How this calculation works
Differentiate the polynomial analytically, evaluate its first two derivatives at the point and build the local tangent. A nonzero tangent slope gives normal slope −1/f′(a); a horizontal tangent has a vertical normal. Compare the tangent’s finite-offset prediction with the actual polynomial.
Tangent: y=f(a)+f′(a)(x−a); curvature κ=|f″(a)|/(1+f′(a)²)^(3/2)A worked example
For y=x² at x=1, the tangent slope is 2 and tangent is y=1+2(x−1). At x=1.1 it predicts 1.2, while the actual polynomial gives 1.21. The normal is y−1=−0.5(x−1).
Good to know
Univariate polynomial degree ≤12. Numerical graph curvature assumes the stated Cartesian units; it is not a physical bending limit. The tangent is locally accurate, not a global replacement. A polynomial graph has no finite vertical tangent.
Reference: OpenStax Calculus: arc length and curvature (opens in a new tab) ↗
A tangent is a local prediction
At a point a, the tangent retains the polynomial’s value and first derivative. It estimates a nearby value with f(a)+f′(a)Δx. It is exact for a linear polynomial and generally approximate for a curved one.
For x² at a=1, the slope is 2. An offset of 0.1 predicts 1.2, while the polynomial gives 1.21. Reducing the offset often makes the local approximation more useful, but the displayed comparison should decide how much error matters for your task.
The interval extrema calculator compares a whole closed interval; a tangent and curvature describe one point and its local neighborhood.
A normal is perpendicular to the tangent
A finite nonzero tangent slope m has normal slope −1/m in matching Cartesian coordinates. When the tangent is horizontal, that slope formula would divide by zero; the normal is instead the vertical line x=a.
The curvature calculation adds the second derivative. For a graph y=f(x), unsigned curvature is |f″|/(1+f′²)^(3/2). It depends on coordinate units and scales, so the visual steepness of independently scaled plot axes is not a measurement of curvature.
Zero curvature is a statement at one point
A zero second derivative gives zero instantaneous graph curvature here. A cubic can have zero curvature at an inflection while bending on either side. It does not make the entire polynomial a straight line.
The Taylor approximation calculator extends the same derivative-matching idea to higher order for its supported functions. The interval extrema calculator instead checks endpoints and stationary candidates over a whole interval, rather than describing just one tangent.
A few questions, answered
Why is the normal vertical for a horizontal tangent?
Perpendicularity requires a vertical line when the tangent slope is zero; a finite reciprocal slope does not exist.
Does zero second derivative prove a maximum or minimum?
No. It may be an inflection or a higher-order stationary case. Inspect the function and the relevant neighborhood or interval.
Does the tangent remain accurate far from the point?
No such guarantee is made. Compare the finite-offset prediction with the actual polynomial and inspect the assumptions.