Ap Calc Theorems

Front 1

Intermediate value theorem

Back 1

Suppose f(x) is continuous on an interval I, and a and b are any two points of I. Then if y is a number between f(a) and f(b), there exists a number c between a and b such that f(c)=y

Front 2

Rolle's Theorem

Back 2

Suppose that y=f(x) is continuous at every point of the clsoed interval [a,b] and is differentiable at every point of its interior (a,b). If f(a)=f(b)=0, then there is at least one number c in (a,b) at which f'(c)=0.

Front 3

Mean Value Theorem

Back 3

Suppose y=f(x) is continuous on a closed interval [a,b] and differentiable on the interval's interior (a,b). Then there is at least one point c in (a,b) at which
(f(b)-f(a))/(b-a)=f'(c).

Front 4

corollary 1 to MVT

Back 4

Functions with zero derivatives are constant-if f'(x)=0 at each point of an interval I, then f(x)=C for all x in I, where C's a constant

Front 5

corollary 2 to MVT

Back 5

Functions with same derivative differ by a constant-is f'(x)=g'(x) at each point of an interval I, then there exists a constant C such that f(x)=g(x)+C for all x in I

Front 6

corollary 3 to MVT

Back 6

The first derivative test for increasing and decreasing- suppose the f is continuous on [a,b] and differentiable on (a,b). If f'>0 at each point of (a,b), then f increases on [a,b]. If f'<0 at each point of (a,b), then f decreases on [a,b]