AP Calculus AB Unit 5: First and Second Derivative Analysis (Monotonicity, Extrema, Concavity, Inflection Points)

Increasing function

A function f is increasing on an interval I if whenever x1 < x2 (both in I), then f(x1) < f(x2).

Decreasing function

A function f is decreasing on an interval I if whenever x1 < x2 (both in I), then f(x1) > f(x2).

First derivative f'(x)

f'(x) measures the instantaneous rate of change or the slope of the tangent line.

When is f increasing?

f is increasing on an interval if f'(x) > 0 on that interval.

When is f decreasing?

f is decreasing on an interval if f'(x) < 0 on that interval.

Constant function

A function is constant on an interval if f'(x) = 0 for all x in that interval.

Critical number

A critical number of function f is a number c in the domain of f such that either f'(c) = 0 or f'(c) is undefined.

Sign analysis

A process used to determine the sign of a derivative on intervals to find increasing/decreasing behavior.

Test interval

An interval created by critical numbers and domain breaks used to test the sign of the derivative.

Inflection point

A point where a function changes concavity.

Concave up

A function is concave up on an interval if the graph bends upward like a cup.

Concave down

A function is concave down on an interval if the graph bends downward like a cap.

Second derivative f''(x)

f''(x) measures the rate of change of slope, helping to determine concavity.

First Derivative Test

A method to classify local extrema by checking the sign of f' around critical points.

Local maximum

A point where f changes from increasing to decreasing, so f(c) is a local maximum.

Local minimum

A point where f changes from decreasing to increasing, so f(c) is a local minimum.

Second Derivative Test

A method that classifies local extrema using the sign of f'' at critical points.

Inconclusive case

When f''(c) = 0, the Second Derivative Test is inconclusive, requiring another method.

Tangent line

A straight line that touches the curve at a point but does not cross it in the immediate vicinity.

Domain break

A point in the domain of a function where the function is not defined.

Sign chart

A chart used to visualize the signs of a function's derivative over different intervals.

Velocity and turning points

When the velocity changes sign, the position of an object reaches a local maximum or minimum.

Terrace point

A point where the derivative does not change sign; it is neither a local max nor min.

Average rate of change

The change in function values divided by the change in input values over an interval.

Slope of tangent line

The slope at a specific point on a curve defined by the first derivative f'(x).

Candidate for inflection point

A point where f''(x) = 0 or f''(x) is undefined, needing sign change confirmation.