Symbolic Differentiation, Numerical Quadrature & Series Workbench

Applies symbolic Product and Chain rules step-by-step, locating stationary extrema at x = 0 and x = 4 plus concavity inflections.

Derives exact rational first and second derivatives, curvature κ(x₀), and bounded global extrema without pole singularities.

Finds all stationary maxima, minima, roots, and concavity sign-flips across [−π, π] with live tangent line visualization.

Classic integration-by-parts identity where terms x/(1+x²) cancel algebraically to leave f′(x) = arctan(x) and f″(x) = 1/(1+x²).

Evaluates ∫₋₂² e^(−x²) dx = √π·erf(2) ≈ 1.76416278 across 6 quadrature rules, revealing exponential accuracy gain.

Shades and integrates the famous ±1.96σ two-tailed statistical threshold to 7-digit Gauss–Legendre precision.

Rapidly chirping optical diffraction integral testing midpoint/trapezoid cancellation errors against 192-node Gauss–Legendre quadrature.

Simultaneously verifies area = 2π ≈ 6.283185, arc circumference = 2π, and solid-of-revolution sphere volume V = (4/3)π(2)³ ≈ 33.5103.

Classic benchmark integral evaluating to exactly 1, illustrating concave-down trapezoidal underestimation vs. midpoint overestimation.

Computes Si(π) ≈ 1.85193705, exceeding π/2 ≈ 1.570796 by the exact 8.95% Fourier Gibbs overshoot ratio.

Overlays odd-degree alternating Maclaurin approximations onto sin(x), showing how each added term hugs another half-wavelength.

Every derivative f^(k)(0) = 1 generates clean reciprocal-factorial coefficients converging across all real x.

Visually demonstrates why Taylor polynomials match ln(1+x) tightly inside (−1, 1] yet diverge wildly for x > 1 due to the pole at x = −1.

Canonical geometric expansion with unit coefficients cₖ = 1 bounded strictly by the vertical asymptote at x = 1.

Synthesizes odd sine harmonics 1, 3, 5, … up to N = 25, measuring RMS energy error and persistent 9% jump-discontinuity ringing.

Contains both odd and even integer harmonics decaying as 1/n, foundational to subtractive audio synthesizer waveforms.

Continuity of the triangle wave accelerates Fourier decay to 1/n², reaching near-visual perfection in just 5 harmonics.

Extracts DC average a₀/2 = 2/π ≈ 0.63662 plus even ripple harmonics cos(2nx)/(4n² − 1) for AC-DC filter capacitor design.