The least single value that a function attains on a given interval. A function f has an
absolute minimum at x = b if f (b)≤f (x) for all x in the domain of f.
The greatest single value that a function attains on a given interval. A function f has an
absolute maximum at x = b if f (b)≥f (x) for all x in the domain of f.
The rate of change of velocity.
Average Rate of Change
For a function f (x), this represents the total change in f over a given interval divided
by the total change in x.
A term applied to a section of the graph where the derivative is decreasing throughout.
A term applied to a section of the graph where the derivative is increasing throughout.
These are points in the domain of f where f'(x) = 0 or f'(x) is undefined.
Critical Point Theorem
If f has a local extremum at x = b, then (b, f (b)) is a critical point.
This is the scalar analog of the vector quantity position. The distance from
a certain point is the absolute value of the position relative to that point. Thus, distances
are always positive.
A horizontal line that the graph of f (x) approaches but never touches as x approaches
positive or negative infinity.
Instantaneous rate of change
This is the rate of change of a function at a single point on its graph. The instantaneous
rate of change of f at x = a is given by the derivative, f'(a).
The collective term for local maxima and local minima.
A function f has a local maximum at x = b if f (b) is the largest value that f attains
"near b." More formally, a function f (x) has a local maximum at x = b if there is an
open interval I in which f (b)≥f (x) for all x in I.
A function f has a local minimum at x = b if f (b) is the smallest value that f
attains "near b." More formally, a function f (x) has a local minimum at x = b if
there is an open interval I in which f (b)≤f (x) for all x in I.
Mean value theorem
is continuous on [a, b]
and differentiable on (a, b)
, then there exists a point c
on [a, b]
|f'(c) = ||
This refers to the relative placement of an object with respect to another object, which is
generally placed at the origin. This is a vector quantity, so it has both a magnitude
and a direction.
A type of motion along a straight line, normally depicted by a number line.
Same as local extrema.
Same as local maximum.
Same as local minimum.
If f is continuous on [a,b] and differentiable on (a,b), and f (a) = f (b), then there is a
c on (a, b) where f'(c) = 0. This is a special case of the mean value theorem.
These are quantities that only express a magnitude. They are often compared against
vector quantities, which have a magnitude and a direction.
Speed is the scalar analog of velocity.
These are quantities that have both a magnitude (indicated by the number) and a direction
(indicated by the presence or absence of a negative sign). They are often compared
against scalar quantities, which only have a magnitude. For example, velocity is
the vector analog of speed, because while speed depends only on the magnitude of
the motion, velocity also depends on the direction. So, whether one is traveling with a
velocity of 3 miles per hour in the positive direction or in the negative direction, one still
has a speed of 3 miles per hour.
This is the rate of change of position.
This is a vertical line that the graph approaches, but never touches. Near a vertical
asymptote, a function approaches positive or negative infinity.
A vertical asymptote occurs at x = c
when the following are all true
1) f (c)
2) f (x) = ∞
or - ∞
3) f (x) = ∞
or - ∞