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| Derivatives: Min, Max, Critical Points.. |
| (Math | Calculus | Derivatives | Extrema/Concavity/Other) |
Asymptotes
Definition of a horizontal asymptote: The line y = y0 is a 'horizontal asymptote' of f(x) if and only if f(x) approaches y0 as x approaches + or - .Definition of a vertical asymptote: The line x = x0 is a 'vertical asymptote' of f(x) if and only if f(x) approaches + or - as x approaches x0 Adaptec scsi card 39160 - ultra160 scsi driver download. from the left or from the right.
Definition of a slant asymptote: the line y = ax + b is a 'slant asymptote' of f(x) if and only if lim (x-->+/-) f(x) = ax + b.
ConcavityDefinition of a concave up curve: f(x) is 'concave up' at x0 if and only ifDefinition of a concave down curve: f(x) is 'concave down' at x0 if and only if
The second derivative test: If f
Definition of a critical point: a critical point on f(x) occurs at x0 if and only if either f '(x0) is zero or the derivative doesn't exist.Extrema (Maxima and Minima)Local (Relative) Extrema
Definition of a local maxima: A function f(x) has a local maximum at x0 if and only if there exists some interval I containing x0 such that f(x0) >= f(x) for all x in I.
Definition of a local minima: A function f(x) has a local minimum at x0 if and only if there exists some interval I containing x0 such that f(x0) <= f(x) for all x in I.
Occurrence of local extrema: All local extrema occur at critical points, but not all critical points occur at local extrema.
The first derivative test for local extrema: If f(x) is increasing (
The second derivative test for local extrema: If
Absolute Extrema
Definition of absolute maxima: y0 is the 'absolute maximum' of f(x) on I if and only if y0 >= f(x) for all x on I.
Definition of absolute minima: y0 is the 'absolute minimum' of f(x) on I if and only if y0 <= f(x) for all x on I.
The extreme value theorem: If f(x) is continuous in a closed interval I, then f(x) has at least one absolute maximum and one absolute minimum in I.
Occurrence of absolute maxima: If f(x) is continuous in a closed interval I, then the absolute maximum of f(x) in I is the maximum value of f(x) on all local maxima and endpoints on I. Alereon cbaf driver download for windows 10.
Occurrence of absolute minima: If f(x) is continuous in a closed interval I, then the absolute minimum of f(x) in I is the minimum value of f(x) on all local minima and endpoints on I.
Alternate method of finding extrema: If f(x) is continuous in a closed interval I, then the absolute extrema of f(x) in I occur at the critical points and/or at the endpoints of I.
(This is a less specific form of the above.)
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0 such that f(x0A Mineral's Streak
) > f(x) for all x in I to the left of x0 and f(x0) < f(x) for all x in I to the right of xA Minute
0.Definition of a decreasing function: A function f(x) is 'decreasing' at a point x0 if and only if there exists some interval I containing x0 such that f(x0) < f(x) for all x in I to the left of x0 and f(x0) > f(x) for all x in I to the right of x0.
The first derivative test: If
Definition of an inflection point: An inflection point occurs on f(x) at x0 if and only if f(x) has a tangent line at x0 and there exists and interval I containing x0 such that f(x) is concave up on one side of x0 and concave down on the other side.
Timer For A Min
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