Ten practical guides
Calculus Concepts for AP and Early College
These pages focus on core definitions, setup habits, and worked examples so you can solve standard calculus problems with clear reasoning.
Calculus topics are foundational for AP preparation, STEM entry courses, and quantitative fields that rely on modeling change and accumulation.
Limits and continuity
Learn how limits describe nearby behavior, how to test continuity, and how to handle common forms like piecewise and removable discontinuities.
Derivative definition
Use the limit definition of the derivative to compute slopes and interpret derivative values as instantaneous rate of change.
Derivative rules
Apply derivative rules by matching expression structure, including constant, power, sum, product, and quotient forms, then simplify clearly.
Chain rule
Differentiate composite functions by identifying inner and outer layers, applying the chain rule in order, and preserving signs and factors.
Implicit differentiation
Differentiate equations containing both x and y by treating y as a function of x, applying chain rule to y terms, and isolating dy/dx accurately.
Related rates
Translate geometry or physics constraints into equations, differentiate with respect to time, and solve for unknown rates.
Optimization
Solve constrained optimization problems by writing one-variable objective functions, finding critical points, and testing candidates.
Integrals and FTC
Connect antiderivatives with accumulation and evaluate definite integrals efficiently using the Fundamental Theorem of Calculus.
u-substitution
Use substitution to simplify integrals by replacing an inner expression and its differential, then integrate in the new variable.
Sequences and series
Understand sequence limits, partial sums, geometric and p-series behavior, and common convergence tests used in introductory calculus.