Mean Value Theorems, Taylor Approximations & L'Hôpital's Rule

1. Rolle's Theorem

Rigorous Proof: By the Extreme Value Theorem, continuous $f$ on $[a, b]$ attains an absolute maximum $M$ and minimum $m$.

• Case 1 (Constant Function): If $M = m$, then $f(x)$ is constant on $[a, b]$. Thus $f'(x) = 0$ for every $x \in (a, b)$, and any $c \in (a, b)$ satisfies the theorem.
• Case 2 (Non-Constant Function): Since $f(a) = f(b)$, at least one of the extrema (say the maximum $M$) must be attained at an interior point $c \in (a, b)$. Since $f$ is differentiable at $c$ and attains a local maximum, Fermat's Interior Extremum Theorem forces $f'(c) = 0 \quad \blacksquare$

2. Lagrange's Mean Value Theorem (MVT)

Geometric Meaning: There exists an interior point $c$ where the instantaneous rate of change (tangent slope) is strictly parallel to the average rate of change (secant slope connecting endpoints $(a, f(a))$ and $(b, f(b))$).

Proof via Rolle's Theorem: Construct the auxiliary function measuring the vertical distance between the curve and the secant line:

Notice $h(a) = 0$ and $h(b) = 0$. Since $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$, $h(x)$ satisfies all hypotheses of Rolle's Theorem on $[a, b]$. Thus $\exists c \in (a, b)$ with $h'(c) = 0$:

1. The Zero-Derivative Constant Function Theorem

Proof: Choose any two points $x_1 < x_2$ in $(a, b)$. By the MVT on $[x_1, x_2]$, $\exists c \in (x_1, x_2)$ such that $f(x_2) - f(x_1) = f'(c)(x_2 - x_1) = 0 \cdot (x_2 - x_1) = 0$. Thus $f(x_1) = f(x_2)$ for all pairs, proving $f(x) \equiv C \quad \blacksquare$

2. Functions Differing by a Constant

This is the fundamental justification for the $+ C$ constant of integration in antiderivatives.

3. Monotonicity Test Criteria

1. Cauchy's Generalized Mean Value Theorem

Proof: By Rolle's Theorem on $g(x)$, $g(b) \ne g(a)$ (otherwise $g'(c) = 0$). Construct $H(x) = [f(b) - f(a)]g(x) - [g(b) - g(a)]f(x)$. Since $H(a) = H(b) = f(b)g(a) - g(b)f(a)$, Rolle's theorem yields $H'(c) = 0 \implies [f(b)-f(a)]g'(c) = [g(b)-g(a)]f'(c) \quad \blacksquare$

2. L'Hôpital's Rule for Indeterminate Forms 0/0 and $\infty/\infty$

Proof for $0/0$: Define $f(c) = g(c) = 0$ to make $f, g$ continuous at $c$. For any $x$ near $c$, by Cauchy's MVT on $[c, x]$ (or $[x, c]$), there exists $x_1$ strictly between $c$ and $x$ such that:

As $x \to c$, $x_1 \to c$ by squeezing. Thus $\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x_1 \to c} \frac{f'(x_1)}{g'(x_1)} = L \quad \blacksquare$

3. Resolution of Other Indeterminate Forms

• Product $0 \cdot \infty$: Rewrite $f \cdot g = \frac{f}{1/g}$ or $\frac{g}{1/f}$ to convert to $\frac{0}{0}$ or $\frac{\infty}{\infty}$.
• Difference $\infty - \infty$: Find a common algebraic denominator or rationalize.
• Exponential Forms $0^0, 1^\infty, \infty^0$: Let $y = [f(x)]^{g(x)}$, take logarithms $\ln y = g(x) \ln f(x)$ (which becomes $0 \cdot \infty$), evaluate limit $L$, and recover $\lim y = e^L$.

1. Linear (Tangent Line) Approximation

The tangent line to $y = f(x)$ at $x = x_0$ provides the optimal first-order linear approximation $L(x) \approx f(x)$ for values of $x$ close to $x_0$:

2. Differentials

Let $y = f(x)$. If $dx = \Delta x$ represents an independent change in the input, the differential $dy$ represents the corresponding change along the tangent line:

While the actual change along the curve is $\Delta y = f(x + \Delta x) - f(x)$, for small $dx$ we have $\Delta y \approx dy$.

3. Error Estimation Metrics

• Absolute Error: $\Delta y \approx dy = f'(x_0) dx$
• Relative Error: $\frac{\Delta y}{y} \approx \frac{dy}{y} = \frac{f'(x_0) dx}{f(x_0)}$
• Percentage Error: $\left( \frac{dy}{y} \right) \times 100\%$