AP Calculus AB study guide
AP Calculus AB · Work through your exam, one topic at a time.
AP Calculus AB is a College Board Advanced Placement course and exam equivalent to a first-semester college course in differential and integral calculus. It is designed for high school students who have already built solid foundations in algebra, geometry, trigonometry, analytic geometry, and elementary functions, including linear, polynomial, rational, exponential, logarithmic, trigonometric, inverse trigonometric, and piecewise-defined functions. The AP Calculus AB study guide has 8 sections and about 1,776 words.
01Limits and Continuity
A limit describes the value a function approaches, not necessarily the value it takes. Master one-sided limits, the existence criterion (left limit equals right limit), and the algebra of limits.
259 words, about 1 min02Differentiation: Definition and Fundamental Properties
The derivative is a limit of a difference quotient: f'(x)= {h 0}f(x+h)-f(x)/h , and at a point a it is the slope of the tangent line. Interpret it as instantaneous rate of change.
218 words, about 1 min03Differentiation: Composite, Implicit, and Inverse Functions
The chain rule differentiates compositions: d/dxf(g(x))=f'(g(x))g'(x) . Layer derivatives from outside in, and expect nested chains.
219 words, about 1 min04Contextual Applications of Differentiation
Derivatives model rates: velocity is the derivative of position, acceleration the derivative of velocity. Interpret the sign and magnitude of a rate in context (increasing/decreasing, speeding up/slowing down).
218 words, about 1 min05Analytical Applications of Differentiation
Use f' to find intervals of increase/decrease and critical points (where f'=0 or undefined). Use the first derivative test or second derivative test to classify extrema.
225 words, about 1 min06Integration and Accumulation of Change
The definite integral is a limit of Riemann sums and represents net accumulation (signed area). Understand left/right/midpoint and trapezoidal approximations and their over/underestimate behavior based on concavity.
224 words, about 1 min07Differential Equations
A differential equation relates a function to its derivatives. A solution satisfies the equation; the general solution includes a constant determined by an initial condition (initial value problem).
204 words, about 1 min08Applications of Integration
Use definite integrals to compute average value, 1/b-a a^b f(x)\,dx , and to find areas between curves: a^b (top-bottom)\,dx or right-minus-left with respect to y . Volumes: disks/washers use [R(x)^2-r(x)^2]\,dx for rotation about an axis; cross-section methods integrate area A(x) .
209 words, about 1 min