Limit
Describe approach behavior without requiring the input to equal the point being approached.
Limits · local change · accumulation · dynamics
Calculus
Calculus studies continuous change by moving between local and accumulated views. Limits make approach behavior precise, derivatives measure instantaneous change, integrals measure accumulated effect, and differential equations describe systems whose evolution is specified through rates of change.
Core spine
Two limiting processes reveal two complementary views of change.
Derivatives arise from shrinking an interval around a point. Integrals arise from refining a partition across an interval. The Fundamental Theorem is the bridge between them.
Derivative
Take a limit of average rates of change to obtain instantaneous rate and local linear behavior.
Integral
Take a limit of finite sums to define accumulated quantity across an interval or region.
Fundamental Theorem
Under suitable conditions, accumulation and instantaneous change are inverse views: differentiating accumulated area recovers the original rate, and integrating a derivative recovers net change.
Local change · derivative
Tangent explorer
f(x) = sin x
x
1.047
f(x)
0.866
f′(x)
0.500
For f(x) = sin x, the derivative is f′(x) = cos x. The dashed tangent line has that actual mathematical slope at the selected point.
Accumulated change · integral
Accumulation explorer
∫₀⁴ [5 − (x−2)²] dx
rectangles6
left sum
14.3704
exact
14.6667
error
-0.2963
The rectangles use left endpoints on [0, 4]. Refining the partition reduces the width of each rectangle and drives this Riemann sum toward the definite integral.
Primary branches · navigation
Choose the kind of change or accumulation the problem asks about.
How the subject expands
Change the domain, output, or unknown and calculus grows with it.
01One variable → many variablesReplace a single input direction with partial and directional derivatives, gradients, multiple integrals, constrained optimization, and geometry in higher-dimensional domains.
02Scalar function → vector fieldWhen outputs have direction as well as magnitude, divergence, curl, circulation, and flux connect local field behavior to integrals over boundaries.
03Known function → unknown trajectoryDifferential equations ask for the function itself when relations among values and derivatives are known, turning calculus into a language for dynamical systems.