Practice Local vs Global Behavior in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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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.

Showing a random 20 of 50 problems.

Example 1

medium

Use the local approximation

\ln(1 + x) \approx x

near

x = 0

to estimate

\ln(1.05)

Example 2

easy

Near

x = 0

\cos(x) \approx 1 - \tfrac{x^2}{2}

. Is this a local or global statement?

Example 3

easy

For

f(x) = \sin(x)

, what are the global maximum and minimum values?

Example 4

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

Example 5

hard

For

f(x) = xe^{-x}

[0, \infty)

, find the global maximum.

Example 6

hard

A function

f

satisfies

f(0) = 10

and has local minima at

x = 1

and

x = 5

with values

f(1) = 2

and

f(5) = -3

. What is the global minimum on

\mathbb{R}

, assuming end behavior

f(x) \to +\infty

x \to \pm \infty

Example 7

medium

f(x) = \sin x

has period

2\pi

. Describe its local behavior near

x = 0

versus its global behavior over all reals.

Example 8

easy

f(x) = -x^2 + 10

has a peak at

x = 0

with value 10. Is this peak a local maximum, a global maximum, or both?

Example 9

easy

Zooming way in on the graph of

f(x) = x^2

x = 2

, it looks almost like a straight line. What is this called?

Example 10

easy

Does knowing

f(0) = 3

tell you the global maximum of

f

Example 11

medium

A model fits data well for

0 \le x \le 10

but predicts a negative population at

x = 50

. Is the failure a local or global (extrapolation) problem, and what is the lesson?

Example 12

easy

True or false: a local maximum is always the global maximum.

Example 13

hard

[1, 4]

, find the global max and min of

f(x) = x + \dfrac{4}{x}

Example 14

challenge

A function is locally increasing at every point in

(0, 10)

yet you are told it is NOT globally increasing on the reals. Construct a simple example and explain the apparent paradox.

Example 15

easy

Which is global: 'slope is

-2

x = 3

' or 'function approaches a horizontal asymptote

y = 0

Example 16

medium

[-3, 3]

, find the global max and min of

f(x) = x^3 - 12x

Example 17

medium

Near

x = 0

f(x) = e^x \approx 1 + x

. Use the local approximation to estimate

e^{0.1}

, and explain why it fails for

x = 5

Example 18

medium

f(x) = x^4 - 4x^2

(slope

4x^3 - 8x

). It has critical points at

x = 0, \pm\sqrt{2}

. Given

f(0)=0

and

f(\pm\sqrt2) = -4

, identify the global minimum.

Example 19

easy

On the closed interval

[0, 4]

f(x) = x

has a global maximum of what value?

Example 20

easy

Does

f(x) = e^x

have a global maximum on

\mathbb{R}

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Related Concepts

Function