AP Calculus AB Unit 5 Notes: Using Derivatives to Understand Function Behavior

Increasing (on an interval)

A function is increasing on an interval if for any $a < b$ in the interval, $f(a) < f(b)$ (outputs rise as x moves left to right).

Decreasing (on an interval)

A function is decreasing on an interval if for any $a < b$ in the interval, $f(a) > f(b)$ (outputs fall as x moves left to right).

First derivative (f′)

The derivative that gives the instantaneous rate of change of f (the slope of the tangent line).

Derivative sign test for increasing/decreasing

If $f'(x) > 0$ on an interval, f is increasing there; if $f'(x) < 0$ on an interval, f is decreasing there.

Horizontal tangent (f′(c)=0)

A point where the tangent slope is zero; it does not automatically mean a local maximum or minimum.

Critical number

An x-value in the domain of f where either f′(x)=0 or f′(x) does not exist.

Test intervals

Intervals formed by splitting the domain at critical numbers (and where f′ is discontinuous), used to test the sign of f′ or f′′.

Sign chart

A chart/analysis showing where a derivative is positive or negative on each test interval to determine behavior (increase/decrease or concavity).

Interval notation (for behavior)

Increasing/decreasing and concavity are stated on intervals (e.g., (1,3)), not “at a point.”

Domain restriction

A limitation where f is not defined at certain x-values; you cannot claim increasing/decreasing or concavity “through” a point not in the domain.

Endpoint (closed interval context)

An endpoint of [a,b] lacks a two-sided neighborhood inside the interval but must still be considered for absolute extrema on [a,b].

Local maximum

A point x=c where f(c) is greater than nearby function values.

Local minimum

A point x=c where f(c) is less than nearby function values.

First Derivative Test

Classifies a critical number c by how $f'$ changes sign around c: $+ \rightarrow -$ gives a local max; $- \rightarrow +$ gives a local min; no sign change gives no local extremum.

No sign change (First Derivative Test outcome)

If $f'$ is the same sign on both sides of a critical number, f has no local maximum/minimum there (it may just flatten).

Concave up

The graph bends like a cup; slopes tend to increase as x increases. Occurs where $f''(x) > 0$.

Concave down

The graph bends like a cap; slopes tend to decrease as x increases. Occurs where $f''(x) < 0$.

Second derivative (f′′)

The derivative of $f'$; measures how the slope changes and is used to determine concavity.

Point of inflection

A point on the graph where concavity changes (from up to down or down to up).

Inflection point candidate

An x-value where $f''(x)=0$ or $f''(x)$ does not exist; it is only an inflection point if concavity actually changes.

Concavity sign test (using f′′)

To find concavity, split the domain at where f′′=0/undefined and test the sign of f′′ on each interval.

Tangent line position and concavity

If f is concave up, tangent lines tend to lie below the graph; if f is concave down, tangent lines tend to lie above the graph.

Second Derivative Test

At a critical point c with $f'(c)=0$ and $f''(c)$ existing: $f''(c) > 0$ implies local min; $f''(c) < 0$ implies local max; $f''(c)=0$ is inconclusive.

Inconclusive (Second Derivative Test)

When f′′(c)=0 (or the needed condition fails), the Second Derivative Test cannot decide; another method (often the First Derivative Test) is needed.

Derivative graph interpretation (f′ graph)

Where f′ is above the x-axis, f is increasing; where f′ is below, f is decreasing; where f′ crosses the x-axis, f′ changes sign and f may have a local extremum.