AP Calculus AB: Study Guide & Free Lessons
Short lessons, worked examples, and practice that connects.
Limits without guessing
A limit describes nearby output behavior; it need not equal a value assigned at the point.
Continuity and local change
Continuity at a point requires a defined function value and a matching two-sided limit. Differentiability is stronger.
Derivative rules and meaning
The derivative is a limiting rate of change. Its units are output units per input unit.
Products, quotients and chains
A composite expression can require several derivative rules. Its structure determines the rule.
Implicit and inverse derivatives
An implicit equation constrains x and y together. When y depends on x, differentiating a y term requires the chain rule.
Position, velocity and acceleration
Velocity is the derivative of position; acceleration is the derivative of velocity. Displacement and total distance are different.
Tangents and derivative tests
A tangent line gives a local linear model, while derivative signs describe increasing and decreasing behavior.
Optimization and related rates
A word problem becomes a calculus problem only after the quantities and constraints are written precisely.
Integrals and numerical accumulation
An integral accumulates signed quantities. A definite integral is a number; an indefinite integral describes a family of antiderivatives.
The fundamental theorem
For continuous f, the derivative of its accumulation from a constant endpoint to x is f(x). A changing endpoint adds a chain-rule factor.
Differential equations and initial values
Use slope relationships, separation and initial conditions to identify a solution.
Area, average value and volume
An application of integration starts with a thin slice or a rate model and the units of what it accumulates.
Limits from evidence
A limit is an inference from nearby behavior, and different kinds of evidence justify different inferences. A conjugate can reveal an algebraic cancellation; two graph branches can establish incompatible approaches; an inequality can control a rapidly oscillating term. None of these arguments is replaced by reading an isolated filled dot. This lesson adds evidence selection to the earlier substitution exercises.
Derivative definitions and structure
A derivative is a limit of a quotient with a particular input increment. A table provides finite secants, not automatically exact derivatives. A piecewise formula must match both values and one-sided slopes to be differentiable. These tasks connect the definition to calculations that can otherwise look like unrelated rules. The final family differentiates an implicit relation twice, where the product rule remains essential.
Trigonometric and inverse-trigonometric rates
Trigonometric derivatives depend on radians and on the inner function. The derivative of tan is sec², while the derivative of sec is sec·tan; these are different outer rules even though both involve secant. Inverse-trigonometric functions have another meaning: arcsin undoes sine on a chosen interval, whereas 1/sin is a reciprocal. Identify the function before choosing a derivative.
Choose and justify a theorem
A theorem is a conditional statement, not a keyword attached to a calculation. The Mean Value Theorem relates a secant slope to an interior derivative under continuity and differentiability assumptions. Rolle’s theorem is its equal-endpoint special case. L’Hôpital’s rule addresses certain indeterminate quotients. Choosing a theorem means verifying its hypotheses and identifying precisely what its conclusion guarantees.
Signs, extrema and concavity
Derivative information describes changes in a function without necessarily giving its height. A sign change from positive to negative in f′ marks a local maximum of f; the opposite marks a local minimum. A sign change in f″ concerns concavity. Keeping the function, first derivative and second derivative separate prevents a zero or sign from being interpreted at the wrong level.
Related rates with geometric constraints
In related rates, the geometry is a relationship among quantities that change with time. Draw the quantities, write the constraint, differentiate with respect to time, and only then substitute the instantaneous measurements. In optimization, the constraint instead reduces an objective to a function of one independent variable. Both techniques depend on preserving the constraint while a quantity changes.
Substitution with consistent variables
Substitution reverses a chain-rule structure. A useful choice of u identifies an inner expression and a differential factor proportional to its derivative. Rewriting only the inner expression while leaving unrelated x-factors behind is not a completed substitution. For a definite integral, the endpoint labels must refer to the same variable as the integrand and differential.
Integral properties and sampled data
An integral combines signed contributions. Linearity distributes constant coefficients, while interval additivity joins adjacent intervals and reversal changes orientation. A sampled-data estimate adds height times width on each chosen interval. These structures remain valid even when the sample spacing is unequal, but shortcuts based on a uniform mesh no longer apply.
Reason about differential equations
A differential equation constrains the derivative of a whole solution. Checking a candidate requires differentiating it and substituting it into both sides on a stated domain. Solving by separation introduces an integration constant and may restrict the interval on which the formula is valid. Initial data selects a solution, while a singularity can limit how far that solution continues.
Areas across intersections and sideways slices
Area between curves is accumulated nonnegative gap. If curves cross, the identity of the upper curve can change, so a single signed integral may cancel part of the desired area. Horizontal slices replace upper-minus-lower with right-minus-left and use dy. Choosing the slice direction is a modeling decision that should make the boundaries and intersections manageable.
Washers and perpendicular cross sections
Volume problems begin with a cross-sectional area, not an automatic formula containing π. Rotation can create disks or washers; prescribed perpendicular sections can instead be triangles, squares or semicircles. The base diagram determines the relevant side or diameter. A correct volume integral multiplies area by a small thickness along the slice direction.
Analyze an accumulated function
An accumulation function can be studied even when its integrand has no convenient antiderivative. The FTC gives a first derivative from the rate, and a second derivative from the rate’s slope when that slope exists. A piecewise rate controls monotonicity of the accumulated amount. Moving endpoints and running averages add chain- or quotient-rule terms beyond the basic FTC form.
Foundation pilot: 400 parameterized original questions across 96 task families, not 400 independently designed scenarios.
A clear route through the material
Follow the topics in order or use the prerequisites to choose a starting point. Read the worked example before focused practice, then use review cards to revisit the idea.
Continue your preparation
Scope: Calculus AB foundation practice