Calculus: Applications of Derivatives (Sections 5.1–5.7)

Vocabulary flashcards covering topics from differential calculus, including the Mean Value Theorem, Extreme Value Theorem, critical points, interval testing, and extrema tests.

Mean Value Theorem (MVT)

A theorem stating that if $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$, there exists at least one value $c$ inside $(a, b)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$.

Critical Number

A value of $x$ in the domain of $f(x)$ where $f'(x) = 0$ or where $f'(x)$ is undefined.

Extreme Value Theorem (EVT) Rule

A rule stating that if a function is continuous on a closed interval $[a, b]$, it is guaranteed to have both an absolute maximum and an absolute minimum.

Increasing Interval

An interval on which the first derivative is positive ($f'(x) > 0$).

Decreasing Interval

An interval on which the first derivative is negative ($f'(x) < 0$).

First Derivative Test (Local Max)

A test establishing a local maximum at a critical number where the sign of $f'(x)$ changes from positive ($+$, increasing) to negative ($-$, decreasing).

First Derivative Test (Local Min)

A test establishing a local minimum at a critical number where the sign of $f'(x)$ changes from negative ($-$, decreasing) to positive ($+$, increasing).

Candidates Test

A procedure used to find absolute/global extrema on a closed interval $[a, b]$ by evaluating and comparing the original function $f(x)$ at critical numbers inside $(a, b)$ and at both endpoints.

Concave Up

A condition describing a section of a function where the second derivative is positive ($f''(x) > 0$).

Concave Down

A condition describing a section of a function where the second derivative is negative ($f''(x) < 0$).

Point of Inflection

A point on the graph of $f(x)$ where $f''(x)$ changes sign, provided that $f(x)$ is continuous at that point.

Second Derivative Test (Local Min)

A classification showing that if $f'(c) = 0$ and $f''(c) > 0$, then a local minimum occurs at $x = c$.

Second Derivative Test (Local Max)

A classification showing that if $f'(c) = 0$ and $f''(c) < 0$, then a local maximum occurs at $x = c$.

Second Derivative Test (Inconclusive)

The condition where $f''(c) = 0$, meaning the second derivative test yields no conclusion and the first derivative test must be used instead.