AP Calculus BC study guide
AP Calculus BC · Work through your exam, one topic at a time.
AP Calculus BC is a College Board Advanced Placement course and exam described as equivalent to a first-semester college calculus course plus the subsequent single-variable calculus course. It is aimed at high school students who have already completed coursework in algebra, geometry, trigonometry, analytic geometry, and elementary functions, and who intend further study in mathematics, engineering, computer science, or economics. The AP Calculus BC study guide has 10 sections and about 2,248 words.
01Limits and Continuity
A limit describes the value a function approaches, not necessarily the value it attains. You must be fluent with one-sided limits, infinite limits, limits at infinity, and the existence criterion: the two-sided limit exists exactly when both one-sided limits agree.
276 words, about 1 min02Differentiation: Definition and Fundamental Properties
The derivative is the limit of a difference quotient and represents instantaneous rate of change and slope of the tangent line. You should be able to compute a derivative from the definition when asked, and to recognize differentiability implies continuity but not conversely (corners, cusps, vertical tangents, and jumps break differentiability).
244 words, about 1 min03Differentiation: Composite, Implicit, and Inverse Functions
The chain rule is the engine of this domain: d/dxf(g(x))=f'(g(x))g'(x) . Layer functions from the outside in and track each inner derivative.
225 words, about 1 min04Contextual Applications of Differentiation
This domain translates derivatives into rates. Interpret ds/dt as velocity and dv/dt as acceleration, with sign indicating direction.
225 words, about 1 min05Analytical Applications of Differentiation
Use derivatives to describe function behavior. Critical points occur where f'=0 or f' is undefined.
253 words, about 1 min06Integration and Accumulation of Change
The definite integral is a limit of Riemann sums and measures net accumulation. The Fundamental Theorem of Calculus links integration and differentiation: a^b f(x)\,dx=F(b)-F(a) when F'=f , and d/dx a^{g(x)}f(t)\,dt=f(g(x))g'(x) .
212 words, about 1 min07Differential Equations
A differential equation relates a function to its derivatives. A solution is a function; an initial condition pins down a particular solution.
218 words, about 1 min08Applications of Integration
Integrals compute geometric and physical quantities. Area between curves is (top-bottom)\,dx or (right-left)\,dy ; choose the orientation that avoids splitting the region.
214 words, about 1 min09Parametric Equations, Polar Coordinates, and Vector-Valued Functions
Parametric curves describe x(t) and y(t) separately; dy/dx=dy/dt/dx/dt when dx/dt 0 . Second derivatives require differentiating the slope with respect to t and dividing by dx/dt again.
164 words, under 1 min10Infinite Sequences and Series
Sequences have limits; series are sums of sequences. Convergence tests form a toolkit: nth-term test (diverges if terms do not go to zero), geometric series ( ar^n converges to a/1-r for r <1 ), p-series (converges for p 1 ), comparison and limit comparison, integral test, ratio and root tests, and alternating series test with error bound.
217 words, about 1 min