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 $f\left(x\right);$ it is given by $$\text{AROC}=\frac{f\left(a+h\right)-f\left(a\right)}{h}.$$

continuous function

a function that has no holes or breaks in its graph

derivative

the slope of a function at a given point; denoted $f^{\prime }(a),$ at a point $x=a$ it is $f^{\prime }(a)=\underset{h\to 0}{\lim }\frac{f\left(a+h\right)-f\left(a\right)}{h},$ providing the limit exists.

differentiable

a function $f\left(x\right)$ for which the derivative exists at $x=a.$ In other words, if $f^{\prime }\left(a\right)$ 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 $f^{\prime }(a)=\underset{h\to 0}{\lim }\frac{f\left(a+h\right)-f\left(a\right)}{h}.$

instantaneous velocity

the change in speed or direction at a given instant; a function $s\left(t\right)$ 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 $s^{\prime }(a)=\underset{h\to 0}{\lim }\frac{s\left(a+h\right)-s\left(a\right)}{h}.$

jump discontinuity

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

left-hand limit

the limit of values of $$f\left(x\right)$$ as $x$ approaches from $a$ the left, denoted $$\underset{x\to a^{-}}{\lim }f(x)=L.$$ 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 $$x\ne a.$$ Both $a$ and $L$ are real numbers.

limit

when it exists, the value, $L,$ that the output of a function $f\left(x\right)$ 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 $$\underset{x\to a}{\lim }f(x)=L.$$

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 $f\left(x\right)$ where the function is discontinuous, but can be redefined to make it continuous

right-hand limit

the limit of values of $$f\left(x\right)$$ as $$x$$ approaches $a$ from the right, denoted $$\underset{x\to a^{+}}{\lim }f(x)=L.$$ 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 $$x\ne a.$$ 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, $$\underset{x\to a}{\lim }f(x)=L$$ if and only if $$\underset{x\to a^{-}}{\lim }f(x)=\underset{x\to a^{+}}{\lim }f(x).$$