12.6.1: Key Terms

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Key Terms

average rate of change: the slope of the line connecting the two points $(a, f (a))$ and $(a + h, f (a + h))$ on the curve of $f (x);$ it is given by $AROC = \frac{f (a + h) - f (a)}{h} .$

continuous function: a function that has no holes or breaks in its graph

derivative: the slope of a function at a given point; denoted $f^{'} (a),$ at a point $x = a$ it is $f^{'} (a) = \lim_{h \to 0} \frac{f (a + h) - f (a)}{h},$ providing the limit exists.

differentiable: a function $f (x)$ for which the derivative exists at $x = a .$ In other words, if $f^{'} (a)$ exists.

discontinuous function: a function that is not continuous at $x = a$

instantaneous rate of change: the slope of a function at a given point; at $x = a$ it is given by $f^{'} (a) = \lim_{h \to 0} \frac{f (a + h) - f (a)}{h} .$

instantaneous velocity: the change in speed or direction at a given instant; a function $s (t)$ represents the position of an object at time $t,$ and the instantaneous velocity or velocity of the object at time $t = a$ is given by $s^{'} (a) = \lim_{h \to 0} \frac{s (a + h) - s (a)}{h} .$

jump discontinuity: a point of discontinuity in a function $f (x)$ at $x = a$ where both the left and right-hand limits exist, but $\lim_{x \to a^{-}} f (x) \neq \lim_{x \to a^{+}} f (x)$

left-hand limit: the limit of values of $f (x)$ as $x$ approaches from $a$ the left, denoted $\lim_{x \to a^{-}} f (x) = L .$ The values of $f (x)$ can get as close to the limit $L$ as we like by taking values of $x$ sufficiently close to $a$ such that $x < a$ and $x \neq a .$ Both $a$ and $L$ are real numbers.

limit: when it exists, the value, $L,$ that the output of a function $f (x)$ approaches as the input $x$ gets closer and closer to $a$ but does not equal $a .$ The value of the output, $f (x),$ can get as close to $L$ as we choose to make it by using input values of $x$ sufficiently near to $x = a,$ but not necessarily at $x = a .$ Both $a$ and $L$ are real numbers, and $L$ is denoted $\lim_{x \to a} f (x) = L .$

properties of limits: a collection of theorems for finding limits of functions by performing mathematical operations on the limits

removable discontinuity: a point of discontinuity in a function $f (x)$ where the function is discontinuous, but can be redefined to make it continuous

right-hand limit: the limit of values of $f (x)$ as $x$ approaches $a$ from the right, denoted $\lim_{x \to a^{+}} f (x) = L .$ The values of $f (x)$ can get as close to the limit $L$ as we like by taking values of $x$ sufficiently close to $a$ where $x > a,$ and $x \neq a .$ Both $a$ and $L$ are real numbers.

secant line: a line that intersects two points on a curve

tangent line: a line that intersects a curve at a single point

two-sided limit: the limit of a function $f (x),$ as $x$ approaches $a,$ is equal to $L,$ that is, $\lim_{x \to a} f (x) = L$ if and only if $\lim_{x \to a^{-}} f (x) = \lim_{x \to a^{+}} f (x) .$

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Key Terms