Chapter 3 Applications of Differentiation(2/2)

• Rolle's Theorem and The Mean Value Thm
• Indeterminate Forms and L'Hospital's Rule
• Antiderivative

# Rolle's Theorem and The Mean Value Thm

# Rolle's Thm

Suppose $f$ satifies that

1. $f:[a,b] \to \mathbb{R}$ is continuous.
2. $f:(a,b) \to \mathbb{R}$ is differentiable.
3. $f(a) = f(b)$
   then $\exists c \in (a,b)$ such that $f'(c) = 0$

# The Mean Value Thm

Suppose $f$ satifies that

1. $f:[a,b] \to \mathbb{R}$ is continuous.
2. $f:(a,b) \to \mathbb{R}$ is differentiable.
   then $\exists c \in (a,b)$ such that $f'(c) = \frac{f(b)-f(a)}{b-a}$.

# Indeterminate Forms and L'Hospital's Rule

# Indeterminate Forms (不定型)

Consider $\lim_{x \to a}\frac{f(x)}{g(x)}$:

1. 若當$x \to a$ 時，同時有 $f(x) \to 0$ 及 $g(x) \to 0$，
   則$\lim_{x \to a}\frac{f(x)}{g(x)}$ 稱為$\frac{0}{0}$ 不定型
2. 若當$x \to a$ 時，同時有 $f(x) \to \infty$ 及 $g(x) \to \infty$，
   則$\lim_{x \to a}\frac{f(x)}{g(x)}$ 稱為$\frac{\infty}{\infty}$ 不定型

# L'Hospital's Rule

Suppose $f$ and $g$ are differentiable and $g'(x) \neq 0$ on $(a,b)$ containing $c$.
Suppose that $\lim_{x \to a}f(x) = 0$ and $\lim_{x \to a}g(x) = 0$ or $\lim_{x \to a}f(x) = \pm \infty$ and $\lim_{x \to a}g(x) = \pm \infty$
Then $\lim_{x\to a}\frac{f(x)}{g(x)} = \lim_{x\to a}\frac{f'(x)}{g'(x)}$ if the limit on the right hand side exists(or is $\pm \infty$)

# Antiderivative

# Define

A function $F$ is called an antiderivative of $f$ on an open interval $I$ if $F'(x) = f(x)$ for all $x \in I$.

# Example

# Reference

• 蘇承芳老師 - 微積分甲（一）109 學年度 - Calculus (I) Academic Year 109