I. Introduction

Calculus I is where you learn the two major tools of calculus: limits and derivatives. You’ll use them to model real systems—motion, growth/decay, optimization, and rates, and will introduce integration as the tool for accumulation and area.

II. Outline

  • Functions & Graphs
    • Function notation, domain/range, transformations, inverse/composition
    • Exponential, logarithmic, and trigonometric functions
    • Reading graphs: intercepts, symmetry, end behavior, piecewise functions
  • Limits & Continuity
    • Limit intuition and limit laws
    • One-sided limits, infinite limits, limits at infinity
    • Algebraic techniques, trig limits
    • Continuity and types of discontinuities
    • Intermediate Value Theorem
  • Derivative Definition & Rules
    • Derivative as a limit; tangent slope and instantaneous rate of change
    • Differentiation rules: power, product, quotient, chain rule
    • Derivatives of trig, exponential, and logarithmic functions
    • Implicit differentiation
    • Higher-order derivatives and interpretation
  • Applications of Derivatives
    • Linearization and differentials
    • Related rates & Optimization
    • Curve sketching toolkit: critical points, increasing/decreasing, concavity, inflection
    • Mean Value Theorem
  • Integrals: The Start of Accumulation
    • Antiderivatives and indefinite integrals
    • Riemann sums and definite integrals
    • Fundamental Theorem of Calculus
    • Numerical integration basics or Newton’s method
    • Basic integration rules + u-substitution
  • Applications of Integration
    • Area between curves (basic setups)
    • Net change / accumulation from a rate
    • Average value of a function
    • Motion from acceleration/velocity via integration

III. Free Books

IV. Video Series

V. See Also