Turn a function into engineering insight: visualize slope, curvature, roots, extrema, inflection points, tangent sensitivity, accumulated area, and bounded numerical optima.
Interactive function viewDrag to pan • wheel/pinch to zoom • double-click/tap to reset
f(x)f′(x)f″(x)Tangent
Roots
—
f(x) = 0 within analysis range
Stationary points
—
f′(x) = 0 within analysis range
Integral
—
Signed definite integral
Absolute area
—
Numerical area between curve and x-axis
Bounded minimum
—
Run numerical optimization
Bounded maximum
—
Run numerical optimization
Function interpretation
f(x) = —
f′(x) = —
f″(x) = —
Critical points
Type
x
f(x)
f′(x)
f″(x)
Interpretation
Run analysis to calculate critical points.
Tangent & local sensitivity
Choose an x-value to evaluate local slope, curvature, and tangent behavior.
Engineering / optimization interpretation
The visualizer compares function value, slope and curvature so you can distinguish where a system is increasing, decreasing, flattening, changing curvature, or reaching a bounded optimum.
Visualize functions with derivatives, roots, extrema, tangents, integrals and bounded numerical optimization in an engineering-focused calculator.
What this tool does
This interactive calculus tool turns a single-variable expression into a graph of the function, first derivative and second derivative. It identifies numerical roots, stationary points, local extrema and inflection points, then adds tangent-line, integral/area and bounded optimization views.
Who this is for
Engineers and analysts exploring response, cost, production, reliability or calibration functions
Students and educators learning derivatives, curvature, integration and optimization
Designers checking sensitivity and bounded operating-point behavior before a detailed model review
How to use this tool
Enter a supported function of x or load a Cubic, Quartic, Oscillation, Response or S-curve preset
Set the analysis range and run Analyze Function
Review the function, derivative curves, critical-point table and engineering interpretation
Enter integral bounds for signed integral and absolute area
Choose a tangent x-value to inspect local slope and curvature, or set bounds to find a numerical minimum or maximum
Toggle display layers and reset the graph view when needed
Inputs
Inputs are a single-variable mathematical expression, x-range, integral bounds, tangent evaluation point, bounded minimum/maximum limits and display toggles for the plotted layers.
Outputs
Outputs include an interactive graph, first and second derivative curves, roots, stationary points, maxima, minima, inflection points, tangent equation and sensitivity, signed integral, absolute area and bounded numerical optimum values.
Limitations and responsible use
Derivative and optimization results are numerical and depend on the expression, range and sampling resolution. Discontinuities, repeated roots, narrow features and highly oscillatory functions may need a tighter range or independent verification.
Example use cases
Enter a supported function of x or load a Cubic, Quartic, Oscillation, Response or S-curve preset
Toggle display layers and reset the graph view when needed
Frequently asked questions
Does it calculate derivatives symbolically?
When supported by the math.js expression engine, it uses symbolic first and second derivatives and numerical evaluation where needed for plotting and analysis.
Does Find Minimum search outside my bounds?
No. The numerical optimizer searches the interval you provide, so the result is a bounded candidate rather than a global optimum over all possible x values.
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