MTH1011 theorems

Intermediate value theorem

Let f:[a,b]—>R be a continuous function.

Then for any y between f(a) and f(b) we can find C€[a,b] : f(C)=y

Finds roots

L’hopitals rule

Let f:I—>R be differentiable functions at a€I, and lim f(t)= lim g(t) (as t—>a)

Then lim f(t)/g(t) = lim f’(t)/g’(t)

Rolles theorem

Let f:[a,b]—> be differentiable on (a,b) and continuous on [a,b], and f(a)=f(b)

Then there exists C€(a,b) : f’(C)=0

Basically a turning point between a and b

Mean value theorem

Let f:[a,b]—>R be differentiable on (a,b) and continuous on [a,b]

Then there exists C€(a,b) :

f’(C)=(f(b)-f(a))/(b-a)

tangents are parallel

increasing-decreasing theorem

let f:I—>R be a real function

Then if f is differentiable on (a,b) we have

i. f is increasing if f’(t)>=0

ii. f is decreasing if f’(t)=<0

iii f is constant if f’(t)=0

for any t€(a,b)

Boltzmann weierstrass theorem

Every bounded sequence contains a convergent subsequence