Theorems Lisa Brady Mrs. Pellissier Calculus AP 28 November 2008

Intermediate Value Theorem If f is continuous on the closed interval [a,b] and M is any number between f(a) and f(b), then there is at least one number c in [a,b] such that f(c) = M.

Mean Value Theorem If f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a number c in (a,b) such that f’(c) = f(b) - f(a) b – a Mean Value Theorem If f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a number c in (a,b) such that f’(c) = f(b) - f(a) b – a

Rolle’s Theorem Let f be continuous on the closed interval [a,b] and differentiable on the open interval (a,b). If f(a) = f(b) then there is at least one number c in (a,b) such that f’(c) = 0. Rolle’s Theorem Let f be continuous on the closed interval [a,b] and differentiable on the open interval (a,b). If f(a) = f(b) then there is at least one number c in (a,b) such that f’(c) = 0.

Extreme Value Theorem If f is continuous on a closed interval [a,b], then f has both a minimum and a maximum on the interval. f(c) is the maximum f(d) is the minimum

Average Value of a Function on an Interval If f is integrable on the closed interval [a,b], then the average value of the interval is. Average Value of a Function on an Interval If f is integrable on the closed interval [a,b], then the average value of the interval is. f(x) Average value ab

Fundamental Theorem of Calculus I The Fundamental Theorem of Calculus: If a function f is continuous on the closed interval [a,b] and F is an antiderivative of f on the interval [a,b], then Fundamental Theorem of Calculus I The Fundamental Theorem of Calculus: If a function f is continuous on the closed interval [a,b] and F is an antiderivative of f on the interval [a,b], then

Fundamental Theorem of Calculus II The Second Fundamental Theorem of Calculus: If f is continuous on an open interval I containing a, then, for every x in the interval, Fundamental Theorem of Calculus II The Second Fundamental Theorem of Calculus: If f is continuous on an open interval I containing a, then, for every x in the interval,

Limit-based Definition of a Derivative The derivative of f at x is given by provided the limit exists. For all x for which this limit exists, f’ is a function of x. Limit-based Definition of a Derivative The derivative of f at x is given by provided the limit exists. For all x for which this limit exists, f’ is a function of x.

Conditions for Continuity Continuity at a Point: A function f is continuous at c if the following three conditions are met. 1) f(c) is defined. 2) lim f(x) exists. 3) lim f(x) = f(c). Continuity on an Open Interval: A function is continuous on an open interval (a,b) if it is continuous at each point in the interval. A function that is continuous on the entire real line (-∞, ∞) is everywhere continuous. Conditions for Continuity Continuity at a Point: A function f is continuous at c if the following three conditions are met. 1) f(c) is defined. 2) lim f(x) exists. 3) lim f(x) = f(c). Continuity on an Open Interval: A function is continuous on an open interval (a,b) if it is continuous at each point in the interval. A function that is continuous on the entire real line (-∞, ∞) is everywhere continuous.

Conditions for Differentiability A function is differentiable on an interval if: 1) It is continuous 2) There is no abrupt change 3) There is no vertical tangent line Conditions for Differentiability A function is differentiable on an interval if: 1) It is continuous 2) There is no abrupt change 3) There is no vertical tangent line