Precalculus 2e

# Key Terms

### Key Terms

average rate of change
the slope of the line connecting the two points $(a,f(a)) (a,f(a))$ and $(a+h,f(a+h)) (a+h,f(a+h))$ on the curve of $f( x ); f( x );$ it is given by $AROC= f( a+h )−f( a ) h . AROC= f( a+h )−f( a ) h .$
continuous function
a function that has no holes or breaks in its graph
derivative
the slope of a function at a given point; denoted $f ′ (a), f ′ (a),$ at a point $x=a x=a$ it is $f ′ (a)= lim h→0 f( a+h )−f( a ) h , f ′ (a)= lim h→0 f( a+h )−f( a ) h ,$ providing the limit exists.
differentiable
a function $f( x ) f( x )$ for which the derivative exists at $x=a. x=a.$ In other words, if $f ′ ( a ) f ′ ( a )$ exists.
discontinuous function
a function that is not continuous at $x=a x=a$
instantaneous rate of change
the slope of a function at a given point; at $x=a x=a$ it is given by $f ′ (a)= lim h→0 f( a+h )−f( a ) h . f ′ (a)= lim h→0 f( a+h )−f( a ) h .$
instantaneous velocity
the change in speed or direction at a given instant; a function $s( t ) s( t )$ represents the position of an object at time $t ,t,$ and the instantaneous velocity or velocity of the object at time $t=a t=a$ is given by $s ′ (a)= lim h→0 s( a+h )−s( a ) h . s ′ (a)= lim h→0 s( a+h )−s( a ) h .$
jump discontinuity
a point of discontinuity in a function $f( x ) f( x )$ at $x=a x=a$ where both the left and right-hand limits exist, but $lim x→ a − f(x)≠ lim x→ a + f(x) lim x→ a − f(x)≠ lim x→ a + f(x)$
left-hand limit
the limit of values of $f( x ) f( x )$ as $x x$ approaches from $a a$ the left, denoted $lim x→ a − f(x)=L. lim x→ a − f(x)=L.$ The values of $f(x) f(x)$ can get as close to the limit $L L$ as we like by taking values of $x x$ sufficiently close to $a a$ such that $x and $x≠a. x≠a.$ Both $a a$ and $L L$ are real numbers.
limit
when it exists, the value, $L, L,$ that the output of a function $f( x ) f( x )$ approaches as the input $x x$ gets closer and closer to $a a$ but does not equal $a. a.$ The value of the output, $f(x), f(x),$ can get as close to $L L$ as we choose to make it by using input values of $x x$ sufficiently near to $x=a, x=a,$ but not necessarily at $x=a. x=a.$ Both $a a$ and $L L$ are real numbers, and $L L$ is denoted $lim x→a f(x)=L. lim x→a f(x)=L.$
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 $f( x ) f( x )$ where the function is discontinuous, but can be redefined to make it continuous
right-hand limit
the limit of values of $f( x ) f( x )$ as $x x$ approaches $a a$ from the right, denoted $lim x→ a + f(x)=L. lim x→ a + f(x)=L.$ The values of $f(x) f(x)$ can get as close to the limit $L L$ as we like by taking values of $x x$ sufficiently close to $a a$ where $x>a, x>a,$ and $x≠a. x≠a.$ Both $a a$ and $L L$ 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 $f(x), f(x),$ as $x x$ approaches $a, a,$ is equal to $L, L,$ that is, $lim x→a f(x)=L lim x→a f(x)=L$ if and only if $lim x→ a − f(x)= lim x→ a + f(x). lim x→ a − f(x)= lim x→ a + f(x).$