The math of change and accumulation
Calculus answers two questions: how fast is something changing right now, and how much has accumulated so far? Derivatives handle the first, integrals the second — and the Fundamental Theorem ties them together.
Limits
The value f(x) approaches as x → a. Limits make “just before” and “just after” precise even when f(a) doesn't exist.
limx→2 (x² − 4) = 0.
Derivatives
The instantaneous rate of change of f at x; the slope of the tangent line. Defined as the limit of slopes of secant lines as the gap shrinks.
d/dx[xⁿ] = n·xⁿ⁻¹
Integrals
The accumulated area under f between two values of x. Approximated by Riemann sums; computed exactly via antiderivatives.
∫xⁿ dx = xⁿ⁺¹/(n+1) + C
Fundamental Theorem
If F'(x) = f(x), then ∫ₐᵇ f(x) dx = F(b) − F(a). Differentiation and integration are inverse operations.
Headline rules
Hands-on tools
Slide a tangent, watch a Riemann sum converge, and animate position-velocity-acceleration.
Limit explorer
Pick a function and a target. See f(x) from both sides as x approaches.
Tangent slider
Pick a function. The tangent line is the limit of the secant as h → 0.
Riemann sum visualizer
Approximate ∫ₐᵇ f(x)dx with rectangles. More slices → better.
Position · velocity · acceleration
An object's position is plotted; its derivative is velocity; the second derivative is acceleration.
Quiz
Flashcards
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For teachers
Print-ready worksheet, answer key, teaching tips and standards alignment.
Teaching tips
Standards alignment
Reference
Formula sheet
Photo gallery
Images sourced from Wikipedia.