Advanced Graphing

Mathematical Principles & Theorems

Analyzes real-valued functions \(y = f(x)\). First derivative \(f'(x) = 0\) identifies critical points (local maxima, minima, saddle points). Second derivative \(f''(x) > 0\) indicates convex concavity up; \(f''(x) < 0\) concave down; \(f''(x) = 0\) locates inflection points. The tangent line at \((x_0, y_0)\) is \(y - y_0 = f'(x_0)(x - x_0)\). Asymptotic behavior evaluated via limits \(\lim_{x \to \pm\infty} f(x)\) and vertical singularities.

Operating Instructions

• Enter mathematical function expressions (e.g., \(x^3 - 3x\), \(\sin(2x)e^{-x/2}\), \(\frac{x^2-1}{x^2+1}\)).
• Configure the domain range \([x_{\min}, x_{\max}]\) and grid resolution.
• Hover over the curve to inspect precise coordinate readouts, instantaneous slopes, and tangent lines.
• Click Analyze to automatically compute roots (zero-crossings), local extrema, and inflection points.