Calculus Theorems

Intermediate Value Theorem

• Let f(x) be continuous and differentiable on XER, if f(a) > 0 and f(b) < 0 then there must exist c such that f(c ) = 0

What is Rolle's Theorem used for?

• Used to prove the uniqueness of a real root by contradiction or the existence of more than one real root (if f’(x) = 0 at 1 point, you have 2 roots, if 2 point, 4, etc because it means you have turning points and the function asses through the x axis again)

What is the Intermediate Value Theorem used for?

• Used to prove the existence of real roots (zeros, same thing as normal roots)

Rolle's Theorem

If:

• f is continuous on [a,b]

• f is differentiable on (a,b)

• f(a) = f(b)

Then there exists c such that f’(c ) = 0

(Since the same y value exists for 2 x values on the same function there must be a turning point)

Mean Value Theorem

Oblique Asymptote Conditions

1) lim as x approaches positive infinity of f(x)/x = m

2) lim as x aporoaches positive infinity of [f(x) - mx] = b