AP Calculus Unit 2 Review Flashcards

Vocabulary and key concepts flashcards covering derivatives, differentiation rules, rates of change, and limit definitions for AP Calculus Unit 2 Review.

Average Rate of Change

The slope of the secant line passing through two points $(a, f(a))$ and $(b, f(b))$ on the graph of $f(x)$, calculated using the formula $\frac{f(b) - f(a)}{b - a}$.

Instantaneous Rate of Change

The slope of the tangent line to a curve at a single point $x = a$, defined as the limit of average rates of change: $f'(a) = \text{lim}_{x \rightarrow a} \frac{f(x) - f(a)}{x - a}$.

Limit Definition of the Derivative

The mathematical definition of the derivative of a function $f(x)$, given by f'(x) = \text{lim}_{\text{\textDelta} x \rightarrow 0} \frac{f(x + \text{\textDelta} x) - f(x)}{\text{\textDelta} x}.

Alternate Form of the Derivative Definition

The formulation of the derivative of $f(x)$ at a specific point $x = a$, expressed as $f'(a) = \text{lim}_{x \rightarrow a} \frac{f(x) - f(a)}{x - a}$.

Power Rule

A differentiation rule stating that for a function $f(x) = x^n$, its derivative is $f'(x) = n x^{n-1}$.

Product Rule

A differentiation rule stating that the derivative of the product of two functions $u(x)$ and $v(x)$ is $\frac{d}{dx}[u(x) v(x)] = u'(x) v(x) + u(x) v'(x)$.

Quotient Rule

A differentiation rule stating that the derivative of the quotient of two functions $\frac{u(x)}{v(x)}$ is \frac{d}{dx}\text{\textleft[}\frac{u(x)}{v(x)}\text{\textright]} = \frac{u'(x) v(x) - u(x) v'(x)}{[v(x)]^2}.

Derivative of Natural Logarithm

The rule stating that the derivative of $y = \text{ln}(x)$ with respect to $x$ is $\frac{dy}{dx} = \frac{1}{x}$ for $x > 0$.

Limit Definition Expression for $f(x) = 2x^2 + 4$

The limit expression that calculates the derivative of $f(x) = 2x^2 + 4$ using the limit definition of the derivative.

Uncle Si Duck Crossing Observation Table

A data table recording the cumulative number of ducks crossing a road at each hour from $t = 0$ to $t = 7$ hours after $9:00\text{ AM}$.

Function and Derivative Values Table

A table providing given values for functions $f(x)$ and $g(x)$ and their derivatives $f'(x)$ and $g'(x)$ at $x = -1$ and $x = 0$.