Local vs Global Behavior Math Example 1

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

Example 1

easy

For

f(x)=x^3-3x

, describe: (a) local behavior near

x=0

using the linear approximation, and (b) global behavior as

x\to\pm\infty

Solution

1
Local at $x=0$ : $f'(x)=3x^2-3$ , $f'(0)=-3$ . Linear approximation: $f(x)\approx f(0)+f'(0)\cdot x = 0+(-3)x=-3x$ near $x=0$ . Locally the function looks like a line with slope $-3$ .
2
Global behavior: for large $|x|$ , the $x^3$ term dominates. $f(x)\to+\infty$ as $x\to+\infty$ ; $f(x)\to-\infty$ as $x\to-\infty$ .
3
Contrast: locally (near $x=0$ ) the function decreases; globally it grows without bound in both directions.

Answer

Local near

x=0

f(x)\approx-3x

(decreasing line); Global:

f(x)\to\pm\infty

with

x^3

dominance

Local behavior captures the function's character near a specific point, often described by a tangent line. Global behavior describes what happens as

x

goes to infinity, usually governed by the highest-degree term.

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 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]?

Example 4 hard

The function [formula] appears bell-shaped. Describe its local behavior at [formula] (using the seco

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Related Concepts

Function