Unit 4: Mean Value Theorem, Rolle's Theorem, and Curve Sketching

Vocabulary practice flashcards covering Rolle's Theorem, the Mean Value Theorem, hypotheses and failure conditions, and derivative relationships used in curve sketching.

Rolle's Theorem

A theorem stating that if a function $f$ is continuous on the closed interval $[a, b]$, differentiable on the open interval $(a, b)$, and $f(a) = f(b)$, then there is at least one number $c$ in $(a, b)$ such that $f'(c) = 0$.

Michel Rolle

A French mathematician (1652–1719) who first published the theorem bearing his name in 1691; he was originally one of the most vocal critics of calculus, claiming it gave erroneous results based on unsound reasoning, before later accepting its usefulness.

Mean Value Theorem (MVT)

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

Geometric Interpretation of the Mean Value Theorem

The principle that on an interval where the Mean Value Theorem applies, there is at least one point where the slope of the tangent line (instantaneous rate of change, or IROC) is equal to the slope of the secant line (average rate of change, or AROC).

Instantaneous Rate of Change (IROC)

The rate at which a function is changing at a specific point, geometrically represented by the slope of the tangent line and calculated using the derivative $f'(x)$.

Average Rate of Change (AROC)

The rate of change of a function over an interval $[a, b]$, calculated as $\frac{f(b) - f(a)}{b - a}$ and geometrically represented by the slope of the secant line connecting $(a, f(a))$ and $(b, f(b))$.

Point of Inflection

A point on the graph of a function where concavity changes from concave up to concave down or vice versa, corresponding to a point where $f''(x) = 0$ or $f''(x)$ is undefined and changes sign.

Relationship Between $x$-Intercepts of $f'(x)$ and $f(x)$

The $x$-intercepts of the first derivative graph $f'(x)$ correspond to the critical numbers that locate the relative extrema (maxima and minima) on the graph of $f(x)$.

Relationship Between $x$-Intercepts of $f''(x)$ and $f(x)$

The $x$-intercepts of the second derivative graph $f''(x)$ identify the $x$-coordinates of the points of inflection on the graph of $f(x)$.

Corner Discontinuity in Differentiability

A point where the slopes from the left and right sides of a curve do not agree (such as $x = 3$ for $f(x) = 3x - |x - 3|$), making the function non-differentiable and causing the Mean Value Theorem to fail.

Cusp Discontinuity in Differentiability

A sharp point where the tangent line becomes vertical with opposing signs on either side (such as $x = 0$ for $h(x) = x^{2/3}$), rendering the function non-differentiable at that point and precluding the use of the Mean Value Theorem.

Vertical Asymptote in Continuity

An infinite discontinuity where function values grow without bound (such as $x = -2$ for $g(x) = \frac{2}{x + 2}$), violating the required condition that a function must be continuous on the closed interval $[a, b]$ for the Mean Value Theorem.

Critical Numbers

Values of $x$ in the domain of a function $f$ where $f'(x) = 0$ or $f'(x)$ does not exist, serving as candidate locations for relative maxima and relative minima.

Concavity Conditions for Curve Sketching

The criteria using the second derivative where a function $f(x)$ is concave up on intervals where $f''(x) > 0$ and concave down on intervals where $f''(x) < 0$.

Monotonicity Conditions for Curve Sketching

The criteria using the first derivative where a function $f(x)$ is increasing on intervals where $f'(x) > 0$ and decreasing on intervals where $f'(x) < 0$.