Rolle's theorem and the mean value theorem have some similarities and differences: - Both theorems deal with continuous functions on closed intervals, but Rolle's theorem requires the function be differentiable on the open interval while mean value theorem does not. - Rolle's theorem states that if a function is continuous on a closed interval and differentiable on the open interval, and if the function values at the endpoints are equal, then there exists a number c in the open interval where the derivative is 0. - The mean value theorem states that if a function is continuous on a closed interval and differentiable on the open interval, there exists a number c in the open interval where the derivative equals the slope of

1. Rolle’s Theorem vs Mean Value Theorem
(i) the function f(x) is continuous on [1,3]. (i) the function f(x) is continuous on [1,3].
(ii) the function f(x) is differentiable on (1,3). (ii) the function f(x) is differentiable on (1,3).
(iii) f(a) = f(b) (iii) f(a) not equal to f(b)
( ) ( )
(iv) f’(c)=0 , then find c . (iv) f’(c) = , then find c .
To check the continuity at point c, you should follow this:
To check the differentiable at point c , you should follow this:
( ) ( )
Use this formula, . Do the limit for left and right side of point c. If the limit is equal for and , so the function is
differentiable at the point c.

2. Question 10
Show that if f is uniformly continuous on an interval, then f is bounded on that interval.
Solution: By contradiction
Suppose f is not bounded on s. For any M, there is an such that | ( )|
For each , there is an such that | ( )| .
The ‘s are a sequence ( ) in the bounded set S. So, ( ) has a convergent subsequence ( ).
So, ( ) is Cauchy. On the one hand, since f is uniformly continuous, ( ) is Cauchy too.
But (| ( ) )| | ( )| . Hence, ( ) is diverges.
This is a contradiction, so our assumption that f is not bounded on S must be false.