Physical Interpretation Of Mean Value Theorem

In mathematics, the intermediate Mean Value Theorem is crucial. Parmeshwara, a mathematician from Kerala, India, devised the existence of the first mean value theorem in the fourteenth century. However, Rolle provided a concise description of this in the seventeenth century: Rolle’s Theorem, which was only verified for polynomials but was not part of the arithmetic. Eventually, Augustin Louis Cauchy presented the current version of the Mean Value Theorem in 1823.

The mean value theorem asserts that there is one place on a curve travelling between two points in which the tangent is perpendicular to the line segment running through the two points. This mean value theorem gave rise to Rolle’s theorem.

Intermediate Mean Value Theorem

The proof of the intermediate mean value theorem appears to be difficult. It can, however, be comprehended in simpler terms.  If there is a constant function f with terminals a and b, the elevation of the points “a” and “b” would be “f(a)” and “f(b),” respectively. If we choose an elevation k between f(a) and f(b), this line must cross the function f at some point (say c), and this point must be between a and b, according to the Intermediate mean value theorem.

Assume that A is the set of all x values in the interval [a, b], with f(x) equal to k.

Because A has a member “a” and is bounded above by the value “b,” it is assumed to be a non-empty set.

As a result of the completeness characteristic, “c” is the smallest score greater than or equal to each member of A. we can state that f(c) = k.

Given that f is continuous. Then let us consider a ε > 0, there exists “a δ > 0” such that

 f(x) – f(c) | < ε for every | x – c | < δ. This gives us

f(x) – ε < f(c) < f(x) + ε

For each x lying within c – δ and c + δ. So, we have values of x lying between c and c -δ, contained in A, such that:

f(c) < (f(x) + ε) ≤ (k + ε) ——– (1)

Similarly, values of x between c and c + δ that is not contained in A, such that

f(c) > (f(x) – ε) > (k − ε) ——–(2)

Combining both the inequality relations, obtain

k – ε < f(c) < k + ε

For every ε > 0   Hence, the intermediate mean value theorem is proved.

State and Prove Mean Value Theorem

The first mean value theorem tells that if a function (f) is ongoing over the finite interval [a,b] and distinguishable over the accessible interval (a,b), there must be at least one point c in the interval (a,b) where f(c) is the function’s mean rate of change over [a,b] and adjacent to the tangent line over [a,b].

Let g(x) be the tangent line connecting (a, f(a)) and (b, f(b)) to f(x). We know that the secant line’s equation is y-y 1 = m (x- x 1).

g(x) – f(a) = f (b) f (a) b a (x-a)

g(x) = f (b) f (a) b a (x-a) + f(a) —–> g(x) = f (b) f (a) b a (x-a) + f(a) ——-> (1)

Let h(x) be equal to f(x) – g. (x)

[f (b) f (a) b a (x-a) + f(a)] h(x) = f(x) – [f (b) f (a) b a (x-a) + f(a)] (Adapted from (1))

h(a) = h(b) = 0, and h(x) is differentiable on [a,b] and continuous on [a,b] (a,b).

Using Rolle’s theorem, there must be some x = c in (a,b) for h'(c) to equal 0.

h'(x) = f'(x) – f (b) f (a) b a

h'(c) = 0 for some c in (a, b). Thus

h'(c) = f'(c) – f (b) f (a) b a = 0.  Hence, the Mean value theorem is proved.

Mean Value Theorem Physical Interpretation

Because (f(b) f(c)) / (b a) is the average change in the function across [a, b], and f(c) is the instantaneous change at ‘c,’ the mean value theorem asserts that the instantaneous change is equal to the average change in the function throughout the interval at some interior point.

The physical constant of the curve connecting these locations has the same slope as the tangent at point (c, f(c)). We know that the slope at that location is the gradient of the tangent.

Slope of the Tangent = Secant’s Slope

f'(c) = f(b)-f(a) / b-a

Conclusion

The Mean Value Theorem concludes that if a function “f” is continuous on the interval [a, b] and also differentiable on the interval (a, b), there exists a point “c” in the interval (a, b) such that

f'(c) = [f(b)-f(a)] /b-a

(i.e.) A point c (a, b) exists where the tangent is parallel to the line passing through the points (a, f(a)) and (b, f(b)). The mean value theorem postulates that a continuous function f(x) is continuous in the interval [a, b] and differentiable in the interval [a, b] (a, b).  The mean value theorem gave Rolle’s theorem, which also discusses the first mean value theorem.