中值定理
2024-01-31 17:02:19 8 举报AI智能生成
中值定理是微积分中的一个重要定理，它指出在一个闭区间上连续且可导的函数，至少存在一个点使得该点的导数为0。这个点被称为函数的极值点或拐点。中值定理在微积分中的应用非常广泛，它可以用来证明许多重要的性质和结论，例如泰勒公式、拉格朗日中值定理等。此外，中值定理还可以用于解决实际问题，例如求解最优化问题、研究函数的性质等。总之，中值定理是微积分中的一个基本而重要的工具，对于深入理解微积分的概念和应用具有重要意义。
作者其他创作
大纲/内容
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)连续函数"><span></span><span></span></span>
最值定理
<span class="equation-text" contenteditable="false" data-index="0" data-equation="闭区间[a,b]，必有最大、最小值"><span></span><span></span></span>
介值定理
<span class="equation-text" contenteditable="false" data-index="0" data-equation="[a,b]，m \leq f(x) \leq M, 则必存在 \eta \in (a,b)，使得f(\eta) = u；其中 m \leq u \leq M"><span></span><span></span></span><br>
平均值定理
<span class="equation-text" contenteditable="false" data-index="0" data-equation="x_1, x_2... \in (a,b)，则m \leq \frac{f(x_1) +f (x_2) +...+f(x_n)}{n} \leq M"><span></span><span></span></span>
零点定理
<span class="equation-text" data-index="0" data-equation="f(a) \cdot f(b) < 0, 则必存在 \eta \in (a,b)，使得f(\eta) = 0" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f'(x)"><span></span><span></span></span>
费马定理
<span class="equation-text" contenteditable="false" data-index="0" data-equation="1.x_0处可导"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="2.x_0取极值"><span></span><span></span></span>
难点
如何证明取极值：
最值定义
<span class="equation-text" contenteditable="false" data-index="0" data-equation="x \in [a,b]，f(x) \leq f(x_0) ，则在x_0点取最大值"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="只要x_0不是端点（即x_0 \in (a,b)）"><span></span><span></span></span>
<b><font color="#e74f4c">这里主要是要求极值点左右两侧有邻域</font></b>
极值定义
<span class="equation-text" contenteditable="false" data-index="0" data-equation="x_0的左右两侧邻域内，有f(x) \leq f(x_0) ，则在x_0点取极大值"><span></span><span></span></span>
罗尔定理
<span class="equation-text" contenteditable="false" data-index="0" data-equation="1.f(x)闭区间连续、开区间可导"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="2.f(a) = f(b)"><span></span><span></span></span>
难点
如何构造辅助函数，使得F(a) = F(b)
辅助函数原型
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(u(x) \cdot v(x))' = u'(x)v(x) + u(x) v'(x)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="辅助函数的作用F(a) = F(b), F'(\eta) = 0"><span></span><span></span></span>
常见辅助函数
<span class="equation-text" contenteditable="false" data-index="0" data-equation="若表达式中有f(x)\cdot f'(x)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="F(x) = (f(x))^2"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="求导后： F'(x) = 2f(x)f'(x)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="若表达式中有[f'(x)]^2 + f(x)f^{{2}}(x)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="F(x) = f(x) \cdot f'(x)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="求导后：F'(x) = [f'(x)]^2 + f(x)f^{{2}}(x)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="若表达式中有f'(x) +f(x)\cdot \phi'(x)"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="F(x) = f(x)e^{\phi(x)}" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="求导后：F'(x) = [f'(x) +f(x)\cdot \phi'(x) ] \cdot e^\phi(x)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="几种特殊形式"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\phi(x) = x"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)+f'(x)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\phi(x) = -x"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)-f'(x)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\phi(x) = kx"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)+kf'(x)"><span></span><span></span></span>
如何证明相等
拉格朗日定理
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f'(\xi) = \frac{f(b)-f(a)}{b-a}，其中 b < \xi < a"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)闭区间连续、开区间可导"><span></span><span></span></span>
几何含义
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{f(b)-f(a)}{b-a}表示a,b两点连线与x轴构成的斜率（tan\theta)"><span></span><span></span></span>
解题信号：<span class="equation-text" contenteditable="false" data-index="0" data-equation="f与f'"><span></span><span></span></span>
柯西中值定理
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{f(b)-f(a)}{g(b)-g(a)} = \frac{f'(\xi)}{g'(\xi)}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)闭区间连续、开区间可导"><span></span><span></span></span>
考法
一般给一个具体，一个抽象函数
泰勒公式
非幂函数都可以用幂函数来表示
解决的问题
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)与f^{(n)}(x) ，其中n \geq 2"><span></span><span></span></span>
带拉格朗日余项的泰勒公式
<span class="equation-text" contenteditable="false" data-index="0" data-equation="要求：x_0的邻域内，n+1阶导数存在"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="存在一个区间范围：可以找到一个\xi"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x) = f(x_0) + f'(x_0)(x-x_0) + ...+\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n+\frac{f^{(n+1)}(\xi)}{(n+1)!}(x-x_0)^{n+1}，其中\xi介于x,x_0之间"><span></span><span></span></span>
带佩亚诺余项的泰勒公式
<span class="equation-text" contenteditable="false" data-index="0" data-equation="要求：x_0处n阶可导"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x) = f(x_0) + f'(x_0)(x-x_0) + ...+\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n+o((x-x_0)^n)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\int_{a}^{b} f(x)\, \mathrm{d}x"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="m \leq \frac{\int_{a}^{b} f(x)\, \mathrm{d}x} {(b-a)} \leq M，则必存在 \eta \in (a,b)，f(\eta) = \int_{a}^{b} f(x)\, \mathrm{d}x / (b-a)" contenteditable="false"><span></span><span></span></span>
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