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AP Exams (Advanced Placement) AP Calculus (AB & BC) Syllabus
The AP Calculus (AB & BC) syllabus for AP Exams (Advanced Placement) in full: 6 chapters, 27 topics and 42 sub-topics, every one listed below, plus 51 AP Calculus (AB & BC) flashcards with answers.
AP Calculus (AB & BC) syllabus: all 6 chapters
Expand any chapter to see its topics and sub-topics.
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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
- Understanding Limits
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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
- Definition and Interpretation of the Derivative
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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
-
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)
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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)
-
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
- Sequences and Series Convergence
AP Exams (Advanced Placement) AP Calculus (AB & BC) flashcards — 51 questions with answers
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.
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).
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 ±∞).
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) = ±∞.
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).
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).
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.
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.
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.
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).
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).
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).
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.
State the Chain Rule.
d/dx[f(g(x))] = f'(g(x))·g'(x).
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.
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).
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.
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.
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).
What are the general steps for solving a related rates problem?
Identify variables and given/wanted rates, write an equation relating the quantities, differentiate both sides with respect to time t, then substitute known values and solve for the unknown rate.
Show the other 31 AP Calculus (AB & BC) flashcards
What is the linearization (tangent line approximation) of f at x=a?
L(x) = f(a) + f'(a)(x − a); it approximates f(x) for x near a.
What is the differential dy and how is it used?
dy = f'(x) dx; it approximates the change in y, Δy ≈ f'(x)·Δx, for a small change Δx in x.
What do the signs of f'(x) tell you about a function's behavior?
f'>0 means f is increasing; f'<0 means f is decreasing; f'=0 or undefined gives critical points (candidates for extrema).
State the First Derivative Test for local extrema.
At a critical point, if f' changes from + to −, it's a local maximum; from − to +, a local minimum; if no sign change, it's neither.
State the Second Derivative Test and what concavity means.
At a critical point where f'(c)=0: if f''(c)>0 it's a local min (concave up), if f''(c)<0 it's a local max (concave down). Concave up means f' is increasing; concave down means f' is decreasing.
What is an inflection point and how do you find it?
A point where concavity changes. Find candidates where f''(x)=0 or is undefined, then confirm f'' changes sign there.
What are the general steps for solving an optimization problem?
Write the quantity to optimize as a function of one variable (using a constraint), find critical points via the derivative, and test endpoints/critical points to identify the absolute max or min.
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].
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).
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.
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.
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.
State the Power Rule for integration.
∫xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ −1; and ∫x⁻¹ dx = ln|x| + C.
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).
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.
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).
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).
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).
State the integration by parts formula.
∫u dv = uv − ∫v du.
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].
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.
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.
What is the average value of a function f on [a,b]?
f_avg = (1/(b−a)) ∫ₐᵇ f(x) dx.
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.
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.
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.
State the arc length formula for y=f(x) on [a,b] (BC).
L = ∫ₐᵇ √(1 + [f'(x)]²) dx.
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.
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.
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.
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.
AP Exams (Advanced Placement) AP Calculus (AB & BC) FAQ
What is in the AP Exams (Advanced Placement) AP Calculus (AB & BC) syllabus?
AP Exams (Advanced Placement) AP Calculus (AB & BC) is split into 6 chapters: Limits and Continuity, Differentiation, Applications of Derivatives, Integration and the Fundamental Theorem, Applications of Integration and Differential Equations and Series, Parametrics, and Polar (BC). Together they contain 27 topics and 42 sub-topics.
How many chapters are in AP Exams (Advanced Placement) AP Calculus (AB & BC)?
6 chapters, containing 27 topics and 42 sub-topics. The largest chapter is Applications of Derivatives, with 6 topics.
Are there flashcards for AP Exams (Advanced Placement) AP Calculus (AB & BC)?
Yes — a 51-card AP Calculus (AB & BC) deck, printed in full in the flashcards section of this page and available in the Examius app with spaced repetition. The free plan covers one exam of your choice, including its flashcards; Examius Pro, a one-time purchase, unlocks every other exam and deck.