## 第三章 微分中值定理与导数应用

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### 3. 第三节 泰勒公式

#### 3.1. 关于$(x – x_0)$的$n$次多项式

$$p_n(x) = a_0 + a_1(x – x_0) + a_2(x – x_0)^2 + \dots + a_n(x – x_0)^n \tag{1}$$

#### 3.2. 替换公式(1)的系数

$$p_n(x) = f(x_0) + f'(x_0)(x – x_0) + \frac{f^{\prime \prime}(x_0)(x – x_0)^2}{2!} + \dots + \frac{f^{(n)}(x_0)(x – x_0)^n}{n!} \tag{2}$$

#### 3.3. 泰勒(Taylor)中值定理

$$f(x) = f(x_0) + f'(x_0)(x – x_0) + \frac{f^{\prime \prime}(x_0)(x – x_0)^2}{2!} + \dots + \frac{f^{(n)}(x_0)(x – x_0)^n}{n!} + R_n(x) \tag{3}$$

$$R_n(x) = \frac{f^{(n+1)}(\xi)}{(n+1)!}(x – x_0)^{n+1}\tag{4}$$

#### 3.5. 佩亚诺型余项

$$\left | R_n(x) \right | = \left | \frac{f^{(n+1)}(\xi)}{(n+1)!}(x – x_0)^{n+1} \right | \leq {\frac{M}{(n+1)!} \left | x – x_0 \right |}^{n+1} \tag{5}$$

$$\lim_{x \to x_0}\frac{R_n(x)}{(x – x_0)^n} = 0$$

$$R_n(x) = o[(x – x_0)^n] \tag{6}$$

$$f(x) = f(x_0) + f'(x_0)(x – x_0) + \frac{f^{\prime \prime}(x_0)(x – x_0)^2}{2!} + \dots +$$
$$\frac{f^{(n)}(x_0)(x – x_0)^n}{n!} + o[(x – x_0)^n] ,\tag{7}$$

$R_n(x)$的表达式$(6)$称为佩亚诺(Peano)型余项,公式$(7)$称为$f(x)$按$(x – x_0)$的幂展开的带有带有佩亚诺型余项的$n$阶泰勒公式.

#### 3.6. 麦克劳林(Maclaulin)公式

$$f(x) = f(0) + f'(0)x + \frac{f^{\prime \prime}(0)x^2}{2!} + \dots + \frac{f^{(n)}(0)x^n}{n!} +$$
$$\frac{f^{(n+1)}({\theta}x)}{(n+1)!}x^{n+1} (0{\lt}\theta{\lt}1) \tag{8}$$

#### 3.7. 带有佩亚诺型余项的麦克劳林公式

$$f(x) = f(0) + f'(0)x + \frac{f^{\prime \prime}(0)x^2}{2!} + \dots + \frac{f^{(n)}(0)x^n}{n!} + o(x^n) \tag{9}$$

#### 3.8. 泰勒公式近似公式

$$f(x) \approx f(0) + f'(0)x + \frac{f^{\prime \prime}(0)x^2}{2!} + \dots + \frac{f^{(n)}(0)x^n}{n!}$$

$$\left | R_n(x) \right | \leq \frac{M}{(n+1)!} \left | x \right|^{n+1} \tag{10}$$

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