Limits, Tangent Lines, and Rates of Change

Vocabulary flashcards covering limits, tangent lines, secant lines, rates of change, velocity, and specialized functions from Calculus Chapter 2.

Tangent Line

A line to the function $f(x)$ at the point $x = a$ that just touches the graph of the function at that point and is parallel in direction to the graph at that point.

Secant Line

A line connecting two distinct points on the graph of a function, such as $P = (a, f(a))$ and $Q = (x, f(x))$, used to approximate the slope of a tangent line.

Parallel Line and Graph at a Point

A condition in which a line and a graph are both moving in the same direction at a specific point.

Average Rate of Change

The ratio of the change in $f(x)$ to the change in $x$ over an interval, given by $\text{A.R.C.} = \frac{f(x) - f(a)}{x - a}$.

Instantaneous Rate of Change

The exact rate at which a function $f(x)$ is changing at a specific point $x = a$, estimated by calculating average rates of change for values of $x$ closer and closer to $a$ from both sides.

Average Velocity

The ratio of the change in position to time traveled over an interval, given by $\text{A.V.} = \frac{\text{change in position}}{\text{time traveled}} = \frac{f(t) - f(a)}{t - a}$.

Instantaneous Velocity

The rate at which the position function $f(t)$ of an object is changing at a single specific time $t = a$.

Difference Quotient Notation with Distance $h$

An alternative formula for slope and rate of change where new points are defined relative to $x$ by a distance $h$, given generally by $\frac{f(x + h) - f(x)}{h}$.

Limit of a Function

Written as $\lim_{x \rightarrow a} f(x) = L$, it means that $f(x)$ can be made as close to $L$ as desired for all $x$ sufficiently close to $a$, from both sides, without actually letting $x$ equal $a$.

Heaviside Function

A step function defined as $H(t) = \begin{cases} 0 & \text{if } t < 0 \\ 1 & \text{if } t \ge 0 \end{cases}$, which approaches different values from the left and right at $t = 0$.