Prove that the expression
is an equivalent form of the mean-value theorem.
Find the value of in terms of and when:
For parts (a) and (b) keep fixed with and find the limit of as tends to 0.
Proof. If is continuous on and differentiable on , then by the mean-value theorem we have
Letting and for some (since ), we have
(This follows since from our definitions, is the distance from . Then, since is somewhere in the interval its value must be plus some portion of the distance to . This portion is then , which is how we know .) Substituting and and ,
Now for parts (a) and (b).
- If , we have , so,
- If , we have . So,