Mean Value Theorems

We need a review of some results from calculus that make an impact on numerical methods and analysis. All of these theorems are included in most calculus textbooks, but are usually not emphasized as much as we will need here, or perhaps not in the way that we will need them.

Mean Value Theorem (Theorem 1): Let f be a given function on interval [𝑎, b] ⊂ ℝ and differentiable on open interval (⊂, b). Then there exists a point ξ ∈ [𝑎, b] such that \[ f' (\xi ) = \frac{f(b) - f(a)}{b-a} . \]
    This theorem leads to a very important equality
\[ f\left( x_1 \right) - f\left( x_2 \right) = f' (\xi ) \left( x_1 - x_2 \right) \]
Thus, the mean value theorem allows us to replace differences of function values with differences of argument values, if we scale by the derivative of the function. For example, we can use the mean value theorem to tell us that
\[ \left\vert \sin x_1 - \sin x_2 \right\vert \leqslant \left\vert x_1 - x_2 \right\vert , \]
because the derivative of the sine is the cosine, which is bounded by 1 in absolute value. Note also that the mean value theorem is simply a special case of Taylor's Theorem, for n = 0.    
Example 1:
A = {{4.1, 1.4}, {9.7, 3.3}};
b = {1.4, 3.3};
LinearSolve[A, b]
{0., 1.}
   ■
End of Example 1
Intermediate Value Theorem (Theorem 2): Let f be a continuous function on closed interval [𝑎, b] and assume that W is a value between f(𝑎) and f(b), that is, either f(𝑎) ≤ W ≤ f(b) or f(b) *le; W ≤ f(𝑎). Then there exists a point c ∈ [𝑎, b] such that f(c) = W.
This theorem is used in the bisection algorithm for finding the roots of functions. Moreover, this is the theorem that tells us that a continuous function is one whose graph can be drawn without lifting the pen off the paper, because it says that between any two function values, we have a point on the curve for all possible argument values.    
Example 2:    ■
End of Example 2
Extreme Value Theorem (Theorem 3): Let f be a continuous function on closed interval [𝑎, b], Then there exists a point m ∈ [𝑎, b] such that f(m) ≤ f(x) for all x ∈ [𝑎, b], and a point M ∈ [𝑎, b] such that f(M) ≥ f(x) for all x ∈ [𝑎, b]. Moreover, f achieves its maximum and minimum values on [𝑎, b] either at the endpoints 𝑎 or b, or at a critical point (where its derivative vanishes).
   
Example 3:    ■
End of Example 3
Integral mean value Theorem (Theorem 4): Let f and g both be continuous functions on closed interval [𝑎, b], and assume further that g does not change sign on [𝑎, b]. Then there exists a point ξ ∈ [𝑎, b] such that \[ \int_a^b g(t)\,f(t)\,{\text d}t = f(\xi ) \int_a^b g(t)\,{\text d}t . \]
   
Example 4:    ■
End of Example 4
Discrete Average Value Theorem (Theorem 5): Let f be continuous function on closed interval [𝑎, b], and consider the baricentric sum \[ S = \sum_{k=1}^n a_k f\left( x_k \right) , \] where each point xk ∈ [𝑎, b], and the coefficients satisfy \[ \sum_{k=1}^n a_k = 1 , \qquad a_k \ge 0. \] Then there exists a point η ∈ [𝑎, b] such that f(η) = S.
   
Example 5:    ■
End of Example 5
Rolle’s, Intermediate-Value Theorem (Theorem 6): If f(x) is continuous and differentiable in the closed interval [𝑎, b] with f(𝑎) = f(b) = 0, then there exists a point ξ ∈ (𝑎, b) where the derivative of function f vanishes.
   
Example 6: The function f(x) = x² − x − 6 has zeroes at x = − 2 and x = 3. According to the Rolle’s theorem, there exists a point ξ between −2 and 3 with f′(ξ) = 0.

Taking derivative, we find f′(x) = 2x − 1. Thereforee, the derivative vanishes at x = ½ ∈ (−2, 3).    ■

End of Example 6

 

 

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