MATH 1500: Rolle’s and Mean-Value Theorem

What does Rolle’s theorem need?

If f(x)

• continuous on closed interval [a.b]

• differentiable on open interval (a,b)

• f(a) = f(b)

Then there will be some number c such that f’(c) = 0.

How do you find out if f(x) agrees with Rolle’s theorem? How to find critical numbers from here?

1. Evaluate the f(x) for continuity and differentiability.

2. Find f(x) for each endpoint for f(a) = f(b).

3. Then, the theorem is applicable.

4. Find critical numbers, i.e. the value of c via the ff:

   • Get f’(x). This will be your f’(c).

   • Set f’(c) = 0.

   • Find critical numbers.

How do you find out if f(x) agrees with Mean-Value theorem? How to find critical numbers from here?

1. Evaluate the f(x) for continuity and differentiability.

2. Then, the theorem is applicable.

3. Find critical numbers, i.e. the value of c via the following:

   • Get f’(x). This will be your f’(c).

   • Set f’(c) = f(b) - f(a) / b-a.

   • Find critical numbers.

How do you approximate some number x using linearization with f(x) with a certain a?

1. L(x) = f(a) + f’(a)(x-a).

2. Solve for L(x) using the equation(s) found in (1) and the number x. For subbing x (e.g. 1.99), you can “simplify.” (e.g. 2 - 0.01).