Quadratic Taylor Models and Curvature

From a Tangent Line to a Quadratic Model

A tangent line gives a function’s value and slope at a chosen point. It is the simplest local model that matches both. But a graph can bend away from its tangent line, and a straight line cannot represent that bending. A quadratic polynomial can: its squared term records the function’s second-derivative information at the center.

This tutorial develops the degree-two Taylor polynomial and compares it with the tangent-line approximation. The focus is local: the polynomial is built to match the function’s value, first derivative, and second derivative at one point. How well it approximates the function nearby depends on appropriate smoothness conditions.

Definition of the Degree-Two Taylor Polynomial

Let \(f\) be a function with a first and second derivative at a point \(a\). The degree-two Taylor polynomial of \(f\) centered at \(a\) is the quadratic polynomial whose value and first two derivatives agree with those of \(f\) at \(a\).

The terms have distinct roles. The constant term makes \(P_2(a)=f(a)\). The linear term supplies the slope \(f'(a)\). The quadratic term supplies the second derivative \(f''(a)\), because differentiating \(\frac{f''(a)}{2}(x-a)^2\) twice gives \(f''(a)\).

The tangent-line approximation, also called the linearization, is \(L(x)=f(a)+f'(a)(x-a)\). Thus the degree-two polynomial keeps the tangent line and adds one correction:

This comparison is the main point: the quadratic model and the tangent line have the same value and slope at the center, while the added term describes the bending captured by the second derivative there.

Worked Example: A Quadratic Function

What the Quadratic Term Says

Since \((x-a)^2\) is positive whenever \(x\ne a\), the sign of the quadratic correction is determined by \(f''(a)\). If \(f''(a)>0\), then \(P_2(x)\) lies above its tangent line for every \(x\ne a\). If \(f''(a)<0\), then \(P_2(x)\) lies below its tangent line. If \(f''(a)=0\), the degree-two polynomial is the tangent line; that fact alone does not tell us what the function does beyond its second-order information at \(a\).

This comparison describes the polynomial model. To draw conclusions about the function’s concavity throughout a neighborhood, we need information about the second derivative throughout that neighborhood. For example, if \(f''(x)>0\) throughout an interval, then \(f'\) is increasing there; if \(f''(x)<0\) throughout an interval, then \(f'\) is decreasing there. A sign at just one point does not establish either neighborhood claim. If \(f''\) is continuous at \(a\) and \(f''(a)>0\), continuity does ensure \(f''(x)>0\) on some interval around \(a\); the corresponding statement holds when \(f''(a)<0\).

Here, “curvature” refers to the bending indicated by the second derivative in the quadratic model. It is not a claim that \(f''(a)\) is always the geometric curvature of the graph: for a graph \(y=f(x)\), geometric curvature involves both \(f''(x)\) and \(f'(x)\).

Why the Formula Works

The coefficients are chosen so that the polynomial and the function agree in their first three pieces of derivative data at the center. This gives a direct justification of the formula.

These three requirements also determine the coefficients uniquely. Suppose a quadratic polynomial has the form \(Q(x)=A+B(x-a)+C(x-a)^2\) and matches \(f\), \(f'\), and \(f''\) at \(a\). Then \(Q(a)=A=f(a)\), \(Q'(x)=B+2C(x-a)\) gives \(B=Q'(a)=f'(a)\), and \(Q''(x)=2C\) gives \(C=f''(a)/2\). Therefore \(Q=P_2\).

Matching derivatives explains how to construct the polynomial. To justify its use as an approximation, a further condition is helpful. Suppose \(f''\) is continuous on an interval containing \(a\) and \(x\). The Fundamental Theorem of Calculus, applied twice, gives

Subtract the proposed quadratic polynomial from both sides. Since \(\int_a^x(x-t)\,dt=(x-a)^2/2\), this gives the exact error identity

To see what this says near \(a\), let \(M\) be the largest value of \(\lvert f''(t)-f''(a)\rvert\) between \(a\) and \(x\). Taking absolute values and integrating the absolute value of the linear factor gives

As \(x\) approaches \(a\), continuity of \(f''\) makes \(M\) approach zero. Thus the error divided by \((x-a)^2\) approaches zero: the error is smaller than a fixed multiple of \((x-a)^2\) sufficiently close to \(a\). This argument explains both the role of the quadratic term and a sufficient condition for \(P_2\) to be a valid local approximation. Merely matching derivative values at a point should not be confused with establishing a particular error bound.

Worked Example: A Negative Second Derivative

Worked Example: Comparing Models for a Reciprocal

Worked Example: A Cubic Function

A Common Mistake: Overstating What One Value Tells You

A common error is to infer the behavior of \(f'\) on an interval from the sign of \(f''\) at a single point. For example, the calculation \(f''(a)>0\) does not, by itself, prove that \(f'\) is increasing throughout a neighborhood of \(a\). The second derivative might be negative at other points arbitrarily close to \(a\).

Another useful distinction is between a model’s position relative to its tangent and the accuracy of that model. The sign of \(f''(a)\) tells us whether the quadratic correction raises or lowers the tangent-line model. It does not, on its own, give a numerical error bound for the actual function. The error identity above supplies such control when \(f''\) is continuous.

When the Quadratic Model Is Useful

The degree-two Taylor polynomial improves on a tangent line when second-derivative information matters over a small range around the center. It can capture a local bend that a straight-line model necessarily misses. Its usefulness in a specific calculation depends on how close \(x\) is to \(a\), the behavior of \(f''\) between those points, and the accuracy required.

The essential comparison is simple: both models share the function’s value and slope at the center, and the quadratic model adds \(\frac{f''(a)}{2}(x-a)^2\). That correction is zero at the center and grows with the square of the distance from it. Its sign describes the quadratic model’s bending relative to the tangent; additional conditions are needed for stronger claims about the function nearby.