude about continuity, and is the converse true?

    Differentiability implies continuity at that point. The converse is false: a function can be continuous but not differentiable (e.g., |x| at x=0).

13. State the Power Rule, Product Rule, and Quotient Rule.

    Power: d/dx(xⁿ)=n·xⁿ⁻¹. Product: (uv)'=u'v+uv'. Quotient: (u/v)'=(u'v−uv')/v².

14. State the Chain Rule.

    d/dx[f(g(x))] = f'(g(x))·g'(x).

15. Give the derivatives of sin x, cos x, tan x, and eˣ.

    d/dx sin x = cos x; d/dx cos x = −sin x; d/dx tan x = sec²x; d/dx eˣ = eˣ.

16. Give the derivatives of ln x, aˣ, and arctan x.

    d/dx ln x = 1/x; d/dx aˣ = aˣ·ln a; d/dx arctan x = 1/(1+x²).

17. What is the process of implicit differentiation?

    Differentiate both sides with respect to x, applying the chain rule to y-terms (multiply by dy/dx), then solve algebraically for dy/dx.

18. What is the formula for the derivative of an inverse function (f⁻¹)'(a)?

    (f⁻¹)'(a) = 1 / f'(f⁻¹(a)), where f(f⁻¹(a)) = a.

19. What is a higher-order derivative, and what does the second derivative tell you?

    A higher-order derivative is the derivative of a derivative. The second derivative f''(x) measures concavity (and acceleration when f is position).

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