Mean Value Theorem

The mean value theorem is one of the most crucial theorems that calculus is based on. From the 14th century till 1823, multiple versions of the mean value theorem have been proposed and the current version we follow was proposed by Augustin Cauchy in 1823.

What does the mean value theorem state?

 The mean value theorem asserts that there is one point on a curve traveling between two points where the tangent is parallel to the secant running through the two locations. This mean value theory gave rise to Rolle’s theorem. Starting with local hypotheses about derivatives at interval points, the mean value theorem is used to prove claims about a function on an interval.  It is widely acknowledged that this theorem is among the most important theorems in analysis, and as a result, all of its applications are of great importance. Listed below are a few of the more popular applications:

• Strictly Increasing and Strictly decreasing functions.

• Leibniz’s Rule

• The symmetry of Second derivatives.

• L’ Hospital’s Rule.

What is the Mean Value Theorem?

According to the mean value theorem, there should be at least one point (c, f(c)) on a curve f(x) passing via two provided points (a, f(a)), (b, f(b)) in which the tangent is parallel to the secant traveling through the two given locations. For a function f(x): [a, b] R that is continuous and differentiable across an interval, the mean value theorem is defined in calculus.

• The function f(x) is continuous across the interval [a, b].

• The function f(x) is differentiable across the interval (a, b). There exists a point c in (a, b).

• f'(c) = f(b)−f(a)b−a

Proof of Mean Value Theorem

Statement: This theorem asserts that if an open interval (such as [a,b]) is differentiable and continuous over the closed interval (a,b), then there must exist at least one point c in the interval (a,b) where f(c) is equal to the average rate of change over [a,b] and parallel to the secant line over the open interval.

Let f be a real valued function on an interval [a, b]. Let c be a point in the interior of [a, b]. That is, c ∈ (a, b). We say that f has a local maximum (respectively local minimum) at c if there is some  > 0 so that f(c) ≥ f(x) (respectively f(c) ≤ f(x)) for every x ∈ (c − , c + ).

f 0 (c) = 0.

f(x) = f(c) + f 0 (c)(x − c) + o(|x − c|),

as x − c −→ 0. Suppose that f’ (c) 0. From the definition of o, we have that there is some δ > 0 so that 

|f(x) − f(c) − f 0 (c)(x − c)| ≤ |f 0 (c)||x − c|/2 ,

Note – that if the function is not differentiable, even at a single point, the conclusion might not even be valid.

The tangent at c is parallel to the secant traveling through the points (a, f(a)), (b, f(b) in this proof. This mean value theorem is used to prove a proposition across a narrow range. Also derived from this mean value theory is Rolle’s theorem.

Mean value theorem for integrals

As a direct result of the Mean Value Theorem (for Derivatives) and the First Fundamental Theorem of Calculus, we get the Mean Value Theorem for Integrals. In other words, a continuous function on a closed, bounded interval has at least one point where its average value is equal to the interval’s average value. This theorem follows that, for each closed interval with one variable that is continuous on its endpoints and differentiable on its endpoints minus its endpoints, there is at least one point in the interval where the product of its value and its length across the interval equals the integral of its value over the interval.

Difference Between Mean Value Theorem and Rolle’s Theorem

The distinction between the mean value theorem and Rolle’s theorem aids comprehension of these two theorems. Both theorems describe f(x) as a continuous function over the interval [a, b] and a differentiable function across the interval [a, b] (a, b). The two referenced locations (a, f(a)), (b, f(b)) are different under the mean value theorem, and f(a) f(b) (b). The points in Rolle’s theorem are specified so that f(a) = f (b).

The slope of the tangent at the point (c, f(c)) is equal to the slope of the secant linking the two points, which is the value of c in the mean value theorem.

Conclusion

In this article, we derived/understood Rolle’s theorem from the mean value theorem of differential calculus. We begin by defining key terms and give some background on Rolle’s theorems before proceeding to derive the theorem.