Key Terms
- absolute extremum
- if has an absolute maximum or absolute minimum at we say has an absolute extremum at
- absolute maximum
- if for all in the domain of we say has an absolute maximum at
- absolute minimum
- if for all in the domain of we say has an absolute minimum at
- antiderivative
- a function such that for all in the domain of is an antiderivative of
- concave down
- if is differentiable over an interval and is decreasing over then is concave down over
- concave up
- if is differentiable over an interval and is increasing over then is concave up over
- concavity
- the upward or downward curve of the graph of a function
- concavity test
- suppose is twice differentiable over an interval if over then is concave up over if over then is concave down over
- critical number
- if or is undefined, we say that is a critical number of
- critical point
- the point a critical point of
- differential
- the differential is an independent variable that can be assigned any nonzero real number; the differential is defined to be
- differential form
- given a differentiable function the equation is the differential form of the derivative of with respect to
- end behavior
- the behavior of a function as and
- extreme value theorem
- if is a continuous function over a finite, closed interval, then has an absolute maximum and an absolute minimum
- Fermat’s theorem
- if has a local extremum at then is a critical point of
- first derivative test
- let be a continuous function over an interval containing a critical point such that is differentiable over except possibly at if changes sign from positive to negative as increases through then has a local maximum at if changes sign from negative to positive as increases through then has a local minimum at if does not change sign as increases through then does not have a local extremum at
- horizontal asymptote
- if or then is a horizontal asymptote of
- indefinite integral
- the most general antiderivative of is the indefinite integral of we use the notation to denote the indefinite integral of
- indeterminate forms
- when evaluating a limit, the forms and are considered indeterminate because further analysis is required to determine whether the limit exists and, if so, what its value is
- infinite limit at infinity
- a function that becomes arbitrarily large as x becomes large
- inflection point
- if is continuous at and changes concavity at the point is an inflection point of
- initial value problem
- a problem that requires finding a function that satisfies the differential equation together with the initial condition
- iterative process
- process in which a list of numbers is generated by starting with a number and defining for
- L’Hôpital’s rule
- if and are differentiable functions over an interval except possibly at and or and are infinite, then assuming the limit on the right exists or is or
- limit at infinity
- the limiting value, if it exists, of a function as or
- linear approximation
- the linear function is the linear approximation of at
- local extremum
- if has a local maximum or local minimum at we say has a local extremum at
- local maximum
- if there exists an interval such that for all we say has a local maximum at
- local minimum
- if there exists an interval such that for all we say has a local minimum at
- mean value theorem
- if is continuous over and differentiable over then there exists such that
- Newton’s method
- method for approximating roots of using an initial guess each subsequent approximation is defined by the equation
- oblique asymptote
- the line if approaches it as or
- optimization problems
- problems that are solved by finding the maximum or minimum value of a function
- percentage error
- the relative error expressed as a percentage
- propagated error
- the error that results in a calculated quantity resulting from a measurement error dx
- related rates
- are rates of change associated with two or more related quantities that are changing over time
- relative error
- given an absolute error for a particular quantity, is the relative error.
- rolle’s theorem
- if is continuous over and differentiable over and if then there exists such that
- second derivative test
- suppose and is continuous over an interval containing if then has a local minimum at if then has a local maximum at if then the test is inconclusive
- tangent line approximation (linearization)
- since the linear approximation of at is defined using the equation of the tangent line, the linear approximation of at is also known as the tangent line approximation to at