Unit 5: Function Analysis Using Derivatives

First Derivative

Represents the instantaneous rate of change or slope of the tangent line of a function f(x).

Critical Number

A number c in the domain of f where f'(c) = 0 or f'(c) is undefined.

Increasing Function

If f'(x) > 0 for all x in an interval (a, b), then f is increasing on [a, b].

Decreasing Function

If f'(x) < 0 for all x in an interval (a, b), then f is decreasing on [a, b].

Constant Function

If f'(x) = 0 for all x in an interval, then f is constant.

Sign Chart

A chart used to determine the sign of f'(x) in intervals defined by critical numbers.

Relative Maximum

A point where f' changes from positive to negative at a critical number c.

Relative Minimum

A point where f' changes from negative to positive at a critical number c.

Point of Inflection (POI)

A point where the graph changes concavity and f''(c) = 0 or is undefined.

Concave Up

When f''(x) > 0 on an interval, indicating the graph is shaped like a cup (holding water).

Concave Down

When f''(x) < 0 on an interval, indicating the graph is shaped like a frown (spilling water).

Second Derivative Test

A method to classify critical points as relative max/min based on the sign of f'' at those points.

Inconclusive Result

When f''(c) = 0, meaning the Second Derivative Test fails and requires further testing.

Local Maximum

Occurs at x=c if f'(x) changes from positive to negative.

Local Minimum

Occurs at x=c if f'(x) changes from negative to positive.

Endpoints in Optimization

Endpoints can be candidates for absolute extrema, but derivatives tests find local extrema only.

Derivative Undefined

Critical points can occur where f'(x) is undefined, such as at sharp corners or cusps.

Cusp Example

Critical points like f(x) = |x| may have relative extrema at points where f' is undefined.

Function Behavior Analysis

Using f'(x) and f''(x) to determine increasing/decreasing and concavity of functions.

First Derivative Test

Determines the nature of a critical point based on the sign change of f' around that point.

Second Derivative Concavity Rule

f''(c) > 0 indicates concave up; f''(c) < 0 indicates concave down.

Critical Points Function Change

Critical numbers are where a function may change direction, influencing its behavior.

Behavior Interpretation

The behavior of f(x) can be interpreted through its first and second derivatives.

Absolute vs Local Extrema

Absolute extrema can occur at endpoints while local extrema are defined through derivative tests.

Testing Intervals

Intervals are tested using values to determine the sign of the first derivative.

Justification Language

Be specific in justifying extrema; avoid vague phrases when explaining changes in behavior.

Cup and Frown Shape Analogy

Use the cup (concave up) and frown (concave down) analogy to remember the behavior of second derivatives.