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Calculus AB: Applications of the Derivative

Terms

Preview of Applications of the Derivative

Rates of Change and Applications to Motion

Absolute minimum  -  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 .
Absolute maximum  -  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 .
Acceleration  -  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 .
Concave Down  -  A term applied to a section of the graph where the derivative is decreasing throughout.
Concave up  -  A term applied to a section of the graph where the derivative is increasing throughout.
Critical points  -  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.
Distance  -  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.
Horizontal asymptote  -  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) .
Local extrema  -  The collective term for local maxima and local minima.
Local maximum  -  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 .
Local minimum  -  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  -  If f is continuous on [a, b] and differentiable on (a, b) , then there exists a point c on [a, b] for which

f'(c) =    

Position  -  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.
Rectilinear motion  -  A type of motion along a straight line, normally depicted by a number line.
Relative extrema  -  Same as local extrema.
Relative maximum  -  Same as local maximum.
Relative minimum  -  Same as local minimum.
Rolle's theorem  -  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.
Scalar quantities  -  These are quantities that only express a magnitude. They are often compared against vector quantities, which have a magnitude and a direction.
Speed  -  Speed is the scalar analog of velocity.
Vector quantities  -  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.
Velocity  -  This is the rate of change of position.
Vertical asymptote  -  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) is undefined
2) f (x) = ∞ or - ∞
3) f (x) = ∞ or - ∞

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