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Precalculus 2e

Key Terms

Precalculus 2eKey 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<a x<a 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).
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