Final Exam (Theorems + Definitions)

Cauchy Mean Value Theorem

If f and g are continuous functions on [a,b] and differentiable on (a,b), then there is a c in (a,b) such that [f(b)-f(a)]g’( c) = [g(b)-g(a)]f’( c)

Bolzano’s Theorem

If f:[a,b]→R is continuous and f(a) and f(b) have opposite signs, then there exists c in (a,b) such that f( c) = 0

Intermediate Value Theorem

Let f:E→R be continuous with E connected. If a,b in E with a < b and y is a number between f(a) and f(b) then there is a number c in (a,b) such that f( c) = y

Rolle’s Theorem

Suppose f:[a,b]→R is continuous on [a,b] and f is differentiable on (a,b). Then if f(a) = f(b) = 0, then there is a c in (a,b) such that f’( c) = 0

Mean Value Theorem

If f:[a,b]→R is continuous on [a,b] and differentiable on (a,b), then there is a c in (a,b) such that f’( c) = f(b)-f(a)/b-a

E c R is said to be connected if

whenever a,b in E, a < b and c in (a,b) we have c in E. All points between two points in the set must also be in the set

Let f:D→R, x_o in D an accumulation point of D. Let T(x) = f(x)-f(x_o)/x-x_o with D_T = D\{x_o}.

If lim x→x_o T(x) exists, then we say that f is differentiable at x_o and f’(x_o) = lim x→x_o (f(x) - f(x_o)/x-x_o) = lim x→x_o T(x)

Let f:D→R. A point x_o in D is a relative maximum of f if and only if

there exists Delta>0 such that x in D and |x-x_o| < Delta implies f(x) <= f(x_o)

Let f:D→R. A point x_o in D is a relative minimum of f if and only if

there exists Delta>0 such that x in D and |x-x_o| < Delta implies f(x) >= f(x_o)