Local vs Global Behavior Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

hard

The function

f(x) = e^{-x^2}

appears bell-shaped. Describe its local behavior at

x=0

(using the second-order Taylor expansion) and global behavior as

x\to\pm\infty

Solution

1
Taylor expansion: $e^{-x^2}=1-x^2+\frac{x^4}{2}-\cdots\approx1-x^2$ near $x=0$ . So locally it resembles an inverted parabola with maximum at $(0,1)$ .
2
Global: as $x\to\pm\infty$ , $-x^2\to-\infty$ , so $e^{-x^2}\to0$ . The horizontal asymptote is $y=0$ . The function decays to zero faster than any polynomial.

Answer

Local at

x=0

f(x)\approx1-x^2

(inverted parabola); Global:

f(x)\to0

x\to\pm\infty

The Gaussian bell curve has a flat maximum locally (looks parabolic) but decays rapidly to zero globally. This super-exponential decay is why it integrates to a finite value despite being defined on all of

\mathbb{R}

About Local vs Global Behavior

Local behavior describes a function's properties near a specific point; global behavior describes its overall properties across the entire domain or as inputs grow without bound.

Learn more about Local vs Global Behavior →

More Local vs Global Behavior Examples

Example 1 easy

For [formula], describe: (a) local behavior near [formula] using the linear approximation, and (b) g

Example 2 medium

For [formula] (with [formula]), describe local behavior near [formula] and global behavior for large

Example 3 easy

For [formula], which term dominates (a) near [formula] and (b) for [formula]?

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Related Concepts

Function