AP Calculus Chapter 3 Review

Vocabulary flashcards covering core AP Calculus Chapter 3 concepts, including critical numbers, the First and Second Derivative Tests, concavity, Points of Inflection, tangent lines, and the Mean Value Theorem.

Critical Numbers

The values in the domain of a function where $f'(x) = 0$ or $f'(x)$ is undefined.

Relative Minimum (First Derivative Test)

Occurs at a point where the first derivative $f'(x)$ changes sign from negative to positive ($f'(x)$ goes from to $+$).

Relative Maximum (First Derivative Test)

Occurs at a point where the first derivative $f'(x)$ changes sign from positive to negative ($f'(x)$ goes from $+$ to).

Extrema on a Closed Interval

Found by analyzing critical numbers and endpoints evaluated in the original function $f(x)$, where the biggest value is the absolute maximum and the smallest value is the absolute minimum.

Mean Value Theorem (MVT)

States that the instantaneous rate of change (derivative) is equal to the average rate of change (slope between two points) at some point between the two points.

Pre-requisites for MVT

The conditions required to apply the Mean Value Theorem: the function must be differentiable and continuous.

Extrema Using $f''(x)$ (Second Derivative Test)

Evaluated at critical numbers: if $f''(x) > 0$, the curve is concave up (CCU) indicating a relative minimum; if $f''(x) < 0$, the curve is concave down (CCD) indicating a relative maximum.

CCD (Concave Down)

The condition of a function's curve on an interval where $f''(x) < 0$.

POI's (Points of Inflection)

Points where $f'(x) = 0$ or undefined and $f'(x)$ changes sign.

$$F'(c)$$

Represents the instantaneous rate of change at $x = 0$.

$$\frac{F(b) - F(a)}{b - a}$$

Represents the average rate of change between $x = a$ and $x = b$.

Horizontal Tangent Line

A tangent line with a slope of zero, indicated by parallel horizontal orientation ($\parallel$).

Vertical Tangent Line

A tangent line occurring where the first derivative is undefined ($f'(x) = \text{undefined}$).