IVT, MVT, ROLLES, NEWTONS

Question 1

ROLLES THEOREM:

Answer 1

• f is continuous on [a,b]
• f is differentiable on (a,b)
  fa=fb
• then there exists c in a,b such that f’c=0 (Ivt)

Question 2

IVT:

Answer 2

• f is continuous over its domain
• show that both roots are greater than/less than 0
• by IVT, there exists a point where f(c)=0, x=c is a solution

Question 3

MEAN VALUE THEOREM:

Answer 3

• assume that f is continuous over [a,b] and differentiable on (a,b)
• there exists c in (a,b) such that

f’c= (f(b)-f(a))/(b-a)

Question 4

NEWTONS METHOD

Answer 4

• x_n+1=x_n-(f(x_n))/(f’x_n))

Question 5

Linear approximation equation

Answer 5

L(x)= f(a)+(x-a)(f’(a))

Question 6

property unique to e

Answer 6

lim x approaches 0
(e^x-1)/x
= 1

THEREFORE

lim x approaches 0
(a^x-1)/x=lna