The mean-value theorem is one of the most important theorems in calculus. However, despite its roles in the development of calculus itself, the mean-value theorem can be used in estimation-related problems. This article provides some examples of how it is applied in that area.

**The Mean-Value Theorem**

If f is a function that is continuous on the closed interval [a,b] and differentiable on the open interval (a,b) then there exists a point ξ ∈ (a,b) such that:

**Example 1**

Give an estimate of the difference between and .

**Answer**

We know that . By the mean-value theorem, we can calculate the range of the difference between and , that is the difference between and . We therefore let f(x) = arcsin x by restricting f to the “new” domain . Note that f is continuous on and f is differentiable on . More specifically, for every . Thus f satisfies the sufficient condition for the consequence in the mean-value theorem. Choose such that . (The existence of such ξ is guaranteed by the mean-value theorem.) From this we get . Since , it can be shown that . It follows that . So, the answer is: .

**Example 2**

Suppose that a > 0. Prove that .

**Answer**

Let a > 0. Consider the real-valued function f defined on the closed interval [0,a] where f(x) = ln (1+x) for every x ∈ [0,a]. It can be shown that f is continuous on [0,a] and f is differentiable on (0,a). Specifically, for every x ∈ (0,a). Thus f satisfies the sufficient condition for the consequence in the mean-value theorem. Choose ξ ∈ (0,a) such that . As a consequence, we have the following.

…………………………………………………………………………………………………………………………………………….. (*)

Since 0 < ξ < a, then 1 < 1 + ξ < 1 + a and this gives:

…………………………………………………………………………………………………………………………………………………………………………………………………………. (**)

Substituting (*) into (**), we get:

**Example 3**

Prove that if n > N^{2} then where n and N are natural numbers)

**Answer**

Let n > N^{2}. Let f be the function on the closed interval [n,n+1] to given by for every x ∈ [n,n+1]. Note that f is continuous on [n,n+1] and f is differentiable on (n,n+1). Specifically, . Thus f satisfies the sufficient condition for the consequence in the mean-value theorem. Choose ξ ∈ (n,n+1) such that . Consequently, . Since n < ξ < n+1, it holds that . From the assumption that n > N^{2} it can be shown that . As a consequence, .

**Example 4**

Suppose that we approximate by 10. By applying the mean-value theorem, give an upper bound of the difference between the exact value of and 10.

**Answer**

Note that 101 and 100 differ by 1. Thus we can apply the proposition in Example 3. Select n = 100 and N = 9 Note that 100 > 9^{2}. By the proposition, we can be sure that . Thus, the difference between the exact value of and 10 is less than .056.