Calculus: Applications of Derivatives and Graphing

Vocabulary flashcards covering extreme value theorems, derivatives and curve sketching, limits at infinity, and optimization.

Absolute Max/Min

The highest and lowest points on the whole graph.

Critical Value (CV)

Any point in the domain where $f'(x) = 0$ or $f'(x)$ does not exist (DNE).

Closed Extreme Value Theorem (CEVT)

If a function is continuous on the closed interval $[a, b]$, there will be an absolute maximum and minimum on $[a, b]$, which can occur at a critical value (CV) or an endpoint (EP).

Mean Value Theorem (MVT)

If a function is continuous on $[a, b]$ and differentiable on $(a, b)$, then there exists a point $c$ such that $f'(c) = \frac{f(b)-f(a)}{b-a}$.

Rolle's Theorem

If a function is continuous on $[a, b]$, differentiable on $(a, b)$, and $f(a) = f(b)$, then there exists a point $c$ such that $f'(c) = 0$.

First Derivative Test (Increasing/Decreasing)

A function is increasing where $f'(x) > 0$ and decreasing where $f'(x) < 0$.

First Derivative Test (Local Extrema)

A local maximum occurs where the function transitions from increasing to decreasing ($\text{Inc} \to \text{Dec}$), and a local minimum occurs where it transitions from decreasing to increasing ($\text{Dec} \to \text{Inc}$).

Concavity Tests (CU / CD)

A graph is concave up (CU) where $f''(x) > 0$ and concave down (CD) where $f''(x) < 0$, with possible inflection points where $f''(x) = 0$ or does not exist (DNE).

Inflection Point (IP)

A point that must exist on the function $f$ where the graph changes concavity from concave up to concave down ($\text{CU} \to \text{CD}$) or from concave down to concave up ($\text{CD} \to \text{CU}$).

Second Derivative Test

If $f'(c) = 0$ and $f''(c) > 0$, then a local minimum occurs at $c$; if $f'(c) = 0$ and $f''(c) < 0$, then a local maximum occurs at $c$.

Limits at Infinity (Rational Strategy)

For limits as $x \to \pm \infty$, the primary goal is to divide each term by the highest degree term found in the denominator.

Limits at Infinity (Non-Fractional Strategy)

Multiply by the conjugate if a root expression is present (and not a fraction); factor if the expression does not contain a root and is not a fraction.

Eight-Step Curve Sketching Guide

The systematic steps to graph a function: 1) Domain, 2) Intercepts, 3) Symmetry, 4) Vertical Asymptotes (VA) and Horizontal/Slant Asymptotes (HA/SA), 5) Increasing/Decreasing, 6) Max/Min, 7) Concavity (CU/CD), and 8) Inflection Points (IP).

Optimization

An application of finding absolute maximums and minimums, typically solved by drawing a picture, setting up one equation to differentiate, and using a second equation to relate two variables to each other.