### 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 $$\text{AROC}=\frac{f\left(a+h\right)\xe2\u02c6\u2019f\left(a\right)}{h}.$$

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

- derivative
- the slope of a function at a given point; denoted ${f}^{\xe2\u20ac\xb2}(a),$ at a point $x=a$ it is ${f}^{\xe2\u20ac\xb2}(a)=\underset{h\xe2\u2020\u20190}{\mathrm{lim}}\frac{f\left(a+h\right)\xe2\u02c6\u2019f\left(a\right)}{h},$ providing the limit exists.

- differentiable
- a function $f\left(x\right)$ for which the derivative exists at $x=a.$ In other words, if ${f}^{\xe2\u20ac\xb2}\left(a\right)$ 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 ${f}^{\xe2\u20ac\xb2}(a)=\underset{h\xe2\u2020\u20190}{\mathrm{lim}}\frac{f\left(a+h\right)\xe2\u02c6\u2019f\left(a\right)}{h}.$

- 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 ${s}^{\xe2\u20ac\xb2}(a)=\underset{h\xe2\u2020\u20190}{\mathrm{lim}}\frac{s\left(a+h\right)\xe2\u02c6\u2019s\left(a\right)}{h}.$

- 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 $\underset{x\xe2\u2020\u2019{a}^{\xe2\u02c6\u2019}}{\mathrm{lim}}f(x)\xe2\u2030\underset{x\xe2\u2020\u2019{a}^{+}}{\mathrm{lim}}f(x)$

- left-hand limit
- the limit of values of $$f\left(x\right)$$ as $x$ approaches from $a$ the left, denoted $$\underset{x\xe2\u2020\u2019{a}^{\xe2\u02c6\u2019}}{\mathrm{lim}}f(x)=L.$$ 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 $$x\xe2\u2030a.$$ 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 $$\underset{x\xe2\u2020\u2019a}{\mathrm{lim}}f(x)=L.$$

- 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 $$\underset{x\xe2\u2020\u2019{a}^{+}}{\mathrm{lim}}f(x)=L.$$ 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 $$x\xe2\u2030a.$$ 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, $$\underset{x\xe2\u2020\u2019a}{\mathrm{lim}}f(x)=L$$ if and only if $$\underset{x\xe2\u2020\u2019{a}^{\xe2\u02c6\u2019}}{\mathrm{lim}}f(x)=\underset{x\xe2\u2020\u2019{a}^{+}}{\mathrm{lim}}f(x).$$