Rolle and Lagrange Theorems: Calculus Principles Explained

Rolle's Theorem

If the function y = f(x) is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), there is at least one x₀ ∈ (a, b) such that f'(x₀) = 0.

Proof of Rolle's Theorem

The continuity of y = f(x) on the closed interval [a, b] implies the existence of an absolute maximum M and an absolute minimum m, according to the Weierstrass theorem. Two cases may occur:

• Case 1: The maximum M is in (a, b), the minimum m is in (a, b), or both are in (a, b).
• Case 2: M and m are at the endpoints a and b.

Suppose Case 1, where M is the value of the function at a point in the open interval (a, b). Since the function is differentiable, the derivative must vanish: if f(x₀) = M, then f'(x₀) = 0. The same occurs if m is reached at a point in the open interval. If both are in the open interval, we get two points where the derivative vanishes.

Now suppose Case 2, where f(a) = M and f(b) = m. Since f(a) = f(b), then M = m. Because all values the function takes on the closed interval [a, b] are between m and M, and these are equal, the function is constant throughout the interval: f(x) = c. If the function is constant, its derivative is zero at all points in the open interval: f'(x) = 0 for all x ∈ (a, b). The point where the derivative vanishes represents a horizontal tangent, parallel to the secant joining the points (a, f(a)) and (b, f(b)).

Lagrange's Mean Value Theorem

Let g be the function y = x, whose graph is the bisector of quadrants 1 and 3. Its derivative y' = 1 does not vanish at x. By g(a) = a and g(b) = b, we have:

Accordingly, f(b) - f(a) = f'(x₀)(b - a) with a < x₀ < b. The slope of the secant joining the points [a, f(a)] and [b, f(b)] equals the slope of the tangent to the curve f(x) at x = x₀. Thus, the two lines are parallel.

Infinitesimals and Limits

Common infinitesimal equivalents:

• ln(a₀ + a₁n + ... + aₖnᵏ) ~ ln(nᵏ)
• sin(w) ~ w
• tan(w) ~ w
• 1 - cos(w) ~ w²/2
• ln(A) ~ A - 1
• A^(1/n) ~ 1/n ln(A)
• eˣ - 1 ~ x
• (1 + x)ᵐ - 1 ~ mx
• x - sin(x) ~ x³/6
• log(x) ~ x - 1
• xᵐ - 1 ~ m(x - 1)

Convergence Criteria

• Stolz Criterion: Aₙ / Bₙ → (Aₙ - Aₙ₋₁) / (Bₙ - Bₙ₋₁)
• Root Criterion: → (w - 1) / w
• Limit 1^∞: Bₙ · (w - 1)
• Limit 0^0: Bₙ · ln(Aₙ)
• Pringsheim Theorem: nᵖ · Aₙ; if < 1, it diverges.
• Raabe Criterion: n[1 - (w₊₁ / w)]; if x > 1, it converges.