Derivatives and Rates of Change Flashcards

Vocabulary flashcards covering key definitions, formulas, interpretations, and examples of derivatives and rates of change from Calculus Section 2.1.

Delta ($\Delta$)

A mathematical symbol representing change, where $\Delta y$ denotes vertical change and $\Delta x$ denotes horizontal change.

Slope of a Line

The ratio of vertical change to horizontal change between two points, calculated as rise over run or $\frac{\Delta y}{\Delta x}$.

Secant Line

A line passing through two points on a curve, such as $(a, f(a))$ and $(a+h, f(a+h))$.

Tangent Line

A line that touches a curve at a single point, having a slope equal to the derivative of the curve at that point.

Definition of the Derivative

The derivative of a function $f$ at a number $a$, denoted by $f'(a)$, defined as $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$.

Alternative Definition of the Derivative

An equivalent definition of the derivative of a function $f$ at a number $a$, given by $f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$.

Derivative

A concept that simultaneously represents the slope of the tangent line to a curve at a point and the instantaneous rate of change.

Difference Quotient

The quotient expression $\frac{f(a+h) - f(a)}{h}$ used to set up the limit when calculating a derivative.

Average Velocity

The rate of change in position over a time interval from $a$ to $a+h$, given by $\frac{f(a+h) - f(a)}{h}$.

Instantaneous Velocity

The derivative of a position function $f(t)$ at time $a$, defined as $v(a) = f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$.

Galileo Falling Object Model

The position function $f(t) = 4.9\,t^2$ representing an object falling due to gravity, which yields an instantaneous velocity of $49\,\text{m/s}$ at $t = 5$.

Hyperbola Tangent Slope Example

The calculation showing that for the hyperbola $y = \frac{3}{x}$ at the point $(3, 1)$, the slope of the tangent line is $y'(3) = -\frac{1}{3}$.