CAL-00 · laboratory
Derivative, Taylor, FTC, trapezoid rule and the Jacobian.
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26 governing sheets
d/dx x^n = n x^{n−1}. Instantaneous slope of a monomial.
Derivatives(uv)' = u' v + u v'. Snapshot at a point.
Derivatives(f∘g)' = f'(g) g'. Composition of rates.
DerivativesT2(x) = f + f' h + ½ f'' h², h = x − a.
Series∫_a^b F' = F(b) − F(a). Net change of an antiderivative.
Integrals∫_a^b f ≈ (b−a)(f(a)+f(b))/2. One panel of the composite rule.
Quadrature∫ ≈ (b−a)/6 (f(a)+4f(m)+f(b)), m midpoint. Parabolic panel.
Quadraturef'(c) = (f(b)−f(a))/(b−a). A slope that matches the chord.
Theoremslim f/g = lim f'/g' when 0/0 or ∞/∞ and the latter limit exists.
LimitsF(x,y) = c ⇒ dy/dx = −Fx / Fy. Slope of a level curve.
Derivativesv = √(ẋ² + ẏ²). Speed along a plane curve.
Parametricκ = |x' y'' − y' x''| / (x'² + y'²)^{3/2}. Turning of a parametric curve.
CurvesJ = ∂(u,v)/∂(x,y) = ux vy − uy vx. Area scale of a map.
Multivariable|∇f| = √(fx² + fy² + fz²). Steepest-ascent rate.
Multivariablediv F = ∂Px/∂x + ∂Py/∂y + ∂Pz/∂z. Source density of a flux.
Vector calculusΔu = u''. Second derivative as a 1D Laplacian.
Vector calculus∫ x^n dx = x^{n+1}/(n+1) + C, n ≠ −1.
Integrals∫_0^x e^{kt} dt = (e^{kx} − 1)/k for k ≠ 0.
IntegralsL ≈ √(1 + (y')²) Δx. One-panel arc of y = f(x).
CurvesdA = r² sin θ dθ dφ. Solid-angle Jacobian on a sphere.
Surfaces(u/v)' = (u' v − u v') / v².
Derivatives∫_a^b f ≈ (b−a) f((a+b)/2). Rectangle of midpoint height.
QuadratureElementary derivative.
CalculusLocal parabola.
CalculusOne step.
CalculusArea element r dr dθ.
Calculus