Why are Taylor polynomial approximations considered local rather than global?

Conditions

• A center point a must be chosen.
• Approximation quality is strongest near a.

Reasoning, step by step

1. Recognize that the coefficients of the Taylor series depend entirely on the derivatives of the function evaluated at the center point a.
2. Observe the graphical behavior of the approximating polynomials near the center, where they closely track the original function.
3. Observe the graphical behavior far from the center, where the polynomial diverges significantly (e.g., a quadratic dropping to negative infinity while a cosine remains bounded).
4. Conclude that the method extracts local information at a point and converts it into local information nearby, rather than providing a global equivalent.

Example

The speaker emphasizes that the computation only depended on the derivative exactly at $a=0$, yet the graph gives a good approximation nearby. Far from 0, the quadratic drops off to minus infinity and is a terrible approximation for the bounded cosine function.

Common misconceptions

• Assuming a low-degree Taylor polynomial works well everywhere because it matches perfectly at the center.
• Treating the approximation symbol as an exact equality for all x, rather than a local relationship near the expansion point.

Watch the explanation

Explore next

• What happens to the Taylor approximations of $\cos (x)$ when the center changes from 0 to pi?
• What are the first and second Taylor polynomial approximations to $\cos (x)$ at $a=0$ shown in the graph?