Definition of a horizontal asymptote: The line y = y_{0} is a "horizontal asymptote" of f(x) if and only if f(x) approaches y_{0} as x approaches + or - .

Definition of a vertical asymptote: The line x = x_{0} is a "vertical asymptote" of f(x) if and only if f(x) approaches + or - as x approaches x_{0} 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.

Concavity

Definition of a concave up curve: f(x) is "concave up" at x_{0} if and only if f '(x) is increasing at x_{0}

Definition of a concave down curve: f(x) is "concave down" at x_{0} if and only if f '(x) is decreasing at x_{0}

The second derivative test: If f ''(x) exists at x_{0} and is positive, then f ''(x) is concave up at x_{0}. If f ''(x_{0}) exists and is negative, then f(x) is concave down at x_{0}. If f ''(x) does not exist or is zero, then the test fails.

Critical Points

Definition of a critical point: a critical point on f(x) occurs at x_{0} if and only if either f '(x_{0}) 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 x_{0} if and only if there exists some interval I containing x_{0} such that f(x_{0}) >= f(x) for all x in I.

Definition of a local minima: A function f(x) has a local minimum at x_{0} if and only if there exists some interval I containing x_{0} such that f(x_{0}) <= 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 (f '(x) > 0) for all x in some interval (a, x_{0}] and f(x) is decreasing (f '(x) < 0) for all x in some interval [x_{0}, b), then f(x) has a local maximum at x_{0}. If f(x) is decreasing (f '(x) < 0) for all x in some interval (a, x_{0}] and f(x) is increasing (f '(x) > 0) for all x in some interval [x_{0}, b), then f(x) has a local minimum at x_{0}.

The second derivative test for local extrema: If f '(x_{0}) = 0 and f ''(x_{0}) > 0, then f(x) has a local minimum at x_{0}. If f '(x_{0}) = 0 and f ''(x_{0}) < 0, then f(x) has a local maximum at x_{0}.

Absolute Extrema

Definition of absolute maxima: y_{0} is the "absolute maximum" of f(x) on I if and only if y_{0} >= f(x) for all x on I.

Definition of absolute minima: y_{0} is the "absolute minimum" of f(x) on I if and only if y_{0} <= 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.

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.)

Increasing/Decreasing Functions

Definition of an increasing function: A function f(x) is "increasing" at a point x_{0} if and only if there exists some interval I containing x_{0} such that f(x_{0}) > f(x) for all x in I to the left of x_{0} and f(x_{0}) < f(x) for all x in I to the right of x_{0}.

Definition of a decreasing function: A function f(x) is "decreasing" at a point x_{0} if and only if there exists some interval I containing x_{0} such that f(x_{0}) < f(x) for all x in I to the left of x_{0} and f(x_{0}) > f(x) for all x in I to the right of x_{0}.

The first derivative test: If f '(x_{0}) exists and is positive, then f '(x) is increasing at x_{0}. If f '(x) exists and is negative, then f(x) is decreasing at x_{0}. If f '(x_{0}) does not exist or is zero, then the test tells fails.

Inflection Points

Definition of an inflection point: An inflection point occurs on f(x) at x_{0} if and only if f(x) has a tangent line at x_{0} and there exists and interval I containing x_{0} such that f(x) is concave up on one side of x_{0} and concave down on the other side