AP Calculus Unit 2 Vocabulary

Vocabulary flashcards covering key terms, concepts, and rules from AP Calculus Unit 2 based on the lecture transcript.

Average Rate of Change

The slope of the secant line created over an interval $[a, b]$, calculated using the formula $\frac{f(b) - f(a)}{b - a}$.

Secant Line

A line connecting two points on a curve, whose slope represents the average rate of change over that interval.

Instantaneous Rate of Change

The rate of change or exact slope of the tangent line at a single specific point on a function.

Tangent Line

A line that touches a graph at exactly one point, having a slope equal to the instantaneous rate of change at that point.

Derivative

The exact instantaneous rate of change of a function at a point, denoted as $f'(x)$ or $\frac{dy}{dx}$.

Derivative Function

A function that gives the instantaneous rate of change at any given point $x$ in the original function instantly.

Point-Slope Form

An equation form used to write the equation of a tangent line, given by $y - y_1 = m(x - x_1)$ where $m$ is the derivative at the point.

Differentiability

A property of a function indicating it has a well-defined slope throughout and is relatively smooth, which allows a derivative function to exist.

Corner or Cusp

A sharp turn on a graph where the slopes on the left and right sides are completely different, making the function non-differentiable at that point.

Vertical Tangent

A location on a graph where the tangent line created is perfectly vertical, causing the derivative to be undefined and the function to be non-differentiable.

Constant Rule

A rule stating that the derivative of any constant is $0$.

Constant Multiple Rule

A rule stating that the derivative of a constant times a function is equal to the constant multiplied by the derivative of that function.

Sum and Difference Rule

A rule stating that the derivative of $f(x) \times \frac{1}{1} \times \text{or} \times g(x)$ as $f(x) \neq g(x)$ expression f(x) \begin{cases} + \ - \begin{cases} \text{is} \begin{cases} f'(x) \begin{cases} + \ - \begin{cases} g'(x). More simply, the derivative of $f(x) + g(x)$ is $f'(x) + g'(x)$, and for $f(x) - g(x)$ is $f'(x) - g'(x)$.

Power Rule

A rule stating that for a function $f(x) = x^n$, the derivative is $n \times x^{n - 1}$.

Product Rule

A rule stating that for a product of two functions $u(x) \times v(x)$, the derivative is $u'(x) \times v(x) + u(x) \times v'(x)$.

Quotient Rule

A rule stating that for a quotient of two functions $\frac{u(x)}{v(x)}$, the derivative is $\frac{u'(x) \times v(x) - u(x) \times v'(x)}{(v(x))^2}$.