Mean Value Theorem & Hopital Rule

Question 1

Intermediate value theorem

Answer 1

Suppose that f(x) is continuous on [a,b] there exists such value c such that;
1. a

Question 2

Rolle’s Theorem - critical point in (a,b)

Answer 2

Suppose f(x) is a function that satisfies all of the following.
f(x) is continuous on the closed interval [a,b].
f(x) is differentiable on the open interval (a,b).
f(a)=f(b)

Then there is a number c such that a

Question 3

Mean Value Theorem

Answer 3

Suppose f(x) is a function that satisfies:
f(x) is continuous on [a,b]
f(x) is differentiable on (a,b)

Then there is a number c such that a

Question 4

L’hopital rule

Answer 4

Used to solve limits when the form in quotient is indeterminate.

It says that when the limit is in quotient form that is 0/0 or INF/INF then we will use the differentiables of f’(x)/g’(x) as long as g’(x) isn’t equal to 0

Question 5

Indeterminate products

Answer 5

When the products of f(x)g(x) isn’t clear at the limit and we get the form of 0*INF then we write

fg=f/(1/g)

Question 6

Indeterminate differences

Answer 6

If we get something of the form INF-INF then we find the common denominator and then apply l’hopital rule

Question 7

Indeterminate powers

Answer 7

If we get something of the form 0^0, INF^0, 1^INF

We can take the natural log on each site or by writing the function as an exponential:

f(x)^g(x)=e^(gxlnfx)