Practice Local vs Global Behavior in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
Quick Recap
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.
Local is "zoom in on one spot"; global is "zoom out to see the whole picture." Near x = 0, \sin(x) \approx x (local linear approximation), but globally it oscillates forever.
Example 1
easyFor 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.
Example 2
mediumFor g(x) = \dfrac{\sin(x)}{x} (with g(0)=1), describe local behavior near x=0 and global behavior for large |x|.
Example 3
easyFor p(x) = x^4 - 100x^2 + 1, which term dominates (a) near x=0 and (b) for |x| = 1000?
Example 4
hardThe 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.